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Mathematical Methods for Economics
Mathematical Methods for Economics
Michael Klein
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Pearson New International Edition Mathematical Methods for Economics Michael Klein Second Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any afﬁliation with or endorsement of this book by such owners. ISBN 10: 1292039183 ISBN 10: 1269374508 ISBN 13: 9781292039183 ISBN 13: 9781269374507 British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library Printed in the United States of America P E A R S O N C U S T O M L I B R A R Y Table of Contents Part I. Introduction Michael Klein 1 Chapter 1. The Mathematical Framework of Economic Analysis Michael Klein 3 Chapter 2. An Introduction to Functions Michael Klein 11 Chapter 3. Exponential and Logarithmic Functions Michael Klein 45 Part II. Matrix Algebra Michael Klein 73 Chapter 4. Systems of Equations and Matrix Algebra Michael Klein 75 Chapter 5. Further Topics in Matrix Algebra Michael Klein 115 Part III. Differential Calculus Michael Klein 143 Chapter 6. An Introduction to Differential Calculus Michael Klein 145 Chapter 7. Univariate Calculus Michael Klein 173 Chapter 8. Multivariate Calculus Michael Klein 211 Part IV. Optimiz; ation Michael Klein 255 Chapter 9. Extreme Values of Univariate Functions Michael Klein 257 I II Chapter 10. Extreme Values of Multivariate Functions Michael Klein 287 Chapter 11. Constrained Optimization Michael Klein 317 Part V. Integration and Dynamic Analysis Michael Klein 361 Chapter 12. Integral Calculus Michael Klein 363 Chapter 13. Difference Equations Michael Klein 407 Chapter 14. Differential Equations Michael Klein 451 Index 489 Part One Introduction Chapter 1 The Mathematical Framework of Economic Analysis Chapter 2 An Introduction to Functions Chapter 3 Exponential and Logarithmic Functions This book begins with a threechapter 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 characteristics 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 introduced in this chapter appear again in later chapters. The final section of Chapter 2 presents 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. From Part One of Mathematical Methods for Economics, Second Edition. Michael W. Klein. Copyright © 2002 by Pearson Education, Inc. All rights reserved. 1 This page intentionally left blank Chapter 1 The Mathematical Framework of Economic Analysis hat are the sources of longrun growth and prosperity in an economy? How does your level of education affect your lifetime earnings profile? Has foreign competition from developing countries widened the gap between the rich and the poor in industrialized countries? Will economic development lead to increased environmental 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 consequences. The importance of economic issues combined with the possibility for alternative modes of economic analysis result in widespread discussion and debate. This discussion and debate takes place in numerous forums including informal conversations, 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 economic 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 economic analysis and also demonstrate the range of uses of mathematics in economics. The parallel presentation of mathematical techniques and economic applications serves several purposes. It reinforces the teaching of mathematics by providing a setting for using the techniques. Demonstrating the use of mathematics in economics helps develop mathematical comprehension as well as hone economic intuition. In this W From Chapter 1 of Mathematical Methods for Economics, Second Edition. Michael W. Klein. Copyright © 2002 by Pearson Education, Inc. All rights reserved. 3 4 Part One Introduction way, the study of mathematical methods used in economics as presented in this book complements your study in other economics courses. The economic applications in this book also help motivate the teaching of mathematics by emphasizing the practical use of mathematics in economic analysis. An effort is made to make the applications reference a wide range of topics by drawing from a cross section of disciplines within economics, including microeconomics, macroeconomics, economic growth, international 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 of economic models and the role of mathematics in economic modeling. Section 1.1 discusses the link between a model and the phenomenon it attempts to explain. This section also discusses why economic analysis typically employs a mathematical framework. Section 1.2 discusses some characteristics of models used in economics and previews 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 be highly 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 simplistic and involve nothing more complicated than the notion that economic variables follow some welldefined pattern over time. An overall evaluation of an economic analysis requires an evaluation of the framework itself, a consideration of the accuracy and relevance of the facts and assumptions used in that framework, and a test of its predictions. A framework based on a formal mathematical model has certain advantages. A mathematical model demands a logical rigor that may not be found in a less formal framework. Rigorous analysis need not be mathematical, but economic analysis lends itself to the use of mathematics because many of the underlying concepts in economics can be directly translated into a mathematical form. The concept of determining an economic equilibrium corresponds to the mathematical technique of solving systems of equations, the subject of Part Two of this book. Questions concerning how one variable responds to changes in the value of another variable, as embodied in economic concepts like price elasticity or marginal cost, can be given rigorous form through the use of differentiation, the subject of Part Three. Formal models that reflect the central concept of economics—the assumption that people strive to obtain the best possible outcome given certain constraints—can be solved using the mathematical techniques of constrained optimization. These are discussed in Part Four. Economic questions that involve consideration of the evolution of markets or economic conditions over time— questions that are important in such fields as macroeconomics, finance, and resource economics—can be addressed using the various types of mathematical techniques presented in Part Five. While logical rigor ensures that conclusions follow from assumptions, it should also be the case that the conclusions of a model are not too sensitive to its assumptions. 4 Chapter 1 The Mathematical Framework of Economic Analysis 5 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 longrun 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 “knifeedge” of stable growth or were marked by ongoing instability. Robert Solow, in a paper published in the mid1950s, 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 relationships 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 function as well as an application 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 subject 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 mathematical technique developed in the 1950s, allowed economists to resolve longstanding 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 1 Solow’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. 5 6 Part One Introduction 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, which bears 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 a classic study of the financial returns to education. It is natural to begin a book of this nature with a discussion of the many advantages of using a formal mathematical method for addressing economic issues. It is important, at the same time, to recognize possible drawbacks of this approach. Any mathematical model simplifies reality and, in so doing, may present an incomplete picture. The comparison of an economic model with a map is instructive here. A map necessarily simplifies the geography it attempts to describe. There is a tradeoff between the comprehensiveness and readability of a map. The clutter of a very comprehensive map may make it difficult to read. The simplicity of a very readable map may come at the expense of omitting important landmarks, streets, or other geographic features. In much the same way, an economic model that is too comprehensive may not be tractable, 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 this question by continuing with our analogy to maps, we recognize that the best map for one purpose is probably not the best map for another purpose. A highly schematic subway 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 particular economic model may be appropriate for addressing some issues but not others. For example, the simple savings relationship posited in many economic growth models may be fine in that context but wholly inappropriate for more detailed studies of savings behavior. The mathematical tools presented in this book will give you access to many interesting ideas in economics that are formalized through mathematical modeling. These tools are used in a wide range of economic models. While economic models may differ 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 the behavior of other variables and through the way the variables interact. The variables used 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 and exogenous variables. Consider a simple demand and supply analysis of the market for the familiar mythical good, the “widget.” The endogenous variables in this model are the price of a widget and the quantity of widgets sold. The exogenous variables in this example include the price of the input to widget production and the price of the good that consumers consider as a possible substitute for widgets. 6 Chapter 1 The Mathematical Framework of Economic Analysis 7 In this example there is an apparently straightforward separation of variables into the categories of exogenous and endogenous. This separation actually represents a central 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 variables into those that are exogenous and those that are endogenous reflects an important 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 endogenous 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 relationships 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 TR TC, where “” means “is identically equal to.” A behavioral equation represents a modeling of people’s actions based on economic principles. The demand equation and supply equation in microeconomics, as well as the investment, money demand, and consumption equations in macroeconomics, all represent behavioral equations. Sometimes these equations reflect very basic economic assumptions such as utility maximization. In other cases, behavioral equations are not derived explicitly from basic economic assumptions 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 QD P G 7 8 Part One Introduction and the supply equation as QS P N, where QD is the quantity of widgets demanded, QS is the quantity of widgets supplied, P is the price of widgets, G is the price of goods that are potential substitutes for widgets, 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 a 1 specific value like 100, 2, 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 QD QS. A simultaneous solution of the demand equation, supply equation, and equilibrium 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 exogenous variables 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 behavioral equations are each linear functions since they take the form y a bx cz, 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 determine 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 bought and 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 restrictive, the discussion of logarithmic functions and exponential functions in Chapter 3 shows that certain nonlinear functions can be expressed in linear form. Also material presented in Chapter 7 shows how to obtain a linear approximation of a nonlinear function. The determination of the solution to this simple linear model may be only the beginning of a deeper economic analysis of the widget market. Such an analysis may 2 8 We return to this model in Chapter 4 where we show how to solve it. Chapter 1 The Mathematical Framework of Economic Analysis 9 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 T . (Q . P), where is the tax rate and (Q . P) 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 differential calculus, which is the subject of Part Three (Chapters 6 through 8). Differential calculus offers a set of tools for analyzing the responsiveness of the dependent variable of a function to changes in the value of one or more of its independent variables. 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 introduction 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 techniques presented in this chapter enable you to address the question of the responsiveness 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 identification of extreme values, that is, the largest or smallest value of a function. 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 markets. Chapter 11 shows how to determine the extreme value of functions when their independent variables are constrained by certain conditions. This technique of constrained optimization explicitly captures the core economic concept of obtaining the best outcome in the face of tradeoffs 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 common 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 payments 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 9 10 Part One Introduction 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 common themes arise in the discussion of difference equations and of differential equations. Chapter 15 concludes this section with a presentation of dynamic optimization, a technique for solving for the optimal time path of variables. Dynamic optimization would enable us to analyze questions like the optimal investment strategy over time for 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 text rather than just to read it passively. When reading this book, keep a pencil and paper at hand, and replicate the chains of reasoning presented in the text. The problems presented at the end of each chapter section are an integral part of this book, and working through 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 the other fields of economics that you are studying. An ability to do this demonstrates a mastery of the material presented here. 10 Chapter 2 An Introduction to Functions unctions 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 logical concept of necessary and sufficient conditions. The final section of this chapter introduces some general forms of functions used extensively in economics. F 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 From Chapter 2 of Mathematical Methods for Economics, Second Edition. Michael W. Klein. Copyright © 2002 by Pearson Education, Inc. All rights reserved. 11 12 Part One Introduction 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 quantitative variables include the dollar value of national income, the number of barrels of imported oil, the consumer price level, and the dollaryen exchange rate. Some quantitative variables, like population, may be expressed as an integer.An integer is a whole number 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. 1 2 Some real numbers can be expressed as ratios of integers, for example, 2, 2.5, or 3 5 . These numbers are called rational numbers. Other real numbers, such as 3.1415. . . and 2, 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 numbers and is written [0, 1.5]. An open interval between any two numbers excludes the endpoints. The open interval between 7 and 10 is written (7, 10). A halfclosed interval or a halfopen interval includes one endpoint but not the other. Notation for halfclosed or halfopen intervals follows from the notation for closed and open intervals. 3 For example, if an interval includes the endpoint 2 but not the endpoint 1, it is writ3 ten as [ 2 , 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, ). The open interval of all positive numbers is written as (0, ). The interval of all real numbers 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,” a set consisting of 46 elements, and “economists who would have liked to have won the Nobel Prize by 2001,” a set with a membership that probably numbers in the thousands. Sets are represented by capital letters. To show that an item is an element of a set, we use the symbol . For example, if we denote the set of all Nobel Prize–winning economists by N, then Paul Samuelson N Milton Friedman N. To show that elements are not members of a set, we use the symbol . For example, Adam Smith N. 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 elements 1 can be described similarly. For example, consider the set of all integers between 2 and 1 5 2. We can describe this set by simply listing its five elements S {1, 2, 3, 4, 5} 12 Chapter 2 An Introduction to Functions 13 Alternatively, we can describe the set by describing the conditions for membership S x x is an integer greater than 1 1 and less than 5 . 2 2 This statement is read as “S is the set of all numbers x such that x is an integer greater 1 1 than 2 and less than 5 2. ” Sets that have an infinite number of elements can be described by stating the condition for membership. For example, the set of all real 1 1 numbers x in the closed interval [ 2, 5 2] can be written as S x 1 1 x5 . 2 2 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 f : X Y. 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: d : N 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 example, 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 univariate function that maps any one element x of the set X to one and only one element y of the set Y is y f (x), 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. 13 14 Part One Introduction The Set of Nobel Laureates in Economics (N) Ragnar Frisch Paul Samuelson Simon Kuznets Kenneth Arrow Wassily Leontief Gunnar Myrdal Tjalling Koopmans Milton Friedman Bertil Ohlin Herbert Simon Theodore Schultz Lawrence Klein James Tobin George Stigler Gerard Debreu Richard Stone Franco Modigliani James Buchanan Robert Solow Maurice Allais Trygve Haavelmo Harry Markowitz Ronald Coase Gary Becker Robert Fogel John Harsanyi Robert Lucas William Vickrey Robert Merton Amartya Sen Robert Mundell James Heckman Jan Tinbergen John Hicks Friedrich von Hayek Leonid Kantorovich James Meade Arthur Lewis William Sharpe Merton Miller Douglass North John Nash Reinhard Selten James Mirrlees Myron Scholes Daniel McFadden The Set of Years in Which the Nobel Prize was Awarded (T ) 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 FIGURE 2.1 The Sets N and T The term f (x) can represent any relationship that assigns a unique value to y for any value of x, such as 1 2 x or 2 y x. y The numbers 12 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 function for different values of its argument. For example, consider a basic Keynesian consumption function that relates consumption, C, to income, I, as C 300 0.6I , 14 (2.1) Chapter 2 TABLE 2.1 I C 0 300 An Introduction to Functions 15 A Consumption Function 1000 900 2500 1800 5000 3300 9000 5700 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 consumption function. 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 consumption function presented previously are (1000, 900), (2500, 1800) and (5000, 3300). Ordered pairs can be plotted in a Cartesian plane (named after the seventeenthcentury 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 horizontal axis, also called the xaxis, 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 yaxis, 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 xcoordinate of the pair (x, y) is called the abscissa, and the ycoordinate 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. C B 1800 900 (2500,1800) (1000,900) 300 A 0 1000 2500 I FIGURE 2.2 A Consumption Function 15 16 Part One Introduction The graph of a function represents all points whose coordinates are ordered pairs of the function. The graph of the consumption function for the domain [0, 2500] is represented by the line AB in Figure 2.2. This graph goes through the first three points identified in Table 2.1 as well as all other points consistent with the consumption function over 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 y f(x) a bx. (2.2) The parameter a is the intercept of the function and represents the value of the function when its argument equals zero. In a graph, the intercept is the point where the function crosses the yaxis. 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 given change in its argument. The slope of the linear function (2.2) evaluated between any two points xA and xB (for xA xB) is (a bxB) (a bxA) f (xB) f (xA) b, xB xA xB xA where f(x)B f(xA) is the change in the value of the function associated with the change in its argument xB xA. 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 the range of the consumption function are restricted to include only positive numbers. Many economic functions include both positive and negative numbers as arguments and values. Graphs of these functions can be represented with other quadrants of the Cartesian plane. In Figure 2.3 the function y 4 2x 2x 2 is presented. You can verify that this function includes the four ordered pairs 1 1 1 1 (2, 8), (2, 2 2), ( 2, 4 2), and (3, 8). Each of these ordered pairs is in a different quadrant of the Cartesian plane, which indicates that the graph of this function passes through 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 variables x1, x2, and x3 is y f (x1, x2, x3). Strictly 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. 1 16 Chapter 2 An Introduction to Functions 17 yaxis (–2,8) (3,8) 7 5 3 y = – 4 – 2x + 2x2 1 –5 –3 –1 1 3 5 7 8 xaxis –1 1 1 – , –2 2 2 –3 –5 1 1 , –4 2 2 FIGURE 2.3 Graph of Function Filling Four Quadrants 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 y f (x1, x2, . . . , xn). A multivariate function with two independent variables is called a bivariate function. Some specific bivariate functions include j 5 4k 3 7h Q K L . and 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 consumption function C 300 0.6I 0.02W, (2.3) where W represents wealth and all variables are expressed in billions of dollars, generates 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 17 18 Part One Introduction zax s C (Consumption) (6000, 0, 3900) (0, 0, 300) (6000, 60000, 5100) xaxis I (Income) (0, 60000, 1500) yaxis W (Wealth) FIGURE 2.4 A Multivariate Consumption Function dimensions when depicting a function of the form z f(x, y) by drawing the xaxis as a horizontal line to the right of the origin, the yaxis as a line sloping down and to the left from the origin, and the zaxis 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 yaxes 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 IW surface. Limits and Continuity It is often necessary in economics to evaluate a function as its argument approaches some value. For example, in the next chapter we will learn how to find the value today of an infinitelylong stream of future payments. In the dynamic analysis presented in Part Five of this book, we solve for the longrun level of a variable. In these cases the argument of the function is time, and we evaluate the value of the function as time approaches 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 correspondence between this mathematical technique and the economic concept of evaluating the effect “at the margin.” In this section we show how to evaluate a function as its 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 lefthand limit, or from larger values of a, giving the righthand limit. LeftHand Limit The lefthand limit of a function f(x) as its argument approaches some number a, written as lim f (x), x→ a 18 Chapter 2 An Introduction to Functions 19 exists and is equal to LL if, for any arbitrarily small number , there exists a small number such that f (x) L L ■ whenever a x a. RightHand Limit The righthand limit of a function f (x) as its argument approaches some number a, written as lim f (x), x→a exists and is equal to LR if for any arbitrarily small number , there exists a small number such that f (x) L R whenever a x a . When the lefthand limit equals the righthand limit, we can simplify the notation by suppressing the superscripts and defining lim f (x) lim f (x) lim f (x) . x→a x→a x→a 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, lim f (x) x→a if, for every N 0, there is a 0 so that f (x) N whenever a x a . Also, we have lim f (x) x→a if, for every N 0, there is a 0 so that f (x) N whenever a x a . Evaluating the limits used in this book involves the following two simple rules. 19 20 Part One Introduction Rules for Evaluating Limits The following two rules are used in evaluating limits: lim m(k x) lim m(k x) mk x→ 0 x→ 0 and k lim (m . x) h 0, x→ where k, m, and h are arbitrary real numbers and m 0. Two applications of these rules are shown below: lim n→ 0 6n 4n2 lim (3 2n) 3 n→ 0 2n and 1 lim t 3 5 5. t→ The limits in these two examples are finite. The following are examples of limits that are infinite: 5 n t 7 10 . lim n→ 5 1 and lim 2 t→ 7 One use of limits in the context of the material presented in this book is to determine 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 righthand limits at x a exist and are equal, lim f (x) lim f (x) lim f (x), x→a x→ a x→ a and the limit as x → a equals the value of the function at that point, lim f (x) f (a). x→a ■ Figure 2.5(a) presents a function that is not continuous at x x0 since, at that value, there is a “hole” in the function and the limit there does not equal the value of the function at x0. When both a lefthand limit and a righthand limit exist, the first part of the definition requires that each approach the same value for the function to be continuous. Figure 2.5(b) presents a total cost curve that is not continuous at 20 Chapter 2 An Introduction to Functions Total cost f (x) x0 21 f (x) x Quantity q0 (a) (b) 3 x (c) FIGURE 2.5 Functions That Are Not Continuous q q0 since the purchase of an additional piece of capital, like a new factory or new equipment, which requires a large onetime cost, is required to increase output above q0. The second part of the definition shows that even if the lefthand limit and the righthand limit of a function exist and are equal at a, it is also necessary for the function to be defined at a for the function to be continuous. This requirement is made clear by considering the function f(x) [1 (x 3)]2 + 5 as x approaches 3. The lefthand limit and the righthand limit are the same since lim x→3 x 3 1 2 5 lim x→3 x 3 1 2 5 . However, this function is not defined at x 3 since the term 10 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 lefthand limit or a righthand limit approaches positive infinity or negative infinity at that point. A function is discontinuous at a point where there is a vertical asymptote. Exercises 2.1 1. Use the notation for intervals to represent each of the following specified intervals of x (a) 5 x 0 (b) 5 x 0 (c) x 100 (d) 100 x (e) x is a positive number greater than zero (f) x is a real number 2. Determine which of the following relationships represent functions. Assume that the interval is the set of real numbers unless otherwise indicated. 21 22 Part One y 5x yx y a x; (0, ) y x2 y2 x 1 (f) y ; (0, ) x3 (g) y2 x4; (0, ) In the following problems, set X represents the domain and Y is the set of values that occur with a potential function, f . Confirm whether the function f can be defined according to the mapping f : X → Y. (a) Set X consists of all the alumni of Anycollege University; set Y is each alumnus’ alma mater. (b) Set X consists of the workers at Busy Firm; set Y is each worker’s social security number. (c) Set X consists of all the people who have shared the prize for BestDressed Celebrity in any given year; set Y consists of the years in which the BestDressed Celebrity prize was shared. (d) Set X is a set of fathers; set Y is the set of their sons. Consider again the functional mapping d : N → T where N is the set of Nobel Prize winners and T is the set of years in which the prizes were won. If an economist wins the prize for a second time, would this still be a valid function? Explain. 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 function is C 75 5Q. (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 produce more than 50 units of output a day. What are the domain and range of the cost function in this scenario? Identify and graph four ordered pairs for each of the following functions. Sketch a graph of each of the functions. (a) y 100 20x over the interval [2, 6] (b) y x x3 over the interval (5, 5) (c) y x2 1 over the interval [100, 100] Evaluate the following limits. 1 (a) y lim 2 x→ x 7 (b) y lim 1 2 x→7 x 7 (a) (b) (c) (d) (e) 3. 4. 5. 6. 7. 22 Introduction Chapter 2 An Introduction to Functions 23 (c) y lim x 1 7 x→1 (d) y lim 1 2x x x→1 1x (Hint: Transform the ratio to remove x from the denominator.) 8. Which functions are continuous over the given intervals? 1 ; (0, ) (a) y 8 x7 4 ax ; (, ) (b) y 7 1 ; (0, ) (c) y 3 x7 (d) y 2x 4 ; [3, 3] 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 2.2 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 generalizing the definitions to multivariate functions. Later chapters also stress the economic interpretation of these properties. Increasing Functions and Decreasing Functions The graph of the consumption function in Figure 2.2 shows that consumption consistently rises as income rises. The value of other functions used in economics may consistently 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, xA and xB, where xB xA. A function is increasing if f(xB) f(xA). A function is strictly increasing if f(xB) f(xA). A function is decreasing if f(xB) f(xA). A function is strictly decreasing if f(xB) f(xA). 23 24 Part One Introduction f (x) f (x) x 0 x 0 Increasing (not strictly) Strictly Increasing (a) (b) f (x) f (x) x 0 x 0 Decreasing (not strictly) Strictly Decreasing (c) (d) FIGURE 2.6 Increasing Functions and Decreasing Functions These definitions show that any strictly increasing function is also an increasing function, and any strictly decreasing function is also a decreasing function. An increasing function, however, may not be a strictly increasing function since an increasing function may have a section where f (xB) f (xA). This is illustrated in Figure 2.6. The increasing function in Figure 2.6(a) has a horizontal section, which precludes it from being a strictly increasing function. Figure 2.6(b) is strictly increasing. Figures 2.6(c) and 2.6(d) present graphs of a decreasing and a strictly decreasing function, respectively. Closely related to these definitions of increasing and decreasing functions are the definitions of a monotonic function, a strictly monotonic function, and a nonmonotonic 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. 24 Chapter 2 An Introduction to Functions 25 nonmonotonic if it is strictly increasing over some interval and strictly decreasing over another interval. Nonmonotonic functions can have the same value for more than one argument. For example, the value of the nonmonotonic function y x2 is 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 onetoone functions and have the following property. OnetoOne Function argument x1 and x2, A function f(x) is onetoone if for any two values of the f (x1) f (x2) implies x1 x2. Any onetoone function has an inverse function. The inverse of the function y f(x) is written as y f 1(x).2 We 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 f 1(x). For example, the inverse of the onetoone function y f (x) 4 2x (2.4) can be found by solving this for x in terms of y to get x 2 1 y 2 and then interchanging x and y to obtain the inverse function y f 1(x) 2 1 x. 2 An important property of a function f(x) and its inverse f 1(x) is f ( f 1(x)) x and f 1( f (x)) x. The term f ( f 1(x)) represents a composite function. The argument of a composite function is itself a function. For example, y g(h(x)) 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 f ( f 1(x)) is f(•), and the inside function is its inverse. For example, using the function (2.4), we have f ( f 1(x)) 4 2 2 1 x x 2 It is important to note that f 1(x) does not mean 1 f (x). 2 25 26 Part One Introduction f (x), f –1(x) y = f –1(x) 45 line y = f (x) 45 x FIGURE 2.7 A Function and Its Inverse and f 1( f (x)) 2 1 (4 2x) x. 2 There is a simple method for graphing the inverse of a function, y f 1(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 onetoone 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 f 1(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 f 1(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 original function become the coordinates (b, a) of the inverse function. This property of inverse functions will be important in our study of logarithmic and exponential functions in Chapter 3. Extreme Values It is often important in economic analysis to identify and characterize the largest or smallest value of a function. For example, we may want to know what price a monopolist 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 interval. 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. 26 Chapter 2 An Introduction to Functions 27 C A B FIGURE 2.8 A Function with Extreme Values The maximum and minimum of a monotonic function within some closed interval occurs at that interval’s endpoints. An illustration of this is provided by the consumption 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 yvalue of its left endpoint, 300, and its global maximum is the yvalue of its right endpoint, 1800. A continuous function that is nonmonotonic 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 (12, 4 12). The function 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 characterize 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 independent variable over that interval. Average Rate of Change The average rate of change of the function y f(x) over the closed interval [xA, xB] is y f (xB) f (xA) . x xB xA We can use this formula to calculate the average rate of change of the univariate consumption function (2.1), C C(I) 300 0.6I, 27 28 Part One Introduction 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 C C(9000) C(1000) 4800 0.6. I 9000 1000 8000 Notice that this average rate of change is constant and equal to the slope of the function. In general, the average rate of change of any linear function 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 [xA, xB]. The average rate of change is equal to (a bxB) (a bxA) y bxB bxA b(xB xA) b. x xB xA xB xA xB xA The average rate of change of a nonlinear function is not constant, but instead depends upon the interval over which the rate of change is defined. For example, consider the function y 1 2 x . 3 The average rate of change of this function over the closed interval [0, 3] is (1 3)(32) (1 3)(02) y 3 1. x 30 3 The average rate of change of this function over the closed interval [0, 6] is (1 3)(62) (1 3)(02) y 12 0 2, x 60 6 while the average rate of change of this function over the closed interval [3, 0] is (1 3)(0 2) (1 3)(32) y 03 1. x 0 (3) 3 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 straight line. Any point (x , y ) on the secant line connecting the two points (xA, yA) and (xB, yB) will satisfy the equation ( y yA) f (xB) f (xA) (x xA), xB xA (2.5) where yA f(xA) and yB f(xB). For any point (x , y ) on the line, x is within the interval [xA, xB], and y is within the interval [yA, yB]. Note that the term in square brackets in the equation is the slope of the secant line. The value of the slope of the secant line represents the average rate of change of the function over the interval 28 Chapter 2 An Introduction to Functions 29 y 35 30 y= 25 1 2 x 3 20 15 B 10 5 C –10 –8 –6 –4 A –2 0 2 4 6 8 10 x FIGURE 2.9 Secant Lines defined by the two endpoints of the secant line. For example, Figure 2.9 is a graph of the equation y (1 3)x2. 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 example of this is that you would get more pleasure from the first cookie than from the fifth cookie at a particular sitting. A utility function that reflects this type of preference cannot 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 yaxis and number of cookies is measured along the xaxis. 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 interval [xA, xB] 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 convex in the interval [xA, xB] 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. 29 30 Part One Introduction y y secant line xA xB x xA Strictly Concave Strictly Concave (a) (b) y x xB y xA xB x xA xB Strictly Convex Strictly Convex (c) (d) FIGURE 2.10 x Strictly Concave and Strictly Convex Functions 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 distinct points x A and x B in that interval, and for all values of in the open interval (0, 1), f (x A (1 )x B) f (x A) (1 )f (x B). Strictly Convex The function f(x) is strictly convex in an interval if, for any two distinct points x A and x B in that interval, and for all values of in the open interval (0, 1), f (x A (1 )x B) f (x A) (1 )f (x B). 30 Chapter 2 An Introduction to Functions 31 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 equation 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 argument of the function in the interval (xA, xB) is x xA (1 )xB, where xA x xB. The value of the function at the point x equals f (x ) f (xA (1 )xB ). 2. Using the definition for the secant line, we find that the value of the secant line at x , which we call y , can be found with (2.5). Using the definition for x , we have ( y y A) f (xB) f (xA) ([xA (1 )xB ] xA) xB xA ™ ( ™ ¤ ) ¤( f xB) f (xA xB xA 1 )(xB xA) (1 )[ f (xB) f (xA)]. Since yA f(xA), if we add f(xA) to each side, we have y f (xA) (1 )f (xB). 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 ), is greater than the value of the secant line at that point, y . Likewise, the definition for a strictly convex function within an interval requires that the value of the function at any point within that interval, f(x ), is less than the value of the secant line at that point, y . Examining the solution for y demonstrates that the algebraic 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 y 20x 2x 2 1 over the interval [0, 4]. Let 2, so x 2. In this case, f (x A) (1 )f (x B) 1 1 0 48 24 2 2 and f (x A (1 )x B) f (2) 32. 1 For 2 , f (x A (1 )x B) f (x A) (1 )f (x B). 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. 31 32 Part One Introduction y y x 0 x 0 Concave Convex (a) (b) FIGURE 2.11 Concave and Convex Functions 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 xA and xB , and for all values of in the open interval (0, 1), f (xA (1 )xB) f (xA) (1 )f (xB). Convex The function f(x) is convex in an interval if, for any two points in that interval xA and xB , and for all values of in the open interval (0, 1), f (xA (1 )xB) f (xA) (1 )f (xB). A function can have a linear segment within an interval and still be concave or convex, as shown in Figures 2.11(a) and 2.11(b), respectively. The functions depicted in these figures fail the requirement of being strictly concave or strictly convex since along the linear segments of the functions f (xA (1 )xB) f (xA) (1 )f (xB). Alternative definitions of concavity and convexity that draw on the tools of calculus are offered in Chapter 7. In that chapter we show how the concavity of a function is important in a number of areas of economic analysis, including consumption theory and 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 logical conclusions follow from certain conditions and the relationships among different categories 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 32 Chapter 2 An Introduction to Functions 33 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 symbols. The symbol ⇒ means “implies,” and therefore, the expression P⇒Q means “if P then Q,” “P implies Q,” or “P only if Q.” The condition P is a sufficient condition 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, f strictly concave ⇒ f concave. 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, P ⇒ Q and P ⇐ Q, which can be written more succinctly as P ⇔ Q. This is read as “P if and only if Q,” “P is equivalent to Q,” or “P is a necessary and sufficient condition for Q.” In the context of concavity, we can use the necessary and sufficient condition expression to write f is concave ⇔ f (x A (1 )x B) f (x A) (1 )f (x B) for xA xB and 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 computers 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 computermanufacturing 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 supplyside effects of an increase in computermanufacturing 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 computers to fall but, in this case, the quantity of computers purchased rises. If we denote a fall in the price of computers as event F, then we have I⇒F and M ⇒ F. 33 34 Part One Introduction S p1 g a p0 p0 b S' h p1 D D D' Q1 S Q0 Q0 Q1 Demand Shift Supply Shift (a) (b) FIGURE 2.12 Demand and Supply for PCs Thus either I or M is sufficient for F. But I is not necessary for F since the price of computers would also fall if M occurred. So we cannot know, when we observe F, if the cause was M or I. If we had information on quantity as well as price, then we could distinguish between the two cases. For example, suppose that we observe both an increase in the numbers 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 M ⇔ S. 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 conclude that there is a shift in demand since, in this example, I ⇔ D. 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 supplyside shock (M) and a demandside shock (I). Exercises 2.2 1. Determine whether the following functions are monotonic, strictly monotonic, or nonmonotonic. (a) y a bx; b 0 (b) y ax 2; a 0 (c) y ax bx 3; a 0, b 0 (d) y a 34 Chapter 2 An Introduction to Functions 35 2. Which of the following functions are onetoone? (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, x f 1( y), where applicable. For the functions that do not have inverses, determine if it is possible to restrict the domain x in order to create a onetoone function that has an inverse. (a) y 14 7x (b) y x 2 6 (c) y x (d) y x 3 4. Prove that the function y 10x 5 has an inverse. Then prove that both the original function and its inverse are inverse functions of each other by showing that f ( f 1( y)) y and f 1( f (x)) x. 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? 1 6. Determine the global maximum and global minimum for the function y 2x 50 over the domain [0, 100]. 7. Sketch the function y 4x 2. 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) [2, 1] (d) [2, 2] 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 y x 2 8x 16 over the interval [1, 5]. Draw a secant line on the function that connects the points xA 1 and xB 3. (a) If x 2 is a point on the secant line, determine the value of y using the equation 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 x 2 over the domain [0, 7]. 35 36 Part One Introduction (a) Assume that xA 1, xB 4, and 0.4. Using the formula x xA (1 )xB, determine the value of x . What is the value of f(x )? (b) Calculate the value of y , which is the value of the secant line at x , using the formula y f (xA) (1 )f (xB). (c) Prove that the above function is strictly concave by demonstrating that f (x ) y . 11. For each of the following sets A and B, state whether A ⇒ B, A ⇐ B, or A ⇔ B. Do any of these sets satisfy the necessary and sufficient condition? (a) A even numbered years, B years in which the Summer Olympic Games are held (b) A a function is monotonic, B a function is increasing (c) A a function is linear, B a function is monotonic (d) A a function is onetoone, B a function has an inverse function 2.3 A MENU OF FUNCTIONS Economists use a variety of types of functions. The choice of the function appropriate for a particular model depends upon two factors. First, the mathematical properties of the function should be able to capture the salient economic characteristics of the activity studied. Second, the function should be as simple as possible to address the question at hand. Thus while the simplicity of a linear function makes it an attractive choice for many applications, in other cases it may prove to be too restrictive. For example, a linear function cannot reflect diminishing marginal utility since it is not strictly concave. In this section we introduce three categories of functions. Some properties of each category are discussed. These functions are used and analyzed more extensively in later chapters. Power Functions A power function takes the general form f (x) kx p, (2.6) where k and p are any constants. The parameter p is the exponent of the function. The use of power functions requires a knowledge of the rules of exponents. Some important rules of exponents and numerical examples of these rules are presented in Table 2.2. Figures 2.13 to 2.15 present the graphs of the power functions that take the form of (2.6). These graphs vary according to the values of the parameters k and p. Figure 2.13 presents three graphs in which p is a positive, even integer: f (x) 2x 2, f (x) 2x 2, and f (x) 2x 4. These graphs illustrate some particular examples of more general points for any power function of the form of (2.6) when p is a positive even integer. • f (0) 0. • If k 0, then f (x) reaches a global minimum at x 0. If k 0, then f (x) reaches a global maximum at x 0. 36 Chapter 2 TABLE 2.2 37 Rules of Exponents Rule Numerical Example x 1 x1 x 4 1 41 4 0 x p An Introduction to Functions 0 1 xp 1 1 42 16 4 3 2 64 8 4 24 1 (4 . 4) . 4 4 3 64 43 4.4.4 1 42 4 . 4 4 4 (4 2) 3 (4 . 4)(4 . 4)(4 . 4) 4 6 4096 4 23 2 (4 . 4)(3 . 3) 12 2 144 4 3 (4 . 4 . 4) 4 3 23 8 3 . . ( ) 2 2 2 2 2 4 2 xm n nx m xaxb xab xa xab xb (x a) b x ab x ay a (xy) a xa x a ya y • These functions are symmetric about the vertical axis. They are strictly convex if k 0 or strictly concave if k 0. The graphs of the power functions f (x) x, f (x) x 3, and f (x) x 3 are presented in Figure 2.14. This figure illustrates some particular examples of more general points for any power function of the form of (2.6) when p is a positive odd integer. • If k 0, then the function is monotonic and increasing. If k 0, then the function is monotonic and decreasing. f (x) y = 2x 2 4 3 2 1 –4 –3 –2 –1 y = 2x 4 1 2 3 4 x –1 –2 –3 y = – 2x 2 –4 FIGURE 2.13 Power Functions in Which the Exponent Is a Positive Even Number 37 38 Part One Introduction f (x) 4 y = x3 y=x 3 2 1 –4 –3 –2 –1 1 2 3 4 x –1 –2 –3 y = – x3 –4 FIGURE 2.14 Power Functions in Which the Exponent Is a Positive Odd Integer f (x) f(x) = x –3 2 1 f(x) = x –2 –2 f (x) = x –2 –1 f (x) = x –3 1 2 x –1 –2 FIGURE 2.15 38 Power Functions in Which the Exponent Is a Negative Integer Chapter 2 An Introduction to Functions 39 • If p 1 and k 0, the function is strictly concave for x 0 and strictly convex for x 0. If p 1 and k 0, the function is strictly convex for x 0 and strictly concave for x 0. Figure 2.15 presents the graphs of the power functions f(x) x2 and f(x) x3. This figure illustrates some particular examples of more general points for any power function of the form of (2.6) when p is a negative integer. • The function is nonmonotonic. • The function is not continuous and has a vertical asymptote at x 0. • If k 0 and p is a negative even integer, then lim kxp and x→0 lim kxp . x→0 • If k 0 and p is a negative even integer, then lim kxp x→0 • If k and lim kxp . x→0 0 and p is a negative odd integer, then lim kxp and x→0 lim kxp . x→0 • If k 0 and p is a negative odd integer, then lim kxp and x→0 lim kxp . x→0 • If k 0 and p is a negative even integer, then kx p is strictly convex for x 0 or for x 0. • If k 0 and p is a negative odd integer, then kx p is strictly convex for x 0 and strictly concave for x 0. Polynomial Functions A univariate polynomial function takes the form y f (x) a0 a1x a2x 2 . . . anx n, (2.7) where the parameters ai, i 0, 1, 2, . . . , n, are real numbers (which may include zero) and the exponents in the polynomial are integers from 1 to n. The degree of the polynomial is the value taken by the highest exponent. The polynomial presented here is an nthdegree polynomial or a polynomial of degree n. A linear function like (2.2) presented earlier is a polynomial of degree 1. A polynomial of degree 2 is called a quadratic function. A polynomial of degree 3 is called a cubic function. The roots of a polynomial are the values of its argument that make the function equal zero. The linear function y a bx 39 40 Part One Introduction has the single root x a b since, at this value of x, y a b(a b) 0. There are, at most, two distinct roots of any quadratic function. These roots are given by the quadratic formula. The roots of a quadratic equation, which are represented by the two values of the argument of the function x1 and x2 , satisfy the relationship 0 ax 2 bx c. To derive the quadratic formula, we first divide each side of this equation by a to get 0 x2 b c x . a a Adding b2 4a2 c a to both sides of this equation gives us b2 c b b2 x2 x . 2 4a a a 4a 2 The right side of this equation is a perfect square; that is, x2 b b2 b x 2 x a 4a 2a 2 . Therefore the roots of a quadratic function satisfy the relationship x 2a b 2 b2 c . 4a2 a We isolate x by taking the square root of each side, multiplying the second term on the right by 4a 4a, subtracting b 2a from each side, and collecting terms to get x1, x2 b b2 4ac , 2a where the symbol means that one root is found by adding the term in the square root sign and the other is found by subtracting that term. For example, the roots of the quadratic equation y f (x) 2x 2 2x 4 represented by the graph in Figure 2.3 are 2 4 (4)(2)(4) 2(2) 1, 2. x1, x2 The graph of this function, therefore, includes the two points (1, 0) and (2, 0). It is straightforward to verify that f (1) f (2) 0. 40 Chapter 2 An Introduction to Functions 41 There are three different possibilities for the roots of a quadratic equation. When b2 4ac 0, then there are two distinct roots. When b 2 4ac 0, then there are two equal roots (also called one multiple root). When b 2 4ac 0, then there are two complex roots.3 In general, a polynomial of degree n has, at most, n distinct roots. There are no simple formulas for finding the roots of cubic or higher degree polynomial functions. Also, the degree of a polynomial function minus one indicates the largest number of “bends” its graph can have and, therefore, the largest number of local minima and maxima that function can have. A linear function has no bends, a quadratic function has one, a cubic function has at most two, and so on. We return to this point in Chapters 6 and 7. Exponential Functions The argument of an exponential function appears as an exponent. The general form of the univariate exponential function is y f (x) kb x, (2.8) where k is a constant and b, called the base, is a positive number. The rules given in Table 2.2 show that f (0) kb 0 k for any base. The sign of the value of this function is the same as the sign of the parameter k for any value of x. When b 1, then bx monotonically increases with x. In this case, lim kbx 0. x → Using the rules given above, we see that 1 1 bx b x b x. Thus the graph of the function (1 b)x is a reflection of the function bx across the yaxis. When 1 b 0, kb x monotonically decreases with x and lim kbx 0. x→ Figure 2.16 presents four different exponential functions. Function A is y 2x. Function B, y (1 2)x, is the reflection of Function A across the yaxis. Function C, y 3(2x), looks much like Function A but lies everywhere above it. Function D, y 3x, has the same intercept as Function A but increases more quickly for positive values of x and decreases more quickly for negative values of x. A complex root is a root is a complex number. A complex number takes the form a b1, where a is the real part of the number and b1 is the imaginary part of the number. 3 41 42 Part One Introduction y Function D y = 3 x 30 25 Function C y = 3(2 x ) 20 15 Function A y = 2 x 10 Function B y = 1– 2 –2 –1 x 5 0 FIGURE 2.16 1 2 3 x Some Exponential Functions Exponential functions are used extensively in economics. The study of these functions and their inverses merits its own chapter. We turn to this study in Chapter 3. Exercises 2.3 1. Evaluate the following. (a) x2x3 (b) x 4y 4 (c) (x 4) 5 (d) x 3y 2 x6 (e) 6 y 2. Condense the following expressions. (a) x a . x b . x c x d (b) x 1 2x 3 2 x1 3 (c) ((x1 3)8)1 2 . x2 x3 4 (d) x 2y 3 . x 3 xy 2 . x 2y 3. Calculate the following and simplify. (a) 16 3 4 (b) 4 3 2 . 2 5 . 16 (c) 2 1 2 . 2 5 2 2 4 x 1 . 2x 2 7x 3 2x 1 (d) x 2 x 2 2x 1 x1 4 {( ) ( ) ( (e) x 1 2x 1 x 1) 4 (2 x)} 5 42 Chapter 2 An Introduction to Functions 43 4. In which quadrant or quadrants does the graph of the power function y xa appear when the domain of x is (, ) and (a) a is a positive even number. (b) a is a positive odd number. (c) a is a negative even number. (d) a is a negative odd number. 5. Find the roots of the following equations. (a) y 6 5x (b) y 6 5x x 2 (c) y 9 6x x 2 (d) y x 3 2x 2 x 2 (Hint: One of the roots is 1.) 6. Find the roots of the function y px 2 2qx r using the method of completing the square, as presented in the text. 7. Graph the functions y 2x2 and y 2x over the interval (0, 1]. How do the values of these functions compare over this interval? What is the yintercept of each function? 8. Using the functional form and your graph from question 7, now assume that xA 1 and xB 4, and plot the functions on the graph. Calculate the average rate of change for each function. Using the formal definitions provided in Section 2.2, determine whether these functions are concave or convex. Consider the specific case of 0.3. 9. Plot on your graph from question 8 the graph of the exponential functions y (1 2)2x and y (1 2)4x over the domain [0, 4]. Do the two curves share a yintercept? Why? Now assume for the first function we have y (1 4)2x. How does this affect the curvature of the graph? 10. How does the graph of the exponential functions from question 9 change when the domain of x is restricted to (3, 0] for both functions? 11. In applying mathematical methods to economic problems, economists utilize a broad menu of functions for the purpose of accurately specifying the properties of the economic variables in question. Choose the appropriate functional type from the following list of functions for that most accurately specifies the properties of each economic relationship described. (Hint: Try sketching a graph of the functions below and see which is appropriate for the relationship described.) i. y ax b; (a 0, b 0) ii. y ax 2 bx c; (a 0, b 0, c 0) iii. y loga (x 1); (a 1) iv. y ax 3 bx 2 cx; (a 0, b 0, c 0) v. y A . xa; (a 0) (a) A production function of one output and one input that exhibits increasing output with an increase in the input but diminishing marginal returns. That is, the rate of change becomes smaller when the input is increased. (Hint: This function is concave and monotonic.) 43 44 Part One Introduction (b) An indifference curve with two goods that exhibits a diminishing marginal rate of substitution (MRS). This means that the more you have of a good, the more you are willing to trade away for the same amount of another good. (Hint: This standard indifference curve is convex and monotonic.) (c) An indifference curve with two goods that are substitutes and that exhibits a constant rate of substitution. (d) A marginal cost function of one output that exhibits diminishing marginal cost when output is small but increasing marginal cost when output gets larger. (Hint: This function is convex but nonmonotonic.) (e) A production function of one output with one input that exhibits increasing returns to scale when output is small and decreasing returns to scale when output is increasing. 12. Consider a function that relates tax revenues R, in billions of dollars, to the average tax rate t such that R 350t 500t2. (a) What tax rate(s) is consistent with raising tax revenues equal to $60 billion? (b) What tax rate(s) is consistent with raising tax revenues equal to $61.25 billion? Summary In this chapter you have begun to build up your “tool kit” of skills and concepts. The material presented in this chapter is used throughout the rest of the book. The rules for exponents and for the quadratic formula appear later in a variety of different contexts. The average rate of change of a function and its relationship to the slope of a secant line form the basis of the calculus chapters. Concepts such as concavity are developed more formally in subsequent chapters. Since subsequent chapters build on the material presented here you should feel comfortable with the concepts, rules, and definitions in this chapter before moving on. As with all other chapters, the best way to do this is by carefully working through the problems offered at the end of each section. 44 Chapter 3 Exponential and Logarithmic Functions he growth of variables through time is an important calculation in many branches of economics. The growth rates of variables like national income, wages, the size of the labor force, the value of a currency, and the price of goods and services are common in economics. This chapter will demonstrate that exponential functions, introduced in the previous chapter, have a special role in economic analysis because of their use in calculating the growth of variables over time. Exponential functions also play an important role in a related problem—the calculation of the present value of a future payment. The analysis of the calculation of growth rates presented in Section 3.1 naturally leads to the discussion of a special type of exponential function, presented in Section 3.2, that is instrumental in growth and present value calculations. Exponential functions are strictly monotonic and, therefore, onetoone. As discussed in Chapter 2, onetoone functions have an inverse. The inverse of an exponential function is called a logarithmic function. The properties of logarithmic functions are discussed in Section 3.3. The discussion in that section, along with the accompanying applications, shows how logarithmic functions have a range of uses in economic analysis. These include the transformation of a nonlinear relationship into a linear expression, which is more easily evaluated, and, as is discussed in later chapters, the specification of an economic function with a constant elasticity. T 3.1 CALCULATING GROWTH An issue of central importance in economics is the rate of growth of income. High rates of growth of income can provide dramatic improvements in the standard of living over time, while low rates of income growth lead to stagnant material progress.As shown in Table 3.1, rates of growth of real income per capita varied widely across countries and regions over the period from 1990 to 1999. The average annual rate of growth of per capita income among countries classified as low income was less than onequarter of the rate among countries classified as high income and just above onesixth of the rate among countries classified as middle income. There were also striking regional differences in the growth rates of income per capita among low and middleincome countries. The East Asia and From Chapter 3 of Mathematical Methods for Economics, Second Edition. Michael W. Klein. Copyright © 2002 by Pearson Education, Inc. All rights reserved. 45 46 Part One Introduction TABLE 3.1 Average Annual Growth Rate of Income per Capita, 1990–1999 Country Group Growth Rate High income countries Middle income countries Low income countries 1.8% 2.3% 0.4% Country or Region (low and middle income countries) East Asia and Pacific region Latin America and Caribbean region SubSaharan Africa China Growth Rate 6.1% 1.7% 0.2% 9.6% Pacific region had the most rapid growth while, on average, countries in SubSaharan Africa experienced a decline in real income per capita.The country with the highest rate of growth of income per capita during this period was China, with a rate of growth of 9.6%.1 While the disparities in growth rates are striking, it is not immediately obvious what the implications are for future levels of income per capita. To address this question, we begin by determining the general relationship between the growth rate of a variable and its level at different moments in time. As a specific example, suppose per capita income in a country equaled $1,000 in 1990 and income grew each year at a rate of 5%. Income in 1991 is found by multiplying $1,000 by 1.05, which gives us $1,050. We use 1.05 because it represents the original level plus a growth of 5% (that is, 1 0.05). In general terms, defining Xt as the level of income in year t, Xt1 as the level of income in year t 1, and r as the growth rate (that is, 0.05), we have Xt1 (1 r)Xt . What was income in 1992? We use the same method, but in this case, the previous year’s income is $1,050 rather than $1,000. Income in 1992 was therefore $1,050 1.05 $1,102.50. Again, generalizing this, we have Xt2 (1 r)Xt1 (1 r)((1 r)Xt) (1 r) 2Xt . The level of income in any year can be determined by repeated application of this calculation. Table 3.2 shows the value of income in different years when there is an initial level of income of $1,000 and a 5% growth rate. Income in any one year is calculated by multiplying the previous year’s income by 1.05. The change in income between one year and the previous year, presented in the third column of Table 3.2, increases with time since income itself grows in each successive year and the change in income between year t and year t 1 equals income in year t times 0.05. The general result in the final row of Table 3.2 follows from the preceding rows. This general result shows that income in any year can be calculated by multiplying the initial year’s income by (1.05)n, where n is the number we add to 1990 to get the 1 These data are from the World Bank’s World Development Report 2000 (New York: Oxford University Press, 2000), Tables 3 and 11. 46 Chapter 3 TABLE 3.2 Exponential and Logarithmic Functions 47 Annual Income with a 5% Growth Rate Year Yt Y Yt Yt 1 1990 1991 1992 1993 1990 n $1,000.00 (1.05) $1,000.00 $1,000.00 (1.05)1 $1,050.00 $1,050 1.05 $1,000 (1.05)2 $1,102.50 $1,120.50 1.05 $1,000 (1.05)3 $1,157.625 $1,000.00 (1.05)n $50.00 $52.50 $55.125 $1,000((1.05)n (1.05)n1) 0 desired year. For example, n 1 for 1991, n 2 for 1992, and so on. This generalization even extends to 1990, itself, as well as earlier years. For 1990, n 0 and 1.050 1 since any number raised to the zero power equals 1. For 1989, n 1 and 1.051 0.95238 (approximately), so income in 1989 was approximately $952.38. Even more generally, the formula for this type of discrete growth, that is, growth that occurs through compounding once per period at the end of each period, is as follows. Growth Formula with Discrete EndofPeriod Compounding The relationship between the value of a variable in period t, Xt , and its level in period t n, Xtn, when it grows by the rate r at the end of each period, is Xtn (1 r) nXt , where r is expressed as a decimal (that is, 5% is expressed as r 0.05) and n can be positive (for periods later than t) or negative (for periods earlier than t). The growth formula allows us to determine the effect of growth rates on the level of income. For example, per capita income in China was $370 in 1990. This means that since China’s rate of growth was 9.6%, per capita income in 1999 was approximately $370 . (1.096)9 $370 . 2.28 $844. Had China instead grown by the average rate for East Asian and Pacific countries during this period (6.1%), its per capita income in 1999 would have been $370 . (1.061)9 $370 . 1.70 $630. The actual level of income in 1999 is 34% higher than this.2 The average rate of growth for all lowincome countries during this period was 0.4%. If per capita income in China 2 The percentage difference in two variables x and y, using y as a reference, equals xy . 100%. y In this case we have $844 $630 . 100% 34%. $630 Alternatively, the reference level could be the average of x and y. Calculating the difference in this way, we have xy (12) (x y) $844 $630 (12)($630 $844) . 100% 29%. 47 48 Part One Introduction Income $1,157.63 $1,102.50 $1,050.00 $1,000.00 0 1980 1981 1982 Year 1983 FIGURE 3.1 Income with Compounding at the End of the Year had grown at this rate, then in 1999 per capita income would have equaled $370 . (1.004)9 $370 . 1.037 $384. Before moving on, it is important to recognize the way in which we are depicting growth in all of these calculations. These calculations assume that the incremental change due to growth occurs only at the end of a period. This type of discrete growth path is reflected in Figure 3.1, which depicts the path of income presented in Table 3.2. The time path of income in this figure is a step function. It is usually more natural to think of continuous growth, which would be reflected in a smooth evolution over time of variables like income or population. In the next section we will see how to calculate growth that is compounded more often than once at the end of a period, including the limiting (and in many instances more natural) case of continuously growing levels. Exercises 3.1 1. Assume that Xt 100. What is the value of Xtn for each of the following values of n and r? (a) n 0, r 4% (b) n 5, r 3% (c) n 1, r 100% (d) n 50, r 4% 2. Now assume that Xt 50. Given the following negative values of n, which indicate time periods that have already passed, and positive values of r, calculate the value of Xtn. (a) n 4, r 8% (b) n 4, r 6% (c) n 1, r 6% (d) n 10, r 2% 3. Now assume that Xtn 25. Given the following values of n and r, calculate the value of Xt . 48 Chapter 3 Exponential and Logarithmic Functions 49 (a) n 3, r 4% (b) n 3, r 7% (c) n 5, r 7% (d) n 0, r 100% (e) n 5, r 2% 4. Suppose you are a farmer that stores grain in a leaky silo so that you lose 2% of your crop each year as a result of dampness and rot. To obtain the future value of X, you still employ the same formula Xtn Xt(1 r) n. If r 2%, what is the value of Xtn as n approaches infinity? That is, what is lim Xt (1r) n ? Will the value of Xtn ever equal zero? n→ 5. Upon graduation, you secure a job with a firm willing to pay you an annual salary of $50,000. Your contract stipulates annual increases equivalent to that year’s rate of inflation plus 2%. Assume that t0 2003. (a) If inflation during your first year is 3%, what will your salary be for 2004? (b) If the inflation rate then falls to 2% for each of the next two years, what salary should you expect to earn in 2006? (c) In 2006, your firm decides that costofliving–based raises are too variable for their budgeting requirements. Instead they offer you a fixed 6% annual increase. What will your salary be in 2012 (t 6)? 6. Assume a firm’s net profits are $50 million in 2000 and are expected to grow at a steady rate of 6% per year through the end of the decade. How much would you expect the firm to earn in 2001? In 2003? Now assume that the firm’s profits have been growing at 6% since 1997. If a negative value of n can be interpreted as the number of time periods before period t, how much did the company earn in 1998? Graph the path of income growth between 1998 and 2003 and explain why the curve gets steeper over time. 7. Productivity improvements are often considered a primary engine of economic growth. In fact, laggin