
Preface
Deirdre N McCloskey (1994)
Formal mathematical modeling in economics has two key advantages. Firstly, formal modeling makes the assumptions explicit. It clarifies intuition and makes arguments transparent. More importantly, it uncovers the limitations of our intuition, delineating the boundaries and uncovering the occasional counterintuitive special case. Secondly, the formal modeling aids communication. Since the assumptions are explicit, participants spend less time arguing about what they really meant, leaving more time to explore conclusions, applications, and extensions. Compare the aftermath of the publication of Keynes' General Theory with that of Von Neumann and Morgenstern's Theory of Games and Economic Behavior. The absence of formal mathematical modeling in the General Theory meant that subsequent scholars spent considerable energy debating ``what Keynes really meant''. In contrast, the rapid development of game theory in recent years owes much to the advantages of formal modeling. Game theory has attracted a predominance of practitioners who are skilled formal modelers. As their assumptions are very explicit, practitioners have had to spend little time debating the meaning of others writings. Their efforts have been devoted to exploring ramifications and applications. Undoubtedly, formal modeling has enhanced the pace of innovation in game theoretic analysis in economics. Economic models are not like replica cars, scaled down versions of the real thing admired for their verisimilitude. A good economic model strips away all the unnecessary and distracting detail and focuses attention on the essentials of a problem or issue. This process of stripping away unnecessary detail is called abstraction. Abstraction serves the same role in mathematics. The aim of abstraction is not greater generality, but greater simplicity. Abstraction reveals the logical structure of the mathematical framework in the same way as it reveals the logical structure of an economic model. Chapter 1 establishes the framework by surveying the three basic sources of structure in mathematics. The order, geometric and algebraic structure of sets are first considered independently. Then, their interaction is studied in subsequent sections dealing with normed linear spaces and preference relations. Building on this foundation, we study mappings between sets or functions in Chapters 2 and 3. In particular, we study functions which preserve the structure of the sets which they relate, treating in turn monotone, continuous and linear functions. In these chapters, we meet the three fundamental theorems of mathematical economics  the (continuous) maximum theorem, the Brouwer fixed point theorem and the separating hyperplane theorem, and outline many of their important applications in economics, finance and game theory. A key tool in the analysis of economic models is the approximation of smooth functions by linear and quadratic functions. This tool is developed in Chapter 4, which presents a modern treatment of what is traditionally called ``multivariate calculus''. Since economics is the study of rational choice, most economic models involve optimization by one or more economic agents. Building and analyzing an economic model involves a typical sequence of steps. First, the model builder identifies the key decision makers involved in the economic phenomenon to be studied. For each decision maker, the model builder must postulate an objective or criterion, and identify the tools or instruments which she can use in pursuit of that objective. Next, the model builder must formulate the constraints on the decision maker's choice. These constraints normally take the form of a system of equations and inequalities linking the decision variables and defining the feasible set. The model, therefore, portrays the decision maker's problem as an exercise in constrained optimization, selecting the best alternative from a feasible set. Typically, analysis of an optimization model has two stages. In the first stage, the constrained optimization problem is solved. That is, the optimal choice is characterized in terms of the key parameters of the model. After a general introduction, Chapter 5 first discusses necessary and sufficient conditions for unconstrained optimization. Then four different perspectives on the Lagrangean multiplier technique for equality constrained problems are presented. Each perspective adds a different insight contributing to a complete understanding. In the second part of the chapter, the analysis is extended to inequality constraints, including coverage of constraint qualification, sufficient conditions and the practically important cases of linear and concave programming. In the second stage of analysis, the sensitivity of the optimal solution to changes in the parameters of the problem is explored. This second stage is traditionally (in economics) called comparative statics. Chapter 6 outlines four different approaches to the comparative static analysis of optimization models, including the traditional approaches based on the implicit function theorem or the envelope theorem. It also introduces a promising new approach based on order properties and monotonicity, which often gives strong conclusions with minimal assumptions. Chapter 6 concludes with a brief outline of the comparative static analysis of equilibrium (rather than optimization) models. The book includes a thorough treatment of some material often omitted from introductory texts, such as correspondences, fixed point theorems and constraint qualification conditions. It also includes some recent developments such as supermodularity and monotone comparative statics. We have made a conscious effort to illustrate throughout with economic examples and where possible to introduce mathematical concepts with economic ideas. Many illustrative examples are drawn from game theory. The completeness of the real numbers is assumed, every other result is derived within the book. The most important results are stated as theorems or propositions, which are proved explicitly in the text. However, to enhance readability and promote learning, lesser results are stated as exercises, answers for which will be available on the internet (see the note to the reader). In this sense, the book is comprehensive and entirely selfcontained, suitable to be used as a reference, a text or a resource for selfstudy. The sequence of the book, preceding from sets to functions to smooth functions, has been deliberately chosen to emphasize the structure of the underlying mathematical ideas. However, for instructional purposes or for selfstudy, an alternative sequence might be preferable and easier to motivate. For example, the first two sections of Chapter 1 (sets and ordered sets) could be immediately followed by the first two sections of Chapter 2 (functions and monotone functions). This would enable the student to achieve some powerful results with a minimum of fuss. A second theme could then follow the treatment of metric spaces (and the topological part of Section 1.6) with continuous functions culminating in the continuous maximum theorem and perhaps the Banach fixed point theorem. Finally, the course could turn to linear spaces, linear functions, convexity and linear functionals, culminating in the separating hyperplane theorem and its applications. A review of fixed point theorems would then highlight the interplay of linear and topological structure in the Brouwer fixed point theorem and its generalizations. Perhaps it would then be advantageous to proceed through Chapters 4, 5 and 6 in the given sequence. Even if Chapter 4 is not explicitly studied, it should be reviewed to understand the notation used for the derivative in the following chapters. The book can also be used for a course emphasizing microeconomic theory rather than mathematical methods. In this case, the course would follow a sequence of topics, such as monotonicity, continuity, convexity and homogeneity, interspersed with analytical tools such as constrained optimization, the maximum, fixed point and separating hyperplane theorems, and comparative statics. Each topic would be introduced and illustrated via its role in the theory of the consumer and the producer. Achieving consistency in notation is a taxing task for any author of a mathematical text. Wherever I could discern a standard notation in the economics literature, I followed that trend. Where diversity ruled, I have tended to follow the notation in Hal Varian's Microeconomic Analysis, since it has been widely used for many years. A few significant exceptions to these rules are explicitly noted. Many people have left their mark on this book, and I take great pleasure in acknowledging their contribution. Foremost amongst my creditors is Graeme Guthrie whose support, encouragement and patient exposition of mathematical subtleties has been invaluable. Richard Edlin and Mark Pilbrow drafted most of the diagrams. Martin Osborne and Carolyn Pitchik made detailed comments on an early draft of the manuscript and Martin patiently helped me understand intricacies of TeX and LaTeX. Other colleagues who have made important comments include Thomas Cool, John Fountain, Peter Kennedy, David Miller, Peter Morgan, Mike Peters, Uli Schwalbe, David Starrett, Dolf Talman, Paul Walker, Richard Watt and Peyton Young. I am also very grateful for the generous hospitality of Eric van Damme and CentER at the University of Tilburg and Uli Schwalbe and the University of Hohenheim in providing a productive haven in which to complete the manuscript during my sabbatical leave. Finally, I acknowledge the editorial team at MIT Press, for their proficiency in converting my manuscript into a book. I thank them all. 