Mathematical methods for economics, 2nd edition

Matrix algebra can be used to conduct comparative static analysis, which evaluates the change in the equilibrium values of a model when the value of one or more exogenous variables chang

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Mathematical Methods for Economics

Michael Klein Second Edition

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Chapter 10 Extreme Values of Multivariate Functions

Exponential and Logarithmic Functions

This book begins with a three-chapter section that introduces some important concepts

and tools that are used throughout the rest of the book Chapter 1 presents background

on the mathematical framework of economic analysis In this chapter we discuss the

advantages of using mathematical models in economics We also introduce some

charac-teristics of economic models The discussion in this chapter makes reference to material

presented in the rest of the book to put this discussion in context as well as to give you

some idea of the types of topics addressed by this book.

Chapter 2 discusses the central topic of functions The chapter begins by defining

some terms and presenting some key concepts Various properties of functions first

intro-duced in this chapter appear again in later chapters The final section of Chapter 2

pres-ents a menu of different types of functions that are used frequently in economic analysis.

Two types of functions that are particularly important in economic analysis are

exponential and logarithmic functions As shown in Chapter 3, exponential functions are

used for calculating growth and discounting Logarithmic functions, which are related to

exponential functions, have a number of properties that make them useful in economic

modeling Applications in this chapter, which include the distinction between annual and

effective interest rates, calculating doubling time, and graphing time series of variables,

demonstrate some of the uses of exponential and logarithmic functions in economic

analysis Later chapters make extensive use of these functions as well.

Part One

The Mathematical Framework

of Economic Analysis

What are the sources of long-run growth and prosperity in an economy? How

does your level of education affect your lifetime earnings profile? Has foreigncompetition from developing countries widened the gap between the rich andthe poor in industrialized countries? Will economic development lead to increased envi-

ronmental degradation? How do college scholarship rules affect savings rates? What is

the cost of inflation in an economy? What determines the price of foreign currency?

The answers to these and similar economic questions have important

conse-quences The importance of economic issues combined with the possibility for

alterna-tive modes of economic analysis result in widespread discussion and debate This

discussion and debate takes place in numerous forums including informal

conversa-tions, news shows, editorials in newspapers, and scholarly research articles addressed to

an audience of trained economists Participants in these discussions and debates base

their analyses and arguments on implicit or explicit frameworks of reasoning

Economists are trained in the use of explicit economic models to analyze

eco-nomic issues These models are usually expressed as sets of relationships that take a

mathematical form Thus an important part of an economist’s training is acquiring a

command of the mathematical tools and techniques used in constructing and solving

economic models

This book teaches the core set of these mathematical tools and techniques The

mathematics presented here provides access to a wide range of economic analysis and

research Yet a presentation of the mathematics alone is often insufficient for students

who want to understand the use of these tools in economics because the link between

mathematical theory and economic application is not always apparent Therefore this

book places the mathematical tools in the context of economic applications These

applications provide an important bridge between mathematical techniques and

eco-nomic analysis and also demonstrate the range of uses of mathematics in ecoeco-nomics

The parallel presentation of mathematical techniques and economic applications

serves several purposes It reinforces the teaching of mathematics by providing a

set-ting for using the techniques Demonstraset-ting the use of mathematics in economics

helps develop mathematical comprehension as well as hone economic intuition In this

Chapter 1

way, the study of mathematical methods used in economics as presented in this bookcomplements your study in other economics courses The economic applications in thisbook also help motivate the teaching of mathematics by emphasizing the practicaluse of mathematics in economic analysis An effort is made to make the applicationsreference a wide range of topics by drawing from a cross section of disciplines withineconomics, including microeconomics, macroeconomics, economic growth, interna-tional trade, labor economics, environmental economics, and finance In fact, each

of the questions posed at the beginning of this chapter is the subject of an application

in this book

This chapter sets the stage for the rest of the book by discussing the nature ofeconomic models and the role of mathematics in economic modeling Section 1.1discusses the link between a model and the phenomenon it attempts to explain Thissection also discusses why economic analysis typically employs a mathematical frame-work Section 1.2 discusses some characteristics of models used in economics and pre-views the material presented in the rest of the book

1.1 ECONOMIC MODELS AND ECONOMIC REALITY

Any economic analysis is based upon some framework This framework may behighly sophisticated, as with a multiequation model based on individuals who attempt

to achieve an optimal outcome while facing a set of constraints, or it may be very plistic and involve nothing more complicated than the notion that economic variablesfollow some well-defined pattern over time An overall evaluation of an economicanalysis requires an evaluation of the framework itself, a consideration of the accu-racy and relevance of the facts and assumptions used in that framework, and a test ofits predictions

sim-A framework based on a formal mathematical model has certain advantages sim-Amathematical model demands a logical rigor that may not be found in a less formalframework Rigorous analysis need not be mathematical, but economic analysis lendsitself to the use of mathematics because many of the underlying concepts in economicscan be directly translated into a mathematical form The concept of determining aneconomic equilibrium corresponds to the mathematical technique of solving systems

of equations, the subject of Part Two of this book Questions concerning how one able responds to changes in the value of another variable, as embodied in economicconcepts like price elasticity or marginal cost, can be given rigorous form through theuse of differentiation, the subject of Part Three Formal models that reflect the centralconcept of economics—the assumption that people strive to obtain the best possibleoutcome given certain constraints—can be solved using the mathematical techniques

vari-of constrained optimization These are discussed in Part Four Economic questions thatinvolve consideration of the evolution of markets or economic conditions over time—questions that are important in such fields as macroeconomics, finance, and resourceeconomics—can be addressed using the various types of mathematical techniques pre-sented in Part Five

While logical rigor ensures that conclusions follow from assumptions, it shouldalso be the case that the conclusions of a model are not too sensitive to its assumptions

It is typically the case that the assumptions of a formal mathematical model are

explicit and transparent Therefore a formal mathematical model often readily admits

the sensitivity of its conclusions to its assumptions The evolution of modern growth

theory offers a good example of this

A central question of economic growth concerns the long-run stability of market

economies In the wake of the Great Depression of the 1930s, Roy Harrod and Evsey

Domar each developed models in which economies either were precariously balanced

on a “knife-edge” of stable growth or were marked by ongoing instability Robert

Solow, in a paper published in the mid-1950s, showed how the instability of the

Harrod–Domar model was a consequence of a single crucial assumption concerning

production Solow developed a model with a more realistic production relationship,

which was characterized by a stable growth path The Solow growth model has become

one of the most influential and widely cited in economics Applications in Chapters 8,

9, 13, and 15 in this text draw on Solow’s important contribution More recently,

research on “endogenous growth” models has studied how alternative production

rela-tionships may lead to divergent economic performance across countries Drawing on

the endogenous growth literature, this book includes an application in Chapter 8 that

discusses research by Robert Lucas on the proper specification of the production

func-tion as well as an applicafunc-tion that presents a growth model with “poverty traps” in

Chapter 13.1

Once a model is set up and its underlying assumptions specified, mathematical

techniques often enable us to solve the model in a straightforward manner even if the

underlying problem is complicated Thus mathematics provides a set of powerful tools

that enable economists to understand how complicated relationships are linked and

exactly what conclusions follow from the assumptions and construction of the model

The solution to an economic model, in turn, may offer new or more subtle economic

intuition Many applications in this text illustrate this, including those on the incidence

of a tax in Chapters 4 and 7, the allocation of time to different activities in Chapter 11,

and prices in financial markets in Chapters 12 and 13 Optimal control theory, the

sub-ject of Chapter 15, provides another example of the power of mathematics to solve

complicated questions We discuss in Chapter 15 how optimal control theory, a

mathe-matical technique developed in the 1950s, allowed economists to resolve long-standing

questions concerning the price of capital

A mathematical model often offers conclusions that are directly testable against

data These tests provide an empirical standard against which the model can be judged

The branch of economics concerned with using data to test economic hypotheses is

called econometrics While this book does not cover econometrics, a number of the

applications show how to use mathematical tools to interpret econometric results For

example, in Chapter 7 we show how an appropriate mathematical function enables us

to determine the link between national income per capita and infant mortality rates in

1Solow’s paper, “A contribution to the theory of economic growth,” is published in the Quarterly Journal of

Economics, 70, no 1 (February 1956): 65–94 The other papers cited here are Roy F Harrod, “An essay in

dynamic theory,” Economic Journal, 49 (June 1939): 14–33; Evsey Domar, “Capital expansion, rate of

growth, and employment,” Econometrica, 14 (April 1946): 137–147; and Robert Lucas, “Why doesn’t capital

flow from rich to poor countries?” American Economic Review, 80, no 2 (May 1990): 92–96.

a cross section of countries An application in Chapter 9 discusses some recent research

on the relationship between pollution and income in a number of countries, whichbears on the question of the extent to which rapidly growing countries will contribute

to despoiling the environment Chapter 8 includes an application that draws from aclassic study of the financial returns to education

It is natural to begin a book of this nature with a discussion of the many tages of using a formal mathematical method for addressing economic issues It isimportant, at the same time, to recognize possible drawbacks of this approach Anymathematical model simplifies reality and, in so doing, may present an incomplete pic-ture The comparison of an economic model with a map is instructive here A map nec-essarily simplifies the geography it attempts to describe There is a trade-off betweenthe comprehensiveness and readability of a map The clutter of a very comprehensivemap may make it difficult to read The simplicity of a very readable map may come atthe expense of omitting important landmarks, streets, or other geographic features Inmuch the same way, an economic model that is too comprehensive may not betractable, while a model that is too simple may present a distorted view of reality.The question then arises of which economic model should be used To answer thisquestion by continuing with our analogy to maps, we recognize that the best map forone purpose is probably not the best map for another purpose A highly schematic sub-way map with a few lines may be the appropriate tool for navigating a city’s subways,but it may be useless or even misleading if used aboveground Likewise, a particulareconomic model may be appropriate for addressing some issues but not others Forexample, the simple savings relationship posited in many economic growth modelsmay be fine in that context but wholly inappropriate for more detailed studies of sav-ings behavior

advan-The mathematical tools presented in this book will give you access to many esting ideas in economics that are formalized through mathematical modeling Thesetools are used in a wide range of economic models While economic models may differ

inter-in many ways, they all share some common characteristics We next turn to a discussion

of these characteristics

1.2 CHARACTERISTICS OF ECONOMIC MODELS

An economic model attempts to explain the behavior of a set of variables through thebehavior of other variables and through the way the variables interact The variablesused in the model, which are themselves determined outside the context of the model,

are called exogenous variables The variables determined by the model are called

endogenous variables The economic model captures the link between the exogenous

and endogenous variables

A simple economic model illustrates the distinction between endogenous andexogenous variables Consider a simple demand and supply analysis of the market forthe familiar mythical good, the “widget.” The endogenous variables in this model arethe price of a widget and the quantity of widgets sold The exogenous variables in thisexample include the price of the input to widget production and the price of the goodthat consumers consider as a possible substitute for widgets

In this example there is an apparently straightforward separation of variables into

the categories of exogenous and endogenous This separation actually represents a

cen-tral assumption of this model—the assumption that the market for the input used in

producing widgets and the market for the potential substitute for widgets are not

affected by what happens in the market for widgets In general, the separation of

vari-ables into those that are exogenous and those that are endogenous reflects an

impor-tant assumption of an economic model Exogenous variables in some models may be

endogenous variables in others This may sometimes reflect the fact that one model is

more complete than another in that it includes a wider set of endogenous variables For

example, investment is exogenous in the simplest Keynesian cross diagram and

endoge-nous in the more complicated IS/LM model In other cases the purpose of the model

determines which variables are endogenous and which are exogenous Government

spending is usually considered exogenous in macroeconomic models but endogenous in

public choice models Even the weather, which is typically considered exogenous, may

be endogenous in a model of the economic determinants of global warming In fact,

much debate in economics concerns whether certain variables are better characterized

as exogenous or endogenous

An economic model links its exogenous and endogenous variables through a

set of relationships called functions These functions may be described by specific

equations or by more general relationships Functions are defined in Chapter 2

In that chapter we describe different types of equations that are frequently used

as functions in economic models For now we identify three categories of

relation-ships used in economic models: definitions, behavioral equations, and equilibrium

conditions

A definition is an expression in which one variable is defined to be identically

equal to some function of one or more other variables For example, profit is total

revenue (TR) minus total cost (TC ), and this definition can be written as

where “” means “is identically equal to.”

A behavioral equation represents a modeling of people’s actions based on

eco-nomic principles The demand equation and supply equation in microecoeco-nomics, as

well as the investment, money demand, and consumption equations in

macroeconom-ics, all represent behavioral equations Sometimes these equations reflect very basic

economic assumptions such as utility maximization In other cases, behavioral

equa-tions are not derived explicitly from basic economic assumpequa-tions but reflect a general

relationship consistent with economic reasoning

An equilibrium condition is a relationship that defines an equilibrium or steady

state of the model In equilibrium there are no economic forces within the context of

the model that alter the values of the endogenous variables

We use our example of the market for widgets to illustrate these concepts The

two behavioral equations in this model are a demand equation and a supply equation

We specify the demand equation for widgets as

Q D     P   G

  TR  TC,

()

and the supply equation as

where Q D is the quantity of widgets demanded, Q Sis the quantity of widgets supplied,

P is the price of widgets, G is the price of goods that are potential substitutes for

widg-ets, and N is the price of inputs used in producing widgets The Greek letters in these

equations,, , , , , and , represent the parameters of the model A parameter is a

given constant A parameter may be some arbitrary constant, as is the case here, or aspecific value like 100, or 7.2

A simple example of an equilibrium condition sets the demand for widgets equal

to the supply of widgets This gives us the equilibrium condition

A simultaneous solution of the demand equation, supply equation, and

equilib-rium condition gives a solution to this model The solution to a model is a set of values

of its endogenous variables that correspond to a given set of values of its exogenousvariables and a given set of parameters Thus, in this case, the solution will show how

the endogenous variables P and Q (where, in equilibrium, Q equals both quantity

demanded and quantity supplied) depend upon the values of the exogenous variables

N and G, as well as the values of the six parameters of the model The values of the

endogenous variables in equilibrium are their equilibrium values.2

The structure of this model is quite simple One reason for this is that the ioral equations are each linear functions since they take the form

behav-,

where y, x, and z are variables and a, b, and c are parameters In this equation y is the

dependent variable, and the variables x and z are the independent variables The

linearity of the behavioral equations enables us to find a solution for the model using

the techniques of linear algebra (also called matrix algebra) presented in Part Two of

this book (Chapters 4 and 5) The techniques in these chapters show how to

deter-mine easily whether a model consisting of several linear equations has a unique

solution Matrix algebra can be used to conduct comparative static analysis, which

evaluates the change in the equilibrium values of a model when the value of one

or more exogenous variables changes For example, an evaluation of the change

in the equilibrium value of the price of widgets and the quantity of widgets boughtand sold in response to a change in the price of the input to widget production would

be a comparative static analysis While the requirement of linearity may seem tive, the discussion of logarithmic functions and exponential functions in Chapter 3shows that certain nonlinear functions can be expressed in linear form Also materialpresented in Chapter 7 shows how to obtain a linear approximation of a nonlinearfunction

restric-The determination of the solution to this simple linear model may be only thebeginning of a deeper economic analysis of the widget market Such an analysis may

require a broader set of mathematical techniques For instance, suppose a tax is

imposed on the sale of widgets The tax revenues from the sale of widgets, T, is given by

the definition

where  is the tax rate and is the value of total widget sales How does a

change in the price of potential substitutes for widgets affect the tax revenues

received from the sale of widgets? Questions of this nature require the use of

differ-ential calculus, which is the subject of Part Three (Chapters 6 through 8) Differdiffer-ential

calculus offers a set of tools for analyzing the responsiveness of the dependent

vari-able of a function to changes in the value of one or more of its independent varivari-ables

These tools are useful in addressing questions such as the responsiveness of the

demand for widgets to changes in their price Chapter 6 provides an intuitive

intro-duction to this subject Rules of univariate calculus are presented in Chapter 7

Chapter 8 presents the techniques of multivariate calculus This chapter builds your

intuition for multivariate calculus by demonstrating the link between it and the

important economic concept of ceteris paribus, that is, “all else held equal.” The

tech-niques presented in this chapter enable you to address the question of the

responsive-ness of tax revenues from the sale of widgets to a change in the price of the inputs to

widget production

An important application of differential calculus in economics is the

identifica-tion of extreme values, that is, the largest or smallest value of a funcidentifica-tion Part Four,

consisting of Chapters 9 through 11, shows how to apply differential calculus in order

to identify extreme values of functions Chapter 9 illustrates how to use the tools of

calculus to identify extreme values of functions that include only one independent

variable An example of an economic application of this technique is the identification

of the optimal price set by a widget monopolist Chapter 10 extends this analysis to

functions with more than one independent variable An application in that chapter

illustrates how the widget monopolist could optimally set prices in two separate

mar-kets Chapter 11 shows how to determine the extreme value of functions when their

independent variables are constrained by certain conditions This technique of

con-strained optimization explicitly captures the core economic concept of obtaining the

best outcome in the face of trade-offs among alternatives Given a target level of

widget production, constrained optimization would be used to determine the optimal

amounts of various inputs

The book concludes with a discussion of dynamic analysis in Part Five Dynamic

analysis focuses on models in which time and the time path of variables are explicitly

included This part begins with Chapter 12, which presents integral calculus A

com-mon use of integral calculus in economics is the valuation of streams of payments over

time For example, the widget manufacturer, recognizing that a dollar received today is

not the same as a dollar received tomorrow, might want to value the stream of

pay-ments from selling widgets at different times Another application of integral calculus,

one not related to time, is the determination of consumer’s surplus from the sale of

widgets We discuss consumer’s surplus in two applications in Chapter 12 Chapters 13

and 14 show how to solve economic models that explicitly include a time dimension In

its discussion of difference equations, Chapter 13 focuses on models in which time is

(Q P)

T   (Q P),

treated as a series of distinct periods In its discussion of differential equations,

Chapter 14 focuses on models in which time is treated as a continuous flow Many mon themes arise in the discussion of difference equations and of differential equa-

com-tions Chapter 15 concludes this section with a presentation of dynamic optimization, a

technique for solving for the optimal time path of variables Dynamic optimizationwould enable us to analyze questions like the optimal investment strategy over timefor a widget maker

A Note on Studying This Material

As you study the material in this book, it is important to engage actively with the textrather than just to read it passively When reading this book, keep a pencil and paper athand, and replicate the chains of reasoning presented in the text The problems pre-sented at the end of each chapter section are an integral part of this book, and workingthrough these problems is a vital part of your study of this material It is also useful to

go beyond the text by thinking yourself of examples or applications that arise in theother fields of economics that you are studying An ability to do this demonstrates amastery of the material presented here

An Introduction to Functions

Functions are the building blocks of explicit economic models You have probably

encountered the term “function” already in your economics education Basic

macroeconomic theory uses, for example, the consumption function, which shows

how consumption varies with income Basic microeconomic theory presents, among

others, the production function, which shows how a firm’s output varies with the level

of its inputs Just as M Jourdain, the title character in Molière’s Le Bourgeois

Gentilhomme, remarked that he had been speaking prose all his life without knowing

it, the material presented in this chapter may make you realize that you have been

using mathematical functions during your entire economics education

An ability to analyze and characterize functions used in economics is important

for a complete understanding of the theory they are used to express The concepts and

tools introduced in this chapter provide the basis for analyzing and characterizing

functions Later chapters of this book will build on the concepts first introduced in this

chapter

This chapter opens with definitions of terms that are important for discussing

functions This section also includes an introduction to graphing functions Section 2.2

discusses properties and characteristics of functions Many of these characteristics are

discussed in the context of graphs There is also a discussion in this section of the

logi-cal concept of necessary and sufficient conditions The final section of this chapter

introduces some general forms of functions used extensively in economics

2.1 A LEXICON FOR FUNCTIONS

A discussion of functions must begin with some definitions In this section we define

some basic concepts and terms We also introduce the way in which functions can be

depicted using graphs

Variables and Their Values

As discussed in Chapter 1, economic models link the value of exogenous variables to the

value of endogenous variables The variables studied in economics may be qualitative or

quantitative A qualitative variable represents some distinguishing characteristic, such as

Chapter 2

male or female, working or unemployed, and Republican, Democrat or Independent.

The relationship between values of a qualitative variable is not numerical Quantitative

variables, on the other hand, can be measured numerically Familiar economic

quantita-tive variables include the dollar value of national income, the number of barrels ofimported oil, the consumer price level, and the dollar-yen exchange rate Some quantita-

tive variables, like population, may be expressed as an integer.An integer is a whole

num-ber like 1, 219,32, or 0.The value of other variables, like a stock price, may fall between

two integers Real numbers include all integers and all numbers between the integers.

Some real numbers can be expressed as ratios of integers, for example, , 2.5, or3

These numbers are called rational numbers Other real numbers, such as

and cannot be expressed as a ratio and are called irrational numbers.

In discussing functions we often refer to an interval rather than a single number.

An interval is the set of all real numbers between two endpoints Types of intervals

are distinguished by the manner in which endpoints are treated A closed interval

includes the endpoints The closed interval between 0 and 1.5 includes these two

num-bers and is written [0, 1.5] An open interval between any two numnum-bers excludes the endpoints The open interval between 7 and 10 is written (7, 10) A half-closed interval

or a open interval includes one endpoint but not the other Notation for

half-closed or half-open intervals follows from the notation for half-closed and open intervals.For example, if an interval includes the endpoint but not the endpoint 1, it is writ-ten as [ , 1) An infinite interval has negative infinity, positive infinity, or both as

endpoints The closed interval of all positive numbers and zero is written as [0,).Theopen interval of all positive numbers is written as (0,) The interval of all real num-bers is written as (, )

Sets and Functions

A set is simply a collection of items The items included in a set are called its elements.

Some examples of sets include “economists who have won a Nobel Prize by 2001,” aset consisting of 46 elements, and “economists who would have liked to have won theNobel Prize by 2001,” a set with a membership that probably numbers in the thou-sands Sets are represented by capital letters To show that an item is an element of aset, we use the symbol  For example, if we denote the set of all Nobel Prize–winning

economists by N, then

To show that elements are not members of a set, we use the symbol  For example,

The set N can be described either by listing all its elements or by describing the

conditions required for membership Sets of numbers with a finite number of elementscan be described similarly For example, consider the set of all integers between and

5 We can describe this set by simply listing its five elements

S{1, 2, 3, 4, 5}

1 2

1 2

Adam Smith N.

Paul Samuelson N Milton Friedman  N.

3 2

3 2

2,

  3.1415 .

2 5 1

2

Alternatively, we can describe the set by describing the conditions for membership

This statement is read as “S is the set of all numbers x such that x is an integer greater

than and less than ” Sets that have an infinite number of elements can be

described by stating the condition for membership For example, the set of all real

numbers x in the closed interval can be written as

The elements of one set may be associated with the elements of another set

through a relationship A particular type of relationship, called a function, is a rule that

associates each element of one set with a single element of another set A function is

also called a mapping or a transformation A function f that unambiguously associates

with each element of a set X one element in the set Y is written as

In this case, the set X is called the domain of the function f , and the set of values that

occur is called the range of the function f

An example of a function is the rule d that associates each member of the Nobel

Prize–winning set N with the year in which he won the prize, an element of the set T :

As shown in Figure 2.1, this function maps James Tobin, a member of N, to 1981, an

element of the set T This function also maps both Kenneth Arrow and Sir John

Hicks, each a member of N, to 1972, an element of T, since Arrow and Hicks jointly

shared the Nobel prize in that year Note that the reverse relationship that associates

the elements of the set T to the elements of the set N is not a function since there are

cases where an element of T maps to two or more separate elements of N For

exam-ple, the year 1972, an element of T, is associated with two elements of N, Arrow and

Hicks

Univariate Functions

A univariate function maps one number, which is a member of the domain, to one and

only one number, which is an element of the range A standard way to represent a

uni-variate function that maps any one element x of the set X to one and only one element

y of the set Y is

which is read as “y is a function of x” or “y equals f of x.” In this case the variable y is

called the dependent variable or the value of the function, and the variable x is called

the independent variable or the argument of the function.

Sx  x is an integer greater than 1

2 and less than 5

1

2

The term can represent any relationship that assigns a unique value to y for any value of x, such as

The numbers and 2 in the first function and the Greek letters  and  in the second

function represent parameters As discussed in Chapter 1, a parameter may be either a

specific numerical value, like 2, or an unspecified constant, like .

Given numerical parameter values, we can find the value of a univariate functionfor different values of its argument For example, consider a basic Keynesian consump-

tion function that relates consumption, C, to income, I, as

C  300  0.6I

1 2

The Set of Years in Which the

The Set of Nobel Laureates in Economics (N) Nobel Prize was Awarded (T )

FIGURE 2.1 The Sets N and T

where all variables represent billions of dollars and “300” stands for $300 billion Table

2.1 reports the value of consumption for various values of income consistent with (2.1)

Graphing Univariate Functions

Table 2.1 illustrates the behavior of the consumption function by providing some values

of its independent variable along with the associated value of its dependent variable This

table presents numbers that can be used to construct some ordered pairs of the

consump-tion funcconsump-tion An ordered pair is two numbers presented in parentheses and separated by

a comma, where the first number represents the argument of the function and the second

number represents the corresponding value of the function Thus each ordered pair for

the function y  f(x) takes the form (x, y) Some ordered pairs consistent with the

con-sumption function presented previously are (1000, 900), (2500, 1800) and (5000, 3300)

Ordered pairs can be plotted in a Cartesian plane (named after the

seventeenth-century French mathematician and philosopher René Descartes) A Cartesian plane, like

the one presented in Figure 2.2, includes two lines, called axes, which cross at a right angle

The origin of the plane occurs at the intersection of the two axes Points along the

horizon-tal axis, also called the x-axis, of the Cartesian plane in Figure 2.2 represent values of the

level of income, which are the arguments of this function Points along the vertical axis,

also called the y-axis, represent values of the level of consumption, which are the values of

this function The coordinates of a point are the values of its ordered pair and represent

the address of that point in the plane The x-coordinate of the pair (x, y) is called the

abscissa, and the y-coordinate is called the ordinate Thus the origin of a Cartesian plane

is represented by the coordinates (0, 0).Two ordered pairs for the univariate consumption

function are represented by points labeled with their coordinates in Figure 2.2

We could continue this exercise by filling in more and more points consistent

with the consumption function Alternatively, we can plot the graph of the function.

TABLE 2.1 A Consumption Function

The graph of a function represents all points whose coordinates are ordered pairs ofthe function The graph of the consumption function for the domain [0, 2500] is repre-

sented by the line AB in Figure 2.2 This graph goes through the first three points

iden-tified in Table 2.1 as well as all other points consistent with the consumption functionover the relevant domain

The consumption function depicted in Figure 2.2 is a particular example of a

linear function A linear function takes the form1

The parameter a is the intercept of the function and represents the value of the

func-tion when its argument equals zero In a graph, the intercept is the point where the

function crosses the y-axis The intercept of the consumption function is 300 The

parameter b is the slope of the graph of the function The slope of a univariate linear

function represents the change in the value of the function associated with a givenchange in its argument The slope of the linear function (2.2) evaluated between any

two points x A and x B(for ) is

where f (x) B  f(x A) is the change in the value of the function associated with the

change in its argument x B  x A This result shows that the slope of a linear function is

constant and equal to the parameter b For example, the slope of the consumption

function presented above is 0.6

Figure 2.2 presents a plane with only one quadrant since the domain and therange of the consumption function are restricted to include only positive numbers.Many economic functions include both positive and negative numbers as argumentsand values Graphs of these functions can be represented with other quadrants of theCartesian plane In Figure 2.3 the function

is presented You can verify that this function includes the four ordered pairs(2, 8), and (3, 8) Each of these ordered pairs is in a differentquadrant of the Cartesian plane, which indicates that the graph of this function passesthrough all four quadrants

Multivariate Functions

A multivariate function has more than one argument For example, the general form of

a multivariate function with the dependent variable y and the three independent

1Strictly speaking, a univariate linear function takes the form y  bx and a function of the form y  a + bx is

called an affine function Following convention, we use the term linear function to mean an affine function.

Note that here we have used subscripts to distinguish among the different independent

variables Even more generally, a multivariate function with n independent variables

denoted x1, x2, and so on, can be written as

A multivariate function with two independent variables is called a bivariate

func-tion Some specific bivariate functions include

The first function includes the dependent variable j, the independent variables k and h,

and the parameters 5, 4, 3, and 7 The second function includes the dependent variable

Q, the independent variables K and L, and the parameters , and

The set of arguments and the corresponding value of a multivariate function can

also be represented by ordered groupings of numbers For example, the bivariate

con-sumption function

(2.3)

where W represents wealth and all variables are expressed in billions of dollars,

gener-ates ordered triples of the form (I, W, C) Two of the ordered triples for this bivariate

consumption function are (5000, 60000, 4500) and (8000, 40000, 5900)

It is also possible to depict a bivariate function in a figure, although this demands

greater drafting skills than the depiction of a univariate function since the surface of a

piece of paper has only two dimensions Nevertheless, we can give the illusion of three

1 2 – , –2

1 2

1 2 , – 4

FIGURE 2.3 Graph of Function Filling Four Quadrants

dimensions when depicting a function of the form z  f(x, y) by drawing the x-axis as a horizontal line to the right of the origin, the y-axis as a line sloping down and to the left from the origin, and the z-axis as a vertical line rising from the origin as shown in Figure 2.4 This figure depicts the multivariate consumption function (2.3) The x- and

y-axes of this graph represent the values of income and wealth, respectively The values

of the function, which are the consumption values, are represented by the heights of

the points in the graphed plane above the I-W surface.

Limits and Continuity

It is often necessary in economics to evaluate a function as its argument approachessome value For example, in the next chapter we will learn how to find the value today

of an infinitely-long stream of future payments In the dynamic analysis presented inPart Five of this book, we solve for the long-run level of a variable In these cases theargument of the function is time, and we evaluate the value of the function as timeapproaches infinity In Part Three of this book we will learn how to evaluate the effect

of a very small change in the argument of a function We show that there is a spondence between this mathematical technique and the economic concept of evaluat-ing the effect “at the margin.” In this section we show how to evaluate a function as its

corre-argument approaches a certain value by introducing the concept of a limit.

The limit of a function as its argument approaches some number a is simply the number that the function’s value approaches as the argument approaches a, either

from smaller values of a, giving the left-hand limit, or from larger values of a, giving the

exists and is equal to L Lif, for any arbitrarily small number , there exists a small

num-ber  such that

Right-Hand Limit The right-hand limit of a function as its argument

appro-aches some number a, written as

exists and is equal to L Rif for any arbitrarily small number , there exists a small

num-ber  such that

When the left-hand limit equals the right-hand limit, we can simplify the notation

by suppressing the superscripts and defining

The limit of a function as its argument approaches some number a equals

positive infinity if the value of the function increases without bound, and the limit

equals negative infinity if the value of the function decreases without bound

Formally,

if, for every there is a so that

if, for every there is a so that

Rules for Evaluating Limits The following two rules are used in evaluating limits:

and

where k, m, and h are arbitrary real numbers and m  0 

Two applications of these rules are shown below:

and

The limits in these two examples are finite The following are examples of limitsthat are infinite:

and

One use of limits in the context of the material presented in this book is to

deter-mine whether a function is continuous Intuitively, a continuous univariate function

has no “breaks” or “jumps.” A more formal definition follows

Continuity A function f (x) is continuous at x  a, where a is in the domain of f, if the left- and right-hand limits at x  a exist and are equal,

and the limit as equals the value of the function at that point,

q  q0since the purchase of an additional piece of capital, like a new factory or new

equipment, which requires a large one-time cost, is required to increase output

above q0

The second part of the definition shows that even if the left-hand limit and the

right-hand limit of a function exist and are equal at a, it is also necessary for the

func-tion to be defined at a for the funcfunc-tion to be continuous This requirement is made

clear by considering the function f (x) 2+ 5 as x approaches 3 The

left-hand limit and the right-left-hand limit are the same since

However, this function is not defined at x  3 since the term is not defined Figure 2.5(c)

illustrates that this function has a vertical asymptote at x  3 A vertical asymptote of a

function occurs at a point when either a left-hand limit or a right-hand limit approaches

positive infinity or negative infinity at that point A function is discontinuous at a point

where there is a vertical asymptote

2 Determine which of the following relationships represent functions Assume that

the interval is the set of real numbers unless otherwise indicated

1 0

(a) y  5x (b) y  x

defined according to the mapping

(a) Set X consists of all the alumni of Anycollege University; set Y is each

alum-nus’ alma mater

(b) Set X consists of the workers at Busy Firm; set Y is each worker’s social

secu-rity number

(c) Set X consists of all the people who have shared the prize for Best-Dressed Celebrity in any given year; set Y consists of the years in which the Best-

Dressed Celebrity prize was shared

(d) Set X is a set of fathers; set Y is the set of their sons.

4 Consider again the functional mapping where N is the set of Nobel Prize winners and T is the set of years in which the prizes were won If an econo-

mist wins the prize for a second time, would this still be a valid function? Explain

5 The total cost of a firm can be expressed as a simple univariate function in which

cost, C, is a function of the firm’s daily output, Q Assume that the total cost tion is C  75  5Q.

func-(a) Calculate the firm’s total cost when Q  10 and Q  25 What are the firm’s

costs if there is no production?

(b) Graph this firm’s total cost function based on your answers to question 5(a).(c) Now assume that the firm faces a capacity constraint and cannot producemore than 50 units of output a day What are the domain and range of the costfunction in this scenario?

6 Identify and graph four ordered pairs for each of the following functions Sketch agraph of each of the functions

(a) y  100  20x over the interval [2, 6]

(b) y  x  x3over the interval (5, 5)

(c) y  x2 1 over the interval [100, 100]

7 Evaluate the following limits

(d)

(Hint: Transform the ratio to remove x from the denominator.)

8 Which functions are continuous over the given intervals?

(a)

(b)

(c)

(d)

9 Is the function presented in question 8(c) continuous over the domain (, 0]?

Explain If the function is not continuous, at what point (or points) in this domain

is the function discontinuous?

2.2 PROPERTIES OF FUNCTIONS

Much of the analysis of economic functions involves characterizing these functions and

understanding the economic relevance of these mathematical characteristics In this

section we introduce a number of properties of functions Many of these properties are

illustrated through the use of graphs, and thus we define and illustrate these properties

in the context of univariate functions In later chapters we return to these properties,

sometimes presenting alternative (though equivalent) definitions and sometimes

gen-eralizing the definitions to multivariate functions Later chapters also stress the

eco-nomic interpretation of these properties

Increasing Functions and Decreasing Functions

The graph of the consumption function in Figure 2.2 shows that consumption

tently rises as income rises The value of other functions used in economics may

consis-tently decrease as the argument of the function increases For example, most

specifications of demand functions have the quantity demanded of a good steadily

decrease as the price of that good increases A function y  f(x) is increasing, strictly

increasing, decreasing, or strictly decreasing if it meets the following criteria for any

two of its arguments, x A and x B , where x B x A

These definitions show that any strictly increasing function is also an increasingfunction, and any strictly decreasing function is also a decreasing function Anincreasing function, however, may not be a strictly increasing function since an

increasing function may have a section where f (x B) f(x A ) This is illustrated in

Figure 2.6 The increasing function in Figure 2.6(a) has a horizontal section, whichprecludes it from being a strictly increasing function Figure 2.6(b) is strictly increas-ing Figures 2.6(c) and 2.6(d) present graphs of a decreasing and a strictly decreasingfunction, respectively

Closely related to these definitions of increasing and decreasing functions are the

definitions of a monotonic function, a strictly monotonic function, and a non-monotonic

function.

A function is:

monotonic if it is increasing or if it is decreasing.

strictly monotonic if it is strictly increasing or if it is strictly decreasing.

0 Increasing (not strictly)

(a)

x

f (x)

0 Strictly Increasing

(b)

x

f (x)

0 Decreasing (not strictly)

(c)

x

f (x)

0 Strictly Decreasing (d)

x

f (x)

FIGURE 2.6 Increasing Functions and Decreasing Functions

non-monotonic if it is strictly increasing over some interval and strictly

decreas-ing over another interval

Non-monotonic functions can have the same value for more than one argument

For example, the value of the non-monotonic function y  x2is 4 for both x 2 and

x 2 In contrast, strictly monotonic functions unambiguously assign only one value

to any argument Therefore, strictly monotonic functions are one-to-one functions and

have the following property

One-to-One Function A function f (x) is one-to-one if for any two values of the

argument x1and x2,



Any one-to-one function has an inverse function The inverse of the function

y  f(x) is written as y  f1(x).2We find the inverse of a function y  f(x) by solving

for x in terms of y and then interchanging x and y to obtain y  f1(x) For example,

the inverse of the one-to-one function

(2.4)

can be found by solving this for x in terms of y to get

and then interchanging x and y to obtain the inverse function

An important property of a function f (x) and its inverse f1(x) is

The term represents a composite function The argument of a composite

function is itself a function For example,

is a composite function with the inside function h(x) and the outside function g (•)

where the symbol • is a placeholder for the argument of the function g The outside

function for the composite function

is f(•), and the inside function is its inverse For example, using the function (2.4), we have

There is a simple method for graphing the inverse of a function, y  f1(x) given the graph of the function y  f(x).This method makes use of the fact that for any given ordered pair (a, b) associated with a one-to-one function, there will be an ordered pair (b, a) associated with its inverse function For example, one ordered pair associated

with the function (2.4) is (2, 8), and an ordered pair associated with its inverse is (8, 2)

To graph y  f1(x), we make use of the 45 line, which is the graph of the function

y  x that passes through all points of the form (a, a) The graph of the function

y  f1(x) is the reflection of the graph of f (x) across the line y  x This is illustrated

in Figure 2.7 You can think of the reflection across y  x as the graph that is created

when you fold the original figure along the 45 line The coordinates (a, b) of the nal function become the coordinates (b, a) of the inverse function This property of

origi-inverse functions will be important in our study of logarithmic and exponential tions in Chapter 3

func-Extreme Values

It is often important in economic analysis to identify and characterize the largest orsmallest value of a function For example, we may want to know what price a monopo-list should charge to obtain the largest amount of profits or what combination of inputs

offers a producer the lowest level of average cost The extreme value of a function

within some interval is the largest or smallest value of that function within that

inter-val The largest value of a function over its entire range is called its global maximum, and the smallest value of a function over its entire range is called its global minimum The largest value within a small interval is called a local maximum The smallest value within a small interval is called a local minimum.

The maximum and minimum of a monotonic function within some closed

inter-val occurs at that interinter-val’s endpoints An illustration of this is provided by the

con-sumption function in Figure 2.2, which has as its domain the closed interval [0, 2500]

The range of this function is then the closed interval [300, 1800] The global minimum

of this strictly increasing function is the y-value of its left endpoint, 300, and its global

maximum is the y-value of its right endpoint, 1800.

A continuous function that is non-monotonic over some interval has at least one

local minimum or at least one local maximum within that interval For example, the

function presented in Figure 2.3 has a global minimum at the point The

func-tion depicted in Figure 2.8 has a local maximum at point A, a local minimum at point

B, and a global maximum at point C.

In Chapters 9 and 10 we will learn how to use calculus to identify and

character-ize extreme values of functions The applications in that chapter illustrate a number of

uses of these techniques in economic analysis

The Average Rate of Change of a Function

There are many concepts in economics that concern the extent to which one variable

changes in response to a change in another variable When these two variables are

linked by a function, as with consumption and income or quantity demanded and price,

the average rate of change can be calculated using that function

The average rate of change of a function over some interval is the ratio of the

change of the value of the dependent variable to the change in the value of the

inde-pendent variable over that interval

Average Rate of Change The average rate of change of the function y  f(x) over

the closed interval [x A , x B] is

over the closed interval [1000, 9000] of the domain It is straightforward to show that

C(1000)  900 and C(9000)  5700 Therefore the average rate of change is

Notice that this average rate of change is constant and equal to the slope of the tion In general, the average rate of change of any linear function

func-y  a  bx over any nonzero interval equals the slope of that function, b This can be shown by using the average rate of change formula over the arbitrary closed interval [x A , x B] Theaverage rate of change is equal to

The average rate of change of a nonlinear function is not constant, but insteaddepends upon the interval over which the rate of change is defined For example, con-sider the function

The average rate of change of this function over the closed interval [0, 3] is

The average rate of change of this function over the closed interval [0, 6] is

while the average rate of change of this function over the closed interval [3, 0] is

The average rate of change of a function over some interval can be depicted

using a secant line A secant line connects two points on the graph of a function with a

(x B , y B) will satisfy the equation

(2.5)

where y A  f(x A ) and y B  f(x B ) For any point (x the interval [x A , x B ], and y A , y B] Note that the term in squarebrackets in the equation is the slope of the secant line The value of the slope of thesecant line represents the average rate of change of the function over the interval

defined by the two endpoints of the secant line For example, Figure 2.9 is a graph of

the equation y 2 The secant line 0A connects points 0 and 3 on the graph, and

the slope of this line is 1 The secant line 0B connects the points 0 and 6 in this figure,

and the slope of this line is 2 The secant line C0 connects points 3 and 0, and its slope

equals 1

Concavity and Convexity

An important concept in economics is “diminishing marginal utility.” A simple

exam-ple of this is that you would get more exam-pleasure from the first cookie than from the fifth

cookie at a particular sitting A utility function that reflects this type of preference

can-not be linear, however, since the constant slope in a linear function implies that each

cookie provides the same amount of utility Instead, utility functions are typically

drawn as bowed, as in Figure 2.10(a) where utility is measured along the y-axis and

number of cookies is measured along the x-axis.

The concavity of a univariate function is reflected by the shape of its graph, and

different categories of concavity can be illustrated through the use of secant lines The

functions depicted in Figures 2.10(a) and 2.10(b) are each strictly concave in the

inter-val [x A , x B] since any secant line drawn in that interval lies wholly below the respective

function The functions depicted in Figures 2.10(c) and 2.10(d) are each strictly

con-vex in the interval [x A , x B] since any secant line drawn in that interval lies wholly

above the function These graphs illustrate that whether a function is strictly concave

or strictly convex is distinct from whether that function is strictly increasing or strictly

decreasing

5 10 15 20 25 30 35

B

A C

y = x1 2

3

x y

FIGURE 2.9 Secant Lines

A formal set of definitions for concavity and convexity are given here.

Strictly Concave The function f (x) is strictly concave in an interval if, for any two tinct points x A and x Bin that interval, and for all values of  in the open interval (0, 1),

(b) (a)

FIGURE 2.10 Strictly Concave and Strictly Convex Functions

How are these definitions linked to the descriptions using secant lines? We can

demonstrate that the two definitions are identical by using some algebra and the

equa-tion for a secant line (2.5) given previously There are three steps

1 For any given value of  within the interval (0, 1), the value of a particular

argu-ment of the function in the interval (x A , x B) is , where

The value of the function at the point x

2 Using the definition for the secant line, we find that the value of the secant line at

x

Since y A  f(x A ), if we add f (x A) to each side, we have

3 The definition of a function that is strictly concave within an interval requires

that the value of the function at any point within that interval, f(x

ater than the value of the secant line at that point, y

a strictly convex function within an interval requires that the value of the

func-tion at any point within that interval, f(x

line at that point, y

braic definitions for a strictly concave interval and a strictly convex interval are

identical to the respective definitions based on the secant line in a graph

A numerical example illustrates this definition Consider the function

over the interval [0, 4] Let so x

and

For ,

In fact, this inequality holds for any value of  between 0 and 1 for this function over the

interval [0, 4] or indeed over any interval Therefore this function is strictly concave

A related pair of definitions for concave and convex functions are given below.

Concave The function f (x) is concave in an interval if, for any two points in that interval x A and x B, and for all values of  in the open interval (0, 1),

Alternative definitions of concavity and convexity that draw on the tools of culus are offered in Chapter 7 In that chapter we show how the concavity of a function

cal-is important in a number of areas of economic analyscal-is, including consumption theoryand production theory

Necessary and Sufficient Conditions

An important concept, one that is used repeatedly throughout this book, is that of

necessary and sufficient conditions This concept enables us to understand what

logi-cal conclusions follow from certain conditions and the relationships among differentcategories of things For example, consider a function that we know to be concave.Can we conclude that the function is also strictly concave? The answer, as illustrated

by Figure 2.11(a), is “no.” If a function is strictly concave, then it is necessarily

concave The fact that a function is concave, however, is not sufficient for concluding

that it is also strictly concave

This concept of necessary and sufficient conditions can be expressed using

sym-bols The symbol ⇒ means “implies,” and therefore, the expression

means “if P then Q,” “P implies Q,” or “P only if Q.” The condition P is a sufficient

con-dition for Q since if P holds, it follows that Q also holds Also, this expression shows that

the condition Q is a necessary condition for P since P cannot hold unless Q also holds.

To illustrate this, consider the relationship between strictly concave and concave

functions The discussion above shows that for the function f,

This expression states that strict concavity is a sufficient condition for concavity

Equivalently, this expression states that concavity is a necessary condition for strict

concavity

In the case of the relationship between strictly concave and concave functions,

the implication runs in one direction only In other examples we can have the direction

of implication running in each direction, that is,

which can be written more succinctly as

This is read as “P if and only if Q,” “P is equivalent to Q,” or “P is a necessary and

suf-ficient condition for Q.” In the context of concavity, we can use the necessary and

suffi-cient condition expression to write

for x A  x Band for any value of  between 0 and 1.

An example drawn from basic microeconomics further illustrates the relationship

between necessary and sufficient conditions Consider the market for personal

comput-ers as depicted by the demand and supply diagrams in Figure 2.12 In this example

we consider only two possible exogenous variables, personal income and

computer-manufacturing productivity A decrease in personal income, which we denote as event

I, shifts the demand curve for computers to the left As depicted in Figure 2.12(a), this

causes the equilibrium to shift from point a to point b, causing the price of computers to

fall and the quantity of computers purchased to decrease Figure 2.12(b) depicts the

supply-side effects of an increase in computer-manufacturing productivity, which we

denote as event M, that causes the equilibrium to shift from point g to point h As

shown in this figure, the rightward shift of the supply curve causes the price of

comput-ers to fall but, in this case, the quantity of computcomput-ers purchased rises If we denote a fall

in the price of computers as event F, then we have

Thus either I or M is sufficient for F But I is not necessary for F since the price of puters would also fall if M occurred So we cannot know, when we observe F, if the cause was M or I.

com-If we had information on quantity as well as price, then we could distinguishbetween the two cases For example, suppose that we observe both an increase in thenumbers of computers purchased and a decrease in the price of computers, which we

label event S In the context of this simple example, the demand and supply analysis

indicates that

On the other hand, if we observe that the price decrease is accompanied by a decrease

in the quantity of computers purchased, which we denote as event D, we could

con-clude that there is a shift in demand since, in this example,

Thus evidence on price and quantity in this simple example (that is, knowing S or D instead of merely F ) enables us to distinguish between a supply-side shock (M) and a demand-side shock (I).

Exercises 2.2

1 Determine whether the following functions are monotonic, strictly monotonic, ornon-monotonic

(a)(b)(c)

2 Which of the following functions are one-to-one?

(a) a function relating countries to their citizens

(b) a function relating street addresses to zip codes

(c) a function relating library call numbers to books

(d) a function relating a student’s identification number to a course grade in a

specific class

3 Determine which of the following functions have inverses Assume their domains

are the set of real numbers Derive the inverse function, where

appli-cable For the functions that do not have inverses, determine if it is possible to

restrict the domain x in order to create a one-to-one function that has an inverse.

(a)

(b)

(c)

(d)

4 Prove that the function has an inverse Then prove that both the

orig-inal function and its inverse are inverse functions of each other by showing that

and

5 Consider the following descriptions of continuous functions with extreme points

(a) Can you draw a graph of a function that has only one extreme point that is a

local minimum but not a global minimum?

(b) Can you draw a graph of a function that has two extreme points, each of which

are local but not global extreme points?

(c) If a function has three extreme points, what are its possible number of minima

and maxima?

6 Determine the global maximum and global minimum for the function

over the domain [0, 100]

7 Sketch the function Draw secant lines and find the average rates of

change for this function over the following intervals for x.

(a) [1, 2]

(b) [1, 3]

(c)

(d)

8 Show that the average rate of change of a strictly increasing function is positive

and that the average rate of change of a strictly decreasing function is negative

9 Sketch the function over the interval [1, 5] Draw a secant line

on the function that connects the points and

(a) If x

tion given in the text

(b) What is the slope of the secant line?

(c) What can you say about this function’s concavity or convexity?

10 Sketch the function y  8  10x  x2over the domain [0, 7]