# Using Mathematics in Economic Analysis

## Peter N. Hess

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For undergraduate courses in Mathematical Economics where students have completed a first course in Calculus and Intermediate Microeconomic Theory and/or Intermediate Macroeconomic Theory. Also serves as a supplement to the standard Intermediate Microeconomic and Macroeconomic Theory courses, graduate programs in Economics, and Master's programs in Business and Public Policy.This text is designed to help students develop mathematical skills that will open up a new dimension of economic analysis, thereby enhancing their understanding of economic theories.

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From the Back Cover:

A first edition that offers a new perspective on mathematical economics. The emphasis throughout the text is not on mathematical theorems and formal proofs, but on how mathematics can enhance our understanding of the economic behavior under study. An efficient and effective writing style, placing a premium on clear explanation, builds confidence as students, move through the text.

Mathematics in undergraduate economics tends to be underutilized to the detriment of student comprehension of economic principles. Usually introductory texts shy away from even basic algebra, and intermediate theory texts, if using calculus at all, often relegate the mathematical treatment to appendices and footnotes. The traditional mathematical economics texts, in contrast, focus on presenting the mathematical techniques with economic applications interspersed throughout, but with no coherent progression of the economic theories.

With this text I seek to convey the utility of mathematics—the conciseness of expression, preciseness of assumptions, and the advantages in manipulation—for economic analysis. The mathematics employed (including derivative and integral calculus, exponential and logarithmic functions, first order difference and differential equations, matrix algebra, linear and nonlinear programming) are motivated by the economics. Accordingly, the coverage of economic theory is organized into three parts: "Analysis of Markets," "Optimization," and "Macroeconomic Analysis." We progress from models of perfectly competitive markets for single commodities, to the optimizing behaviors of the firms and households operating in markets characterized by perfect and imperfect competition, to the aggregate markets of macroeconomic models. Mathematical concepts and techniques are introduced at appropriate points in the presentation of the economic theories and then reinforced throughout the text. The primary objective of Using Mathematics in Economic Analysis is for students to develop mathematical skills that can open up a new dimension of economic analysis, thereby enhancing their understanding of economic theories.

FEATURES

The emphasis throughout the text is not on mathematical theorems and formal proofs, but on how mathematics can enhance our understanding of the economic behavior under study. The careful development of economic topics strives for a coherence not often found in mathematical economic texts. A premium is placed on clear explanation, with a blending of mathematical, verbal, and graphical exposition. Numerical illustrations of the economic models are used extensively to aid understanding. Frequent practice problems are provided throughout each chapter, with answers at the end of the chapter to give immediate feedback to students, bolstering confidence and building a secure foundation as they progress through the material. The end-of-chapter exercises are comprehensive and challenging. Moreover, key terms in economics (in bold) and mathematics (in italics) are listed at the end of the chapter (with page references), and a glossary of concise definitions is provided at the end of the text.

ORGANIZATION

In the first chapter, the construct of an economic model and the concept of equilibrium are discussed and illustrated with the example of a competitive market for a commodity. Then Chapter 2 provides a review of some basic mathematics; including sets, derivatives, limits, and integrals.

In Chapter 3, which begins Part I of the text, "Analysis of Markets," we use derivatives to establish conditions for the static stability of market equilibria and for comparative static analysis, inverse functions to illustrate different market adjustment mechanisms, and integral calculus to find consumers' surplus and producers' surplus. We shift to dynamic analysis in Chapter 4, beginning with first order difference equations. As a bridge between the discrete time of period analysis and continuous time we review exponential and logarithmic functions. The discussion of compound interest leads us to natural exponential and natural logarithmic functions, which are part of the solution of the first order differential equations also used in the dynamic modeling of a competitive market. In Chapter 5 we expand beyond a single competitive market to general competitive equilibrium. To build up to the matrix algebra needed for solving systems of linear equations, we first review vectors. A technique known as Cramer's rule is introduced, which will prove especially useful in the comparative static analysis for systems of equations.

In Part II, "Optimization," we look behind the market to the underlying optimizing behavior of firms and consumers. Chapter 6 develops the short run decision rules for a firm in selecting the profit-maximizing levels of output and labor employment. The powerful Implicit Function Theorem is introduced to enhance our ability to do comparative static analysis. Perfectly competitive and monopolistically competitive firms are addressed in Chapter 7, along with the concept of price elasticity of demand. Monopolies and monopsonies—at the other end of the theoretical spectrum—are the topics of Chapter 8. We will see how a monopolist may practice price discrimination to bolster profits and how a union can diminish, if not negate, the power of a monopsonist in a labor market. Chapter 9 completes our modeling of market structures by examining duopolies and oligopolies, where the firms in the market, few in number, are actively engaged in competition. The introduction to game theory follows naturally from the discussion of the firms' interdependent decision-making.

The progression from free optimization with one independent variable (e.g., perfectly competitive firms in Chapter 7) to two and more independent variables (e.g., a monopolist with two plants in Chapter 8) allows us to illustrate partial differentiation and the Implicit Function Theorem for the analysis of systems of equations. We then turn to constrained optimization. In Chapter 10 the long run decision of a firm in selecting the cost-minimizing input combination for producing a given output is modeled using the Lagrange multiplier method. With linear programming and nonlinear programming in Chapter 11, we extend the analysis to incorporate optimization under more than one constraint and to illustrate the concept of duality. Chapter 12 concludes Part II of the text with the application of constrained optimization to the theory of consumer behavior. In particular, the labor-leisure tradeoff of a household is modeled.

Part III, "Macroeconomic Analysis," parallels the first part of the text in that the treatment advances from static analysis to comparative statics to dynamic analysis. Chapter 13 begins with a discussion of input-output models. Then a Simple Keynesian Model is developed and applied to two countries to illustrate a phenomenon known as the foreign repercussions effect. In Chapter 14 we build up from the Simple Keynesian model (product market only), to the IS-LM model (addition of the money market), to the Aggregate Demand-Aggregate Supply Model (incorporation of the labor market and aggregate supply constraints). Again partial differentiation and matrix algebra are extensively used in the derivation of the multipliers associated with the various versions of the macromodels. The policy implications of the derived multipliers are discussed. The coverage of growth rates and growth models in the concluding Chapter 15 reinforces the use of natural exponential and natural logarithmic functions and first order differential equations, as well as derivative and integral calculus.

ALTERNATIVE ORGANIZATION

Covering all fifteen chapters in one semester may be challenging. Twelve chapters may be a more reasonable objective. Two basic options for using the text are: (I) Chapters 1-12-which emphasizes the microeconomics (leaving out Part III, "Macroeconomic Analysis"); and (II) Chapters 1-9 and 13-15-which allows for coverage of both microeconomics and macroeconomics (leaving out Chapters 10-12 on constrained optimization). Skipping the dynamic analyses in Chapters 4 and 15 would further reduce options I and II to eleven and ten chapters respectively.

INSTRUCTOR'S MANUAL

Available with the text is an Instructor's Manual, which provides answers to all of the end-of-chapter problems. The practice problems in the text are designed for students to work through themselves as they progress through each chapter. The end-of-chapter problems, of varying degrees of difficulty and often with several parts, can be assigned as homework exercises.

AUDIENCE

Using Mathematics in Economic Analysis can be adopted effectively for different audiences. I believe the text is most appropriate for an undergraduate coursed in mathematical economics where students have completed a first course in calculus and intermediate microeconomic theory and, perhaps, intermediate macroeconomic theory. The text, or individual chapters, may also serve as a supplement to the standard intermediate microeconomic and macroeconomic theory courses. And, the text might function well in the summer review courses given by graduate programs in economics and in Master's programs in business and public policy.

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