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work with these more sophisticabxl. more realistic. and more intererring models.

xxii PREFACE

math-for-economists text. Each chapter begins with a discussion of the economic

motivation for the mathematicel concepts presented. On the other hand, this is

a honk on mathematics for economists, not a text of mathematical economics.

We do not feel that it is productive TV learn advanced mathematics and advanced

economics at the same time. Therefore, have focused on presenting an intro-

WC

duction to the mathematics that students need in order to work with more advanced

economic models.

4. Economics is a dynamic tield; economic theorists are regularly introducing

or using new mathematical ideas and techniques to shed light on economic theory

and econometric analysis. As active researchers in economics, we have tried to

make many of these new approaches available to students. In this book we present

rather complete discussions of topics at the frontier of economic research, topics

like quasiconcave functions, concave programing, indirect utility and cxpendi-

ture functions, envelope theorems, the duality between cost and production, and

nonlinear dynamics.

5. It is important thal studentsofeconomics understand what constitutesa solid

proof-a skill that is learned, not innate. Unlike most other texts in the field, WC

try to present careful proofs of nearly all the mathematical results presented-so

that the reader can understand better both the logic behind the math techniques

used and the total structure in which each result builds upon previous results. In

many of the exercises, students arc asked tu work wt their own proofs, often by

adapting proofs presented in the text.

COORDINATION WITH OTHER COURSES

Often the material in this course is taught concurrently with courses in advanced

micro- and macroeconomics. Students arc sometimes frustrated with this arrange-

ment because the micro and macro courses usually start working with constrained

optimization or dynamics long before these topics can be covered in an orderly

mathematical presentation.

We suggest a number of strategies to minimize this frustration. First, we have

tried to present the material so that a student can read each introductory chapter

in isolation and get a reasonably clear idea of how to work with the material of

that chapter, even without a careful reading of earlier chapters. We have done

this by including a number of worked exercises with descriptive figures in every

introductory chapter.

Often during the first t w o weeks of our first course on this material, we present

a series of short modules that introduces the language and formulation of the more

advanced topics so that students can easily reed selected parts of later chapters on

their own. or at least work out some problems from these chapters.

Finally, we usually ask students who will be taking our course to be farnil-

iar with the chapters on one-variable caIcuIus and simple matrix theory before

classes begin. We have found that nearly every student has taken a calculus coursr

and nearly two-thirds have had some matrix algebra. So this summer reading

reqwrment- sometimes supplemented by a review session ,just before classes

begin ˜ is helpful in making the mathematical backgrounds of the students in the

cc˜ursc more homo:eneous.

ACKNOWLEDGMENTS

It is a ˜ICIISUIC to acknorvled˜!c the wluablc suggestion\ and c˜mmcnts of our

colleagurs. students and reviewers: colleagues such as Philippe Artzner. Ted

Bergstrom Ken Binmore. Dee Dcchert. David Easlry. Leonard Herk. Phil How-q.

Johli Jacquer. Jan Kmenta. James Koopman. Tapan Mitra. Peter Morgan. John

hachhar. Scott Pierce. Zxi Safra. Hal Varian. and Henry Wan: students such as

Katblccn .A˜;derson. Jackie Coolidge. Don Dunbar. Tom Gorge. Kevir Jackson.

Da4 Meyer. Ann Simon. David Simon. and John Woodcrs. and the countless

classrs ilr Coimell and Michigan who struggled through early drafts: reviewer\ such

aвЂ™ Richard Anderson. Texas A 8: M Univrr\it);: .James Bergin. QueenвЂ˜s Uniwr-

slty: Brian Binger. University of Arirona: Mark Feldman. University of Illinois

Roger Folxm˜. San Jose State University: Femidn Handy. York University: John

McDonald. Lnivcrsit!; of Illinois: Norman Ohst. Michigan State Lniversity: John

Kile!;. L;nivcrsity of California at Los Angeles: and Myrna Wooders. llniversity

of Toronta We appreciate the assistance of the people at W.W. Norton. especialI>

Drake McFeely. Catberinc Wick and Catherine Von Novak. The order of the au-

thor\ on tbc cover of thih book merely rcHccts our decision to use different nrdel-s

ior different hooks that w c write.

CHAPTER 2

One-Variable Calculus:

Foundations

A central goal of economic theory is to express and llnderstand relationships he-

tween economic variables. These relationships are described mathematically by

functions. If we arc interested in the effect of one economic variable (like govern-

ment spending) on one other economic variable (like gross national product), we

are led to the study of functions of a single variable-a natural place to begin our

mathematical analysis.

The key information about these relationships between economic variables

concerns how a change in one variable affects the other. How does a change in the

money supply affect interest rates? Will a million dollar increase in government

spending increase or decrease total productionвЂ™! By how muchвЂ™? When such rela-

tionships are expressed in terms of linear functions, the effect of a change in one

variable on the other is captured by the вЂњslopeвЂќ of the function. For more general

nonlinear functions, the effect of this change is captured by the вЂњderivativeвЂќ of

the function. The derivative is simply the generalization of the slope to nonlinear

functions. In this chapter, we will define the derivative of a one-variable function

and learn how to compute it, all the while keeping aware of its role in quantifying

relationships between variables.

2.1 FUNCTIONS ON RвЂ™

Vocabulary of Functions

The basic building blocks of mathematics are numbers and functions. Jn working

with numbers, we will find it convenient to represent them geometrically as points

on a number line. The number line is a line that extends infinitely far to the right

and to the left of a point called the origin. The origin is identiticd with the number

0. Points to the right of the origin represent positive numbers and points to the

left represent negative numbers. A basic unit of length is chosen, and successive

intervals of this length are marked off from the origin. Those to the right are

numbered +l, +2. +3, etc.: those to the left are numbered I, -2, -3. etc. One

can now represent any positive real number on the line by finding that point to

chosen units is that

the rifihr of the origin whose distance from the origin in the

number. Negative numhcrs we represented in the same manner, but by moving

to the kft. Consequently, every real number is represented by exactly one point

on the lint, and each point on the line represents w˜c and only one number. See

Figure 2.1. We write R1 for the set of all real numbers.

N >

-6 -5 4 -3 -2 -1 0 1 2 3 4 5 6

Figure

lint RвЂ™. 2.1

The mmther

A function is simply a rule which assigns a numher in RвЂ™ to each number in

RвЂ™. For example. there is the function which assigns to any number the numher

f(x)

which is one unit larger. We write this function as = I + I. To the number 2

it assigns the number 3 and to the numhcr ˜3/2 it assigns the number l/2. We

wile lhcsc assignments as

f(2) = 3 and f(-3/2) = I /2,

The function which assigns to any numhcr its double can he written as g(x) = 2x.

Write ˜(4) = 8 and ,&3) = -6 to indicate that it assigns 8 to 4 and -6 to -3,

rcspcctively.

˜=,rl, and i; = 2˜V,

x is called the independent variable. or in eco-

respectively. The input vxiahlc

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