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: 1-292-03918-3 ISBN 10: 1-269-37450-8 ISBN 13: 978-1-292-03918-3 ISBN 13: 978-1-269-37450-7 British Library Cataloguing-in-Publication 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 Chapter The Mathematical Framework of Economic Analysis Michael Klein Chapter An Introduction to Functions Michael Klein 11 Chapter Exponential and Logarithmic Functions Michael Klein 45 Part II Matrix Algebra Michael Klein 73 Chapter Systems of Equations and Matrix Algebra Michael Klein 75 Chapter Further Topics in Matrix Algebra Michael Klein 115 Part III Differential Calculus Michael Klein 143 Chapter An Introduction to Differential Calculus Michael Klein 145 Chapter Univariate Calculus Michael Klein 173 Chapter Multivariate Calculus Michael Klein 211 Part IV Optimization Michael Klein 255 Chapter 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 The Mathematical Framework of Economic Analysis Chapter An Introduction to Functions Chapter Exponential and Logarithmic Functions This book begins with a three-chapter section that introduces some important concepts and tools that are used throughout the rest of the book Chapter 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 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 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 This page intentionally left blank Chapter The Mathematical Framework of Economic Analysis hat are the sources of long-run 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 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 of Mathematical Methods for Economics, Second Edition Michael W Klein Copyright © 2002 by Pearson Education, Inc All rights reserved 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 well-defined 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 Chapter The Mathematical Framework of Economic Analysis It is typically the case that the assumptions of a formal mathematical model are explicit and transparent Therefore a formal mathematical model often readily admits the sensitivity of its conclusions to its assumptions The evolution of modern growth theory offers a good example of this A central question of economic growth concerns the long-run stability of market economies In the wake of the Great Depression of the 1930s, Roy Harrod and Evsey Domar each developed models in which economies either were precariously balanced on a “knife-edge” of stable growth or were marked by ongoing instability Robert Solow, in a paper published in the mid-1950s, showed how the instability of the Harrod–Domar model was a consequence of a single crucial assumption concerning production Solow developed a model with a more realistic production relationship, which was characterized by a stable growth path The Solow growth model has become one of the most influential and widely cited in economics Applications in Chapters 8, 9, 13, and 15 in this text draw on Solow’s important contribution More recently, research on “endogenous growth” models has studied how alternative production relationships may lead to divergent economic performance across countries Drawing on the endogenous growth literature, this book includes an application in Chapter 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 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 long-standing questions concerning the price of capital A mathematical model often offers conclusions that are directly testable against data These tests provide an empirical standard against which the model can be judged The branch of economics concerned with using data to test economic hypotheses is called econometrics While this book does not cover econometrics, a number of the applications show how to use mathematical tools to interpret econometric results For example, in Chapter we show how an appropriate mathematical function enables us to determine the link between national income per capita and infant mortality rates in Solow’s paper, “A contribution to the theory of economic growth,” is published in the Quarterly Journal of Economics, 70, no (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 (May 1990): 92–96 Chapter 14 Differential Equations 479 The term k2s2 e t ␭2 is common to the solutions for x(t) and y(t) Thus we can find the joint time path of x(t) and y(t) by solving to remove this term This gives us the unique linear saddlepath y(t) ϭ (␭2 Ϫ a) b x(t) ϩ ™ ␣bc Ϫ ␤ab Ϫ (␭2 Ϫ a)(␤b Ϫ ␣d) , (ad bc)b Ô which shows the relationship between x(t) and y(t) for any t One of the variables in this system may evolve slowly and the other must jump such that the saddlepath relationship between the two variables is always satisfied Notice that the dynamics of this system are monotonic since there is a linear relationship between y(t) and x(t) along the saddlepath We can obtain the specific solution to this model, given an initial value, such as x(0), and by solving for k2s2 For example, evaluating the solution at t ϭ 0, given x(0), we have k2s2 ϭ (␭2 Ϫ a) b x(0) Ϫ (␭2 Ϫ a)(␤b Ϫ ␣d) b(ad Ϫ bc) This value for k2s2 can be used in the general solution to get a specific solution Systems That Are Not Diagonalizable The previous results depend upon our ability to find a matrix that, when multiplied by A, results in a diagonal matrix In general, we cannot find a diagonal matrix when the two characteristic roots of A are identical because tr (A)2 ϭ det (A) In this case the single characteristic root is ␭ϭ tr (A) a ϩ d ϭ 2 For example, the system x˙(t) ϭ x(t) ϩ y(t) ϩ ␣ y˙(t) ϭ y(t) ϩ ␤ cannot be diagonalized since the trace of the matrix of the system is and the determinant of the matrix of the system is and tr Ϫ det ϭ In general, the solution to the system (14.14) when there is a repeated characteristic root because (a ϩ d)2 ϭ 4(ad Ϫ bc) is ␤b Ϫ ␣d ad Ϫ bc ␭Ϫa k2 ␣c Ϫ ␤a (k1 ϩ k t) ϩ ϩ b b ad Ϫ bc x(t) ϭ k1e t␭ ϩ k2te t␭ ϩ y(t) ϭ ΄΂ ΃ ΅ (14.20) for b The system will be globally stable if ␭ ϭ (aϩd)͞2 Ͻ In this case the pair of values defining the steady state is called a stable node or a stable focus If ␭ = (a ϩ d)͞2 Ͼ 0, then the system is globally unstable, and the pair of values defining the steady state is called an unstable node or an unstable focus 479 480 Part Five Integration and Dynamic Analysis In the case where tr (A)2 Ͻ det (A), the system will include two complex roots since they will include the term ͙Ϫ1 For example, if the matrix of the system is ¢ › Ϫ Aϭ then the two characteristic roots are ␭1, ␭2 ϭ Ϫ9 Ϫ , Ϫ1 Ϯ ͙1 Ϫ 4(1͞4 ϩ 9͞4) ϭ Ϫ Ϯ 3͙Ϫ1 2 These two roots are complex numbers As discussed in Chapter 13, a complex number takes the form r ϩ m͙Ϫ1, where r is called the real part and m is called the imaginary part of the number Thus, for the two roots given above, each has a real part equal to Ϫ2, while the imaginary part of one of the roots is and the imaginary part of the other root is Ϫ3 A detailed study of the analytic solution in this case is beyond the scope of this book, but we note two important results, which we draw on in the next section where we discuss qualitative solutions with phase diagrams First, a system of differential equations with complex roots is stable if the real parts of the characteristic roots of the system are negative Second, the dynamics of the system are oscillatory We can verify these results by noting that the solution to the system in the case of two complex roots is x(t) ϭ e rt [k1 cos (vt) ϩ k2 sin (vt)] ϩ y(t) ϭ e rt ΄ (r Ϫ a)k1 ϩ vk2 b ␤b Ϫ ␣d ad Ϫ bc cos (vt) ϩ (r Ϫ a)k2 Ϫ vk1 b ΅ sin (vt) ϩ ac Ϫ ba , ad Ϫ bc where k1 and k2 are constants that will depend upon the initial conditions and vϭ ͙4 det (A)2 Ϫ tr (A)2 , which is a real number The important point with respect to stability is that the sine function, sin (vt), and the cosine function, cos (vt), are bounded; therefore, x(t) and y(t) are bounded if r Ͻ 0, and x(t) and y(t) are not bounded if r Ͼ (see Figure 13.8, which presents a graph of the sine function and the cosine function) If r Ͻ 0, then the steady state point is called the stable focus or spiral sink, and both x(t) and y(t) move toward the steady state point over time from any initial point The steady state point is called an unstable focus or source if r Ͼ 0, and both x(t) and y(t) move away from the steady state point over time from any initial point other than the steady state point Also, the sine and cosine functions are periodic functions, that is, the values of the functions oscillate Therefore the presence of these functions in the solution means that there will be an oscillatory path for x(t) and y(t) Next we investigate the qualitative aspects 480 Chapter 14 Differential Equations 481 of the dynamics in the case of roots that are complex numbers, along with the other cases discussed here, by using a two-dimensional phase diagram Two-Variable Phase Diagram We can study a system of two differential equations by using a two-variable phase diagram A phase diagram is drawn in a plane with y(t) on one axis and x(t) on the other axis The combination of points x(t) and y(t) such that x˙(t) ϭ represents one line in the phase diagram, and the combination of points such that y˙(t) ϭ represents another line in the diagram The point where these two schedules cross represents the steady state point of the system, (xϱ, yϱ) We can analyze dynamic stability in a phase diagram by considering the four regions defined by the x˙(t) ϭ schedule and the y˙(t) ϭ schedule In Figure 14.7 we study three different phase diagrams and consider the dynamics in the regions we refer to as N, E, S, and W, representing the directions north, east, south, and west of the steady state point In each of the phase diagrams in Figure 14.7, the x˙(t) ϭ schedule is positively sloped, and the y˙(t) ϭ schedule is negatively sloped Therefore, in terms of the elements of A in (14.15), a and b are of different signs and c and d are of the same sign The phase diagrams differ in the individual parameters of A, a difference that gives rise to a different type of dynamic stability in each case The phase diagram in Figure 14.7(a) is globally stable since the sign pattern of the elements of A is aϽ0 cϽ0 bϾ0 d Ͻ 0, and, therefore, the trace of A, which equals a ϩ d, is negative and its determinant, which equals ad Ϫ bc, is positive In regions N and W, the dynamics force x to increase, and, in regions S and E, the dynamics force x to decrease In regions N and E, the dynamics force y to decrease, and, in regions S and W, the dynamics force y to increase Any initial point, like that labeled in the figure, leads to a movement toward the steady state point at the intersection of the x˙(t) ϭ and y˙(t) ϭ schedules The figure includes a possible time path from point to the steady state The phase diagram in Figure 14.7(b) is globally unstable since the sign pattern of the elements of A is aϾ0 cϾ0 bϽ0 d Ͼ 0, and, therefore, the trace of A is positive and its determinant is positive In regions N and W, the dynamics force x to decrease, and, in regions S and E, the dynamics force x to increase In regions N and W, the dynamics force y to increase, and, in regions S and E, the dynamics force y to decrease At any initial point other than the steady state where the x˙(t) ϭ and y˙(t) ϭ schedules intersect, the dynamics force x(t) and y(t) away from the steady state Thus the only stable solution in models that are globally unstable is an immediate jump to the new steady state with a change in any exogenous 481 482 Part Five Integration and Dynamic Analysis FIGURE 14.7 Two-Variable Phase Diagram 482 Chapter 14 Differential Equations 483 variables Any other initial point, such as the points labeled 2, 3, 4, or 5, result in a time path diverging from the steady state as shown in the figure The phase diagram in Figure 14.7(c) is saddlepath stable since the sign pattern of the elements of A is aϾ0 cϽ0 bϽ0 d Ͻ 0, and, therefore, the determinant of A is negative In regions N and W, the dynamics force x to decrease, and, in regions S and E, the dynamics force x to increase In regions S and W, the dynamics force y to increase, and, in regions N and E, the dynamics force y to decrease This figure also includes the saddlepath, which passes through the steady state point represented by the intersection of the x˙(t) ϭ and y˙(t) ϭ schedules The forces of motion dictate the arrows of motion drawn on the saddlepath The slope of the saddlepath is dy(t) ␭2 Ϫ a ϭ , dx(t) b which is positive since the characteristic root is negative, a is positive, and b is negative At any moment, the pair of values of x(t) and y(t) are on the saddlepath and either at the steady state or moving toward it For example, points and are on the saddlepath, and the subsequent time path from either of these points is along the saddlepath toward the steady state The dynamics in the globally stable and saddlepath cases are monotonic In the cases where the system cannot be diagonalized, the dynamics may be oscillatory The two graphs in Figure 14.8 illustrate possible stable trajectories for a system with one repeated root Figure 14.8(a) illustrates the trajectories for the homogeneous system x(t) ␭ y(t) x(t) , y(t) Ô Ô ¤ ␭ FIGURE 14.8 Oscillatory Dynamics 483 484 Part Five Integration and Dynamic Analysis where ␭ Ͻ The steady state of this system is the origin The two possible initial points are labeled A and B Note that, along either one of these trajectories, from A to (0, 0) or from B to (0, 0), the time path of x(t) oscillates, while the time path of y(t) is monotonic Figure 14.8(b) illustrates the time path of a homogeneous system with two complex roots that have negative real parts The trajectories in this figure spiral in from the initial point C towards the steady state point (0, 0) This spiral trajectory represents oscillatory paths for both x(t) and y(t) Phase diagrams with saddlepath properties are used frequently in macroeconomic models that assume perfect foresight One such model is presented in the following application Exchange Rate Overshooting The flexible price exchange-rate model presented in the application in Section 14.2 has the property that a given change in the money supply leads to a proportional change in the exchange rate Exchange rates are very volatile, however, a characteristic not well captured by the flexible price exchange-rate model An alternative model developed by Rudiger Dornbusch is able to account for exchange-rate volatility.11 The simplest version of Dornbusch’s model consists of four equations We assume perfect foresight, so there is no distinction between the expected value of a variable and its actual value The interest parity relationship i(t) ϭ i* ϩ ␧˙(t), where i(t) is the nominal interest rate at moment t, i* is the foreign interest rate, which is assumed constant, and ␧˙(t) is the derivative of the exchange rate with respect to time The exchange rate is defined as units of domestic currency per one unit of foreign currency, so an increase in ␧ represents a depreciation of the domestic currency The money demand equation m Ϫ p(t) ϭ Ϫ␮i(t) where m is the logarithm of the money supply and p(t) is the logarithm of the price level at time t, is similar to the one presented in Section 14.2 The simplest version of an income determination equation y(t) ϭ ␤(␧(t) ϩ p* Ϫ p(t)) has income as a function of the logarithm of the real exchange rate, which is the sum of the logarithm of the nominal exchange rate and the foreign price level, p*, minus the domestic price level An increase in the real exchange rate, which represents a real depreciation of the currency, stimulates output by making exports cheaper and by making imports more expensive The final equation is a Phillips curve–type relationship that links inflation, p˙(t), to the difference between actual income, y(t), and potential income, y, p˙(t) ϭ ␪(y(t) Ϫ y) , 11 Rudiger Dornbusch, “Expectations and exchange-rate dynamics,” Journal of Political Economy (December 1976): 1161–1176 484 Chapter 14 p(t) Differential Equations 485 p(t) saddlepath pи (t) = sp ∼ m εи (t) = pи (t) = Ϫ m εи (t) = ε (t) ε (t) Dornbusch Overshooting Model The Effect of a Money Supply Increase (a) (b) FIGURE 14.9 Dornbusch Overshooting Model and Effect on Money Supply where ␪ is a parameter that reflects the responsiveness of inflation to excess aggregate demand We simplify the algebra by setting i*, p*, and y equal to zero After some algebra, we express this model as the system £ ␧˙(t) p(t) Ô m ( ) ␧ t ␮ ␮ ϩ ( ) p t Ô Ê The determinant of the matrix of this system is Ϫ␤␪͞␮ The fact that the determinant is negative indicates that the dynamics are characterized by saddlepath stability The exchange rate is the jump variable, and the price level is the slowly evolving variable The phase diagram of this system is presented in Figure 14.9(a) The ␧˙(t) ϭ schedule is horizontal and intersects the vertical axis at the point where p(t) ϭ m The p˙(t) ϭ passes through the origin, and its slope equals The saddlepath passes through the steady state point (␧ϱ, pϱ) We can determine this steady state point by solving the system when ␧˙(t) ϭ and p˙(t) ϭ 0, which gives us ␧ϱ ϭ pϱ ϭ m The slope of the saddlepath is dp(t) ϭ ␭2 ␮ , d␧ (t) where ␭2 is the negative root of the system This shows that the saddlepath has a negative slope since ␮ is positive ෂ Figure 14.9(b) shows the effect of an increase in the money supply from m to m This change causes the ␧˙(t) ϭ schedule to shift up and the saddlepath to shift up and to the right At the moment the money supply increases, the exchange rate jumps to 485 486 Part Five Integration and Dynamic Analysis point from its initial steady state of point Over time the system moves along the saddlepath to the new steady state point 2, with the exchange rate falling (that is, appreciating) and the price level rising Thus the exchange rate overshoots its new ෂ steady state level of ␧ϱ ϭ m in the short run This overshooting, which does not occur in the flexible price monetary model presented in Section 14.2, is a possible reason for the observed volatility in exchange rates A Note on Second-Order Differential Equations A second-order linear differential equation takes the form d 2x(t) dx(t) ϭa ϩ bx(t) ϩ c dt dt or, equivalently, with the dot notation, xă(t) ax˙(t) ϩ bx(t) ϩ c This equation can be solved with the techniques developed in this section Define the new variable y(t), where y(t) ϭ dx(t)͞dt Using this definition, we can rewrite the single second-order linear differential equation as the system of first-order differential equations dy(t) ϭ ay(t) ϩ bx(t) ϩ c dt dx(t) ϭ y(t) dt or, in matrix form, as x˙(t) ϭ y˙(t) b x(t) a y(t) c Ô Ô ¤ ™¤ This system can be solved with the techniques discussed in this section The stability of the solution to a second-order differential equation follows from the results described previously Define the ϫ matrix in the above system as A and, consider the case where there are two distinct real roots The trace of A is negative and its determinant is positive if a Ͻ and b Ͻ In this case both characteristic roots are negative, and the solution to the second-order differential equation is globally stable The trace of A is positive and its determinant is positive if a Ͼ and b Ͻ In this case both characteristic roots are positive, and the solution to the second-order differential equation is globally unstable The determinant of A is negative if b Ͼ In this case one characteristic root is positive, and one is negative As t → ϱ, the positive characteristic root will dominate the solution, and x(t) → ϱ or x(t) → Ϫϱ Exercises 14.3 Determine the steady state values for the variable x(t) and y(t) for each system of equations 486 Chapter 14 Differential Equations 487 (a) System I: x˙(t) ϭ Ϫx(t) ϩ y˙(t) ϭ 3x(t) Ϫ 6y(t) Ϫ (b) System II: 1 x(t) Ϫ y(t) y˙(t) ϭ 13x(t) ϩ y(t) Ϫ 56 x˙(t) ϭ (c) System III: x(t) ϩ y(t) Ϫ 25 2 1 y˙(t) ϭ x(t) Ϫ y(t) ϩ 2 x˙(t) ϭ What are the stable dynamics of each system presented in question 1? Construct a phase diagram for each system, and show the path to the steady state that would result from some point away from the steady state, if such a path exists Determine the explicit stable solution to System II presented in question Find the stable solution to System III in question when the initial value of the system is x ϭ 0, y ϭ Consider a variant of the Dornbusch overshooting model in which the money demand equation is m(t) Ϫ p(t) ϭ Ϫ␮i(t) ϩ ␸y(t) and otherwise the model is the same as in the text (a) Determine the conditions under which the model is saddlepath stable (b) Is it possible to have “undershooting” rather than “overshooting” in this model (that is, can the saddlepath have a positive slope)? Consider the macroeconomic model u˙(t) ϭ Ϫ␤(m Ϫ p(t)) ϩ ␴(u Ϫ ␪(t)) p˙(t) ϭ ␪(u Ϫ ␪(t)), where u(t) is the unemployment rate, p(t) is the logarithm of the price level, m is the exogenously determined money supply, and u is the natural rate of unemployment The first equation shows that unemployment responds to monetary policy and also, all else equal, unemployment tends towards its natural rate The second equation is a Phillips curve (a) Determine the steady state of the model (b) What type of dynamics characterize this model? (c) Construct a phase diagram for this model (d) Use your phase diagram to trace out the effects of a one-time increase in the ෂ money supply from m to m 487 488 Part Five Integration and Dynamic Analysis Calculate the characteristic roots of the following second-order linear differential equations Determine whether each equation is stable or unstable (a) xă(t) 4x(t) x(t) (b) yă(t) 2y(t) y(t) (c) ză(t) 2z(t) z(t) Summary This chapter has continued our study of dynamic analysis with a study of differential equations Differential equations are found in fields of economics in which the value of variables at different moments in time is important, such as finance, macroeconomics, and international economics Many of the concepts discussed in this chapter are similar to those discussed in Chapter 13 in which dynamic analysis is studied in the context of discrete time models Differential equations and phase diagrams are important for the material presented in the next chapter in which we solve for the optimal time path of variables in dynamic problems 488 Index Page references followed by "f" indicate illustrated figures or photographs; followed by "t" indicates a table A Accounting, 79, 122, 127, 233-234, 281, 284, 422, 459 accuracy, 4, 220 addresses, 35, 219 Advances, 211, 235, 350 Advantages, 1, 4, Advertising, 292-294, 299-300, 307 local, 307 product, 292 Affect, 3, 9, 43, 66, 106, 110, 128, 145-146, 195-196, 212, 214, 226, 229, 231, 242, 250, 274, 277, 298, 347, 352, 464 Africa, 46 Age, 152, 212 Aggregate demand, 75, 86, 97, 425-426, 433, 485 Aggregate supply, 75 Application, 3-6, 9, 46, 58, 60, 62, 66-67, 86, 94, 107, 110, 119, 128, 132, 146, 165, 171, 173, 178, 184, 186-187, 189, 192-194, 210, 216, 218, 225, 228-230, 235-237, 243, 253, 267-268, 270, 276, 282, 284, 291-292, 294, 297, 304-305, 307, 347, 351, 353, 361-362, 383, 390-391, 393, 395, 416, 421-422, 425, 427, 429, 432, 434-435, 441, 467-468, 470-471, 484 Applications, 1, 3-5, 9-10, 20, 27, 36, 45, 56, 60, 65, 71, 73, 143-144, 197, 210, 253, 255-256, 257, 272, 286, 287, 290, 294, 315, 332, 343, 359, 361-362, 363, 388, 396, 405, 407, 417, 425, 449, 451, 463, 465 Arbitrage, 470-471 arguments, 3, 15-17, 23, 60, 143-144, 145, 173, 180, 211-212, 214-215, 221, 224, 230, 232, 236, 238-239, 242, 244-245, 250-251, 256, 276, 288-289, 305, 308, 310-311, 314-315, 317-319, 325-327, 334-338, 340, 349-350, 354 logical, 144 Asia, 45-46, 157 Assets, 50-52, 56, 59, 185-186, 279, 389, 401, 407 current, 407 Atlas, 192 attention, 123 attributes, 234-235 AU, 229 audience, Austria, 280 Authority, 277 express, 277 Autonomous consumption, 112, 170 Available, 56, 233, 271, 317, 323, 328, 334, 349-350, 353, 358, 388, 422, 467 Average costs, 332 B Balance of trade, 434, 465 Balanced budget, 188 Bank deposits, 184 Bank failures, 185 Banks, 55, 58, 184-186, 278 Basic form, 170 Behavior, 6, 15, 70, 143, 200, 205, 255, 257, 274, 286, 430 Benefits, 285, 393, 403 Bid, 465 Bond market, 59 Bond prices, 57, 389 interest, 57, 389 Bonds, 59-60, 272, 389, 432, 470-471 payment, 59 Borrowing, 170 brackets, 28, 104, 175, 177, 328, 335, 337 Brand, 292, 300 Brands, 300 Brazil, 394, 396 Britain, 276, 466 British pound, 429 Budget, 188, 196, 300, 317-318, 320-323, 326-328, 334, 339, 341-343, 345-349, 351-352, 357 defined, 328, 348 Budget deficit, 188, 196 Budget line, 321-322 Budget surplus, 188 Budgeting, 49 Bulgaria, 394 C Capacity, 22, 317, 388 Capital, 5, 12, 21, 59, 64-65, 71, 77, 89, 113, 128-130, 146, 157, 160, 196, 209, 218-220, 224-225, 229, 233-235, 237, 241, 249, 252, 281-282, 284, 290, 299, 328, 331-332, 342, 344-345, 347, 357-358, 387-388, 421-422, 429, 458-459, 470-471 definition of, 64, 157, 225, 235 fixed, 328, 331-332 growth, 5, 71, 196, 220, 233-235, 281-282, 284, 388, 421-422, 458-459 human, 157, 219 working, 12, 157 Capital flows, 196, 219-220, 388, 470 Capital gains, 471 capital letters, 12, 89 Caribbean, 46 cause and effect, 294 CEA, 453 Central banks, 55, 184, 278 Certainty, 204 Ceteris paribus, 9, 77, 80-81, 85-86, 125, 128, 131-132, 144, 211-212, 214, 218 Character, 11, 431 Checkable deposits, 185-186 Children, 347, 351, 353 China, 46-47, 49, 59 Chinese, 54 citations, 433 Classification, 83, 312 Collapse, 186 Columbia, 217, 279 Columns, 73, 75, 89-94, 100, 105-107, 117, 119, 137, 308-309, 312-313 Commercial banks, 55, 185 Commitment, 277-278 Commodities, 343 Comparative advantage, 128, 130, 132 Compensation, 82, 284-285, 299, 307 Compete, 292, 300 Competition, 3, 109-111, 282, 285, 292, 352 perfect, 110, 282 Competitiveness, 427, 465-466 Conditions, 4, 7, 9, 11-13, 32-33, 76, 110, 132, 134-135, 146, 160, 188, 204, 215, 228, 256, 264-266, 268, 276-277, 285, 287-289, 291, 293-294, 301-308, 310-315, 323-328, 330, 333-340, 343-344, 346, 348-350, 354-355, 357-359, 436, 448, 457, 480, 487 Consideration, 4, 161, 299 Consistency, 277 Constant returns to scale, 241 Constraints, 4, 278, 317-318, 325, 332-334, 337, 339-340, 342, 345, 348, 350, 352-355, 357-359 CHECK, 337, 340, 357 Construction, 5, 358 Consumer behavior, 70 Consumer Price Index, 68 Consumer spending, 425 Consumers, 6, 76, 80-82, 146, 192-195, 197-198, 283, 287, 290, 292, 363, 402 Consumption, 7, 11, 14-18, 23, 27, 32, 79, 86, 97, 112, 122, 127, 131-132, 148, 163, 170, 200-202, 209-210, 244, 248, 255-256, 280-282, 284, 317, 320, 322-323, 325-326, 333-334, 336, 344-349, 351-354, 358, 362, 407, 426, 441 autonomous, 112, 170 consumer, 317, 346, 351, 353, 362 Consumption function, 11, 14-18, 23, 27, 79, 112, 170 changes in, 27, 170 Continuity, 18, 20 Contract, 49, 323 Control, 5, 184 Convergence, 386, 411, 419, 421-422, 429, 435 conversations, conversion, 62 cookies, 29, 204-205, 209, 243-244, 247, 340-341 Cooperation, 130, 299 Copyright, 1, 3, 11, 45, 73, 75, 115, 143, 145, 173, 211, 255, 257, 287, 317, 361, 363, 407, 451 Costs, 22, 60, 70-71, 198, 272, 274, 276, 283, 285, 300-301, 317, 320, 323, 326, 331-332, 341, 351, 358, 378, 391 Countries, 3, 5-6, 35, 45-47, 49-50, 59, 70, 109, 111, 128, 130, 157, 192, 195, 218-220, 229, 240, 267, 297, 349, 351, 387-388, 422-423, 427, 429, 434, 470, 473 CPI, 68 Credit, 278 Cross-price elasticity of demand, 228 Crowding out, 429 Currency, 3, 45, 184-186, 196, 253, 466, 470-471, 484 Curves, 43, 75, 239, 245, 249-250, 253, 321-322, 329-331, 390, 394 indifference, 245, 249, 253, 321-322 slope of, 249-250, 253, 321-322 supply, 75, 253, 330 Customers, 292 D Damage, 267 data, 5, 46, 59, 69-70, 94, 157, 192, 212, 267, 294, 394, 429, 433 Debt, 69, 188-189 defined, 189 Debt crisis, 69 Decision making, 318 Deflation, 469 Degree of risk, 206, 210 Demand, 6-9, 23, 33-34, 73, 75-81, 83-84, 86, 97, 114, 122, 127, 143, 160, 180, 182, 189-192, 194-195, 197-198, 224-225, 228-230, 251-252, 267-268, 272-274, 279, 282-283, 285, 291-294, 300-301, 345, 363-364, 390-392, 402-403, 425-426, 429-431, 433, 467-469, 471, 473-474, 484-485, 487 aggregate, 75, 86, 97, 363, 425-426, 433, 485 change in, 8-9, 77-78, 81, 83-84, 86, 143, 180, 189, 194, 198, 225, 228, 251-252, 345, 390, 392, 402, 425, 430, 484 currency, 471, 484 derived, 7, 390 elastic, 190, 197, 274 excess, 402-403, 429, 474, 485 for labor, 224-225, 229, 345 for money, 180 inelastic, 190, 197, 274 price elasticity of, 228, 294 prices and, 283, 473 unit elastic, 190, 197 Demand curve, 33, 75, 160, 190-191, 197, 283, 363, 390-392, 403 labor, 283 market, 33, 197, 283, 363 Demand schedule, 83, 224-225 Dependent variables, 408, 477 Deposits, 184-186 Depreciation, 240, 281-282, 284, 378, 422, 459, 489 471-472, 484 Depression, 5, 184-185, 253 Derivatives, 153-154, 159, 163, 165, 171, 173-176, 184, 187, 199, 201, 203, 211, 214-216, 220-224, 227-228, 230, 232-233, 235, 238-241, 243, 246-247, 249, 253, 264, 266, 287-288, 304-306, 308, 311-312, 314-315, 330, 332, 336, 350, 354, 452 Detailed analysis, 440 Determinant, 106-110, 112-114, 115-124, 126-129, 133-135, 139, 141, 212, 309-311, 313, 335-340, 437, 448, 470, 477, 479, 481, 483, 485-486 composite, 212 Devaluation, 434, 473 Developing countries, 3, 70, 109, 111, 220, 267 diagrams, 33, 220, 420-421, 456-457, 480-481, 484, 488 Differentiation, 4, 173-174, 177, 182-184, 190, 198-199, 210, 211, 215-216, 218, 223, 242, 253, 255-256, 285, 300, 315, 371, 373, 378-380, 382, 405 product, 173-174, 184, 218, 285 Diminishing marginal returns, 43, 209, 229 Diminishing returns, 200, 202 Direct investment, 387-388 Discount rate, 346, 353, 378, 393, 402-403 Discrimination, 212-214, 273 Disposable income, 230 Distance, 149, 166, 208, 244, 294, 366, 464 Distribution, 80, 192, 393, 396-401, 403-404 Diversity, 320, 352 Dividends, 433 Dollar, 9, 12, 59, 161, 197, 228, 240, 347, 389, 429, 464 exchange rates, 429 Dollars, 15, 17, 44, 79, 158, 178, 251, 267, 291, 366, 378 Duopoly, 276, 283 Durable goods, 378 Duty, 145 Dynamics, 134, 279, 408-410, 421, 426-427, 429, 434, 441-443, 448-449, 453-454, 468-470, 478-481, 483-485, 487 E Earnings, 3, 94, 152, 216-217, 228-229 test, 94 Eastern Europe, 387-388 Economic analysis, 1, 3-5, 7-9, 26-27, 32, 45, 70-71, 77, 114, 143-144, 145, 154, 165, 179, 189, 196, 245, 253, 257, 317, 330, 465, 474 Economic cost, 391 Economic development, 3, 157, 268 Economic environment, 83 Economic factors, 231 Economic forces, Economic growth, 4-6, 49, 233, 235, 256, 267-268, 280, 421-422 in developing countries, 267 rates, 5, 233, 280 Economic models, 1, 3-4, 6-7, 9, 11, 62, 76, 115, 145, 211, 255, 257, 318, 425, 432, 449, 457, 463, 474 Economic policy, 393 Economic principles, Economic questions, 3-4, 405 Economic variables, 4, 43 Economics, 1, 3-7, 9-10, 11, 14, 18, 23, 27, 29, 42, 45, 60, 65, 73, 75, 94, 115, 130, 141, 143-144, 145, 154, 161, 171, 173, 177, 189, 192, 198, 202, 210, 211-212, 230, 253, 255-256, 257, 267, 272, 274, 276, 278, 280, 286, 287, 294, 299, 317, 324, 332, 343, 359, 361, 363, 379, 385, 388, 391, 396, 407, 417, 422, 429, 433, 451, 465, 470-471, 488 Economy, 3, 49, 54, 82, 97, 129-130, 132, 170, 188, 196, 199, 218, 225, 233, 276-277, 280-281, 283-284, 349, 391, 393-394, 403, 421-423, 426, 429, 433-435, 459, 467, 473, 484 Ecuador, 70 Education, 1, 3, 6, 11, 45, 49, 73, 75, 94, 115, 143, 145, 157, 173, 211-212, 216-217, 255, 257, 287, 299, 307, 317, 347-348, 361, 363, 407, 451 Education level, 299, 307 Effective interest rate, 55-56, 58 Efficiency, 196, 284 Elasticities, 63, 189-190, 192, 194, 197, 344 490 Elasticity of demand, 198, 228, 273, 291, 293-294 cross-price, 228 income, 228, 294 price, 198, 228, 273, 291, 293-294 Elections, 231 Employment, 5, 82-84, 157 Employment opportunities, 157 Endowments, 130 endpoints, 12, 27, 29, 203, 257 English, 153 Environment, 6, 83-84, 267, 277 natural, 6, 277 Equilibrium, 4, 7-8, 33, 75-78, 80-82, 84-86, 98, 102, 108, 125, 127, 193, 198, 274-275, 283, 410, 413, 425, 427, 429, 452, 456 long-run, 410, 429, 452 market, 7-8, 33, 76, 85, 108, 127, 193, 198, 274, 283, 425, 429 Nash, 274 Equilibrium price, 75-76, 78, 80-81, 125, 198, 274, 283 Equity, 362 Europe, 113, 157, 387-388 economic development, 157 European Monetary System, 55 Evaluation, 4, 8, 118-119, 125, 173, 182, 267, 369, 384-385, 473 evidence, 34, 130, 212-213, 278, 282, 292, 294, 351 Excess demand, 403, 429 Excess supply, 429 Exchange, 12, 62, 64, 97-98, 131, 196, 240, 362, 427, 429, 434, 466, 470-472, 474, 484-486 Exchange rate, 12, 62, 64, 97-98, 131, 196, 240, 362, 427, 429, 434, 466, 471-472, 474, 484-486 appreciation, 471 depreciation, 240, 471-472, 484 fixed, 466 flexible, 484, 486 Exchange rates, 429, 470, 484, 486 U.S dollar and, 429 Expansion, 5, 117-122, 126, 128, 141, 207-208, 210, 279, 311, 313 expect, 49, 59, 69-70, 146, 150, 164-165, 219, 432-433, 468 Expectations, 277-278, 429-432, 435, 467-468, 470, 478, 484 Expectations of inflation, 278 Expected return, 432, 471 Expenditures, 70, 170, 279, 294, 343-344, 347-348, 351 defined, 348 Experience, 59, 69, 82, 94, 147, 157, 198, 212, 216-217, 228-229, 299, 307, 422 Exports, 97-98, 129, 131, 196, 240, 276, 434, 484 Expropriation, 220 External environment, 84 F Factors of production, 109, 113, 128, 233-234, 237, 245, 344 Failure, 105, 267 Family, 157, 255, 347, 349, 351, 362, 375 FAST, 351 Feature, 201 Federal Reserve, 402 Fields, 4, 10, 73, 144, 286, 359, 451, 488 Finance, 4, 80, 115, 141, 161, 345, 348, 361, 363, 401, 407, 451, 465, 488 Financial markets, 5, 429 Financial Times, 55 Firms, 83-84, 86, 224, 273-276, 282-283, 285, 290-292, 299-300, 363, 405, 407 Fiscal policy, 188, 211 expansionary, 211 Fixed capital, 331-332 Fixed costs, 283, 300, 378 Flat tax, 161-162 Flexible prices, 469 Food, 128-130, 132, 342, 351 production, 128-130, 342 Forecasts, 251 Foreign bonds, 470-471 Foreign competition, Foreign direct investment, 387 Foundations, 70, 330 fractions, 110 Free trade, 276 Frequency, 52-53 Fund, 279 G Game theory, 274 GDP, 70, 188-189 Gender, 212-214 Geography, Germany, 280 Gini coefficient, 393-396, 404 Global warming, GNP, 192 Gold, 434, 465-467, 473 Gold standard, 465 Golden Rule, 280, 282, 284, 422 Goods, 8, 44, 45, 76, 83, 102, 109, 113, 125, 128-130, 132, 219, 233, 245, 272, 281, 287, 292, 317-318, 322, 345, 349-353, 358, 363, 378, 402, 404, 427, 429, 434, 466 basket of, 429 complementary, 358 free, 353 public, 292 substitute, 76, 125, 318, 358 Government, 7, 59, 79-80, 112-113, 122, 127-128, 131, 161, 163, 188-189, 230, 276, 279, 281, 283, 298, 307, 317, 425-426, 435, 443 Government purchases, 79-80 Government spending, 7, 112, 122, 127-128, 131, 230, 425-426, 435, 443 multiplier, 128, 425-426 Graphs, 11, 16, 23-24, 29, 36-37, 39, 75, 155, 161, 272, 329, 421, 456, 483 Great Depression, 5, 184-185, 253 Greece, 280 Greenhouse gases, 388 Gross domestic product, 70 Group, 46, 70, 212, 291 groups, 212, 214 Growth accounting, 233 Growth rate, 46-47, 49, 54-55, 66-67, 178, 180, 188-189, 196, 378, 393, 403 H Health care, 70, 253 Health care costs, 70 Health insurance, 285 Heckscher-Ohlin theory, 128-129 Home country, 474 Human capital, 157, 219 Hungary, 280, 394 Hyperinflation, 199, 279-280, 429-430, 468 hypothesis, 431 I III, 43, 51, 226-227, 246, 341, 487 illustration, 27, 119 Image, 196 country, 196 Implementation, 115 Imports, 97-98, 109, 129, 131, 167-168, 196-197, 240, 285, 403, 434, 484 Inc., 1, 3, 11, 45, 73, 75, 115, 143, 145, 173, 211, 255, 257, 287, 317, 361, 363, 407, 451 Income, 5-6, 11-12, 14-15, 18, 23, 27, 33, 45-50, 54, 70, 79-80, 86, 97-98, 112, 122, 127-128, 131-132, 145, 157-159, 161-162, 164-165, 170-171, 188, 192, 197, 228, 230, 236-237, 240-241, 251-253, 257, 267-268, 270, 272, 279-281, 284, 294, 297-299, 317, 323, 345-348, 350, 352-353, 358, 362, 363, 378, 386-387, 390-396, 403, 421-423, 425-426, 430, 433-435, 441-443, 446, 459, 484 decrease in, 33, 80, 131, 294, 347, 350, 353, 390-391 differences in, 45, 128, 422-423 disposable, 230 increase in, 33, 128, 131-132, 145, 157-159, 164, 170, 192, 228, 240-241, 251-252, 267, 279, 284, 345, 347, 390, 425, 435, 484 market, 5-6, 33, 79, 127, 197, 284, 294, 363, 425, 433 national, 5, 12, 45, 50, 79, 86, 97, 112, 122, 127, 131-132, 145, 157-158, 165, 170-171, 188, 236-237, 240-241, 253, 281, 284, 297, 347, 392-393, 395, 422, 441, 459 per capita, 5, 45-48, 54, 70, 157-159, 192, 197, 253, 257, 267-268, 281, 284, 294, 297, 459 permanent, 347, 363, 392-393, 422-423, 426, 441 personal, 33, 162, 251 Income effect, 157 Income inequality, 362, 393, 396 Income tax, 161-162, 279, 298-299 Income tax revenues, 299 Independent variables, 8-9, 16-17, 217, 245, 294, 408, 451 India, 128, 158-159, 165, 218-220, 229, 396 Indifference curves, 253, 321-322 Indonesia, 49, 59 Industry, 82-84, 110, 130, 275-276 Inequality, 31, 65, 303, 317-318, 354, 356-357, 359, 362, 393-394, 396 Inflation, 3, 49, 67-69, 71, 184, 189, 199, 231-233, 276-280, 282, 298-299, 307, 362, 391-393, 403, 430-432, 467-469, 473, 484-485 anticipated, 391 costs of, 391 expectations of, 278 hyperinflation, 199, 279-280, 430, 468 unanticipated, 277 unemployment and, 184, 232-233, 277 Inflation expectations, 430 Inflation rate, 49, 67-69, 71, 199, 231, 276-279, 298-299, 392-393, 430-432, 467-468 Information, 34, 67, 94, 105, 203, 270, 292, 318, 358, 402, 467 Inheritance, 58 Innovation, 277 Input costs, 358 Insurance, 285 Integration, 361-362, 363-364, 366, 368-374, 376, 378-388, 390, 392-396, 398, 400, 402-405, 408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428-430, 432, 434, 436, 438, 440, 442, 444, 446, 448, 452, 454, 456, 458-464, 466, 468, 470, 472, 474, 476, 478, 480, 482, 484, 486, 488 Interest, 1, 50-60, 62, 70-71, 75, 79-80, 97, 122, 127, 131-132, 170-171, 180, 182, 196, 220, 230, 272, 279, 345, 347-348, 352, 364-365, 372-373, 376-378, 386, 388-389, 391-392, 401-403, 430, 432, 470-471, 474, 484 Interest rate, 51, 53-60, 70, 75, 79-80, 97, 131-132, 170-171, 180, 220, 230, 272, 279, 345, 347-348, 352, 364-365, 372-373, 376-378, 386, 388-389, 391-392, 401, 430, 432, 470-471, 474, 484 current, 55, 57, 364, 432, 470, 474 risk, 474 Interest rates, 1, 53, 55-56, 59, 71, 122, 127, 171, 180, 182, 196, 389, 401-402 money demand and, 180 nominal, 180 real, 53, 122, 127, 171 International capital, 429, 470 International trade, 4, 109, 114, 128-129, 141, 402, 433 nature of, theories of, 128-129 Internet, 363 Inventory, 272 Investment, 7, 10, 54, 59, 79, 86, 97, 112, 122, 127, 131, 157, 170, 180, 182, 219, 230, 255, 281, 284, 362, 387-388, 401-402, 407, 422, 441, 459 government, 7, 59, 79, 112, 122, 127, 131, 230, 281 gross, 422 net, 281, 402, 422 Investment spending, 281 Investors, 58, 70, 220, 407, 432-433, 470 Israel, 171 J Japan, 240 exports, 240 Job creation, 82-84, 86-87 Job loss, 82 Jobs, 82, 157 levels, 157 service, 157 K Knowledge, 36 L Labor, 4, 45, 59-60, 64-65, 71, 83-84, 86, 109-110, 113, 128-130, 132, 144, 157-159, 161, 165, 171, 196, 198, 209, 212, 218, 220, 224-225, 229, 233-235, 237, 241, 249, 252-253, 281-285, 290, 294, 299, 318, 323, 328, 331, 342, 344-345, 349, 357-358, 402 trends in, 235 Labor demand, 83-84, 86, 224-225 labor force, 45, 157-159, 165, 234, 282-284 Labor market, 212 Labor supply, 83-84, 157, 323, 349 Latin America, 46 letters, 8, 12, 14, 63, 76, 89, 122, 215 Level of economic development, 157 Liabilities, 185 Liquidity, 333-334, 339, 354, 432 Living standards, 267 Loans, 55 London, 465 Long-run average cost, 328-329 Long-run average cost curve, 329 Long-run total cost, 332 Loss, 82, 277-278, 393 expected, 277-278 income, 393 reduction, 393 M Macroeconomic policy, 184 Macroeconomics, 4, 7, 115, 141, 144, 282, 345, 361, 451, 488 use of, 4, 115, 361 Malaysia, 268 Manufacturers, 113 Manufacturing, 33, 83, 157, 299, 422 Manufacturing employment, 83 Manufacturing firms, 299 Margin, 18, 144, 145, 153-154, 171, 173, 199 Marginal benefit, 217, 328, 354 Marginal change, 164, 198 Marginal cost, 4, 44, 160, 164, 173, 180, 195, 199, 273-274, 276, 282-283, 291, 293-294, 364, 378 average cost and, 195 Marginal propensity to consume, 79, 112, 154 Marginal rate of substitution, 44, 252 Marginal revenue, 154, 160, 164, 173, 179-180, 282, 363-364, 378 Marginal utility, 29, 36, 143, 173, 198-202, 209, 243, 247, 328 Market demand, 283, 292-293 Market economies, Market equilibrium, 85, 127 Market share, 276 Market value, 59 Markets, 4-5, 9, 161, 290-291, 299, 429 Markup, 294 Massachusetts, 300-301, 307 Matrices, 73, 75, 87, 89-90, 93-94, 96-102, 105, 107-108, 111, 114, 115, 117-119, 121, 308, 315, 474 meaning, 170, 179 Measurement, 62 message, 268 negative, 268 positive, 268 Mexico, 68-70, 159, 165, 268, 394 Microeconomics, 4, 7, 33, 144, 160, 179, 361, 363, 390 Minors, 118, 123-124, 308-313, 315, 337-340 Monetary policy, 211, 232-233, 240, 276-277, 487 expansionary, 211, 232 Money, 7, 62, 64, 71, 79, 122, 127, 145, 180, 182, 184-186, 189, 228, 253, 272, 276, 278-279, 328, 347, 350, 353, 391-392, 429-432, 435, 443, 465-474, 484-485, 487 demand for, 79, 228, 272, 429 M1, 435 properties of, 62, 429 Money demand, 7, 122, 180, 272, 279, 391-392, 429-431, 467-469, 471, 473-474, 484, 487 inflation and, 279, 392, 468 transaction, 272 Money market, 71, 79, 127 Money market account, 71 Money multiplier, 184-186, 228, 253 Money supply, 122, 127, 145, 180, 184-186, 189, 253, 279, 430-431, 435, 443, 465-474, 484-485, 487 changes in, 127, 145, 180, 189 Monopolistic competition, 352 Monopoly, 273, 276 Mozambique, 171 Multipliers, 328, 354, 358 money, 328 using, 328, 358 Museums, 290 N Nash equilibrium, 274 National income, 5, 12, 45, 50, 79, 86, 97, 112, 122, 127, 131-132, 145, 158, 165, 170-171, 188, 236-237, 240-241, 253, 281, 284, 297, 392-393, 395, 422, 441, 459 measuring, 393 National production, 237, 241 Nations, 195 Natural rate of unemployment, 277, 487 Negative relationship, 148, 277, 283 Net present value, 378, 402 NPV, 402 Netherlands, 429 Newspapers, Nominal interest rate, 180, 279, 391, 430, 471, 474, 484 NPV, 402 O Occurrence, 398 OECD, 130 Organization for Economic Cooperation and Development, 130 Offer, 5, 49, 118, 341, 378, 388-389, 471 Offsets, 155 Oil, 12, 211 Open economy, 434 Operations, 88, 98 opinion polls, 391 Opportunities, 157 Opportunity cost, 279, 471 of holding money, 279, 471 Opportunity costs, 317 Organization, 130, 292 Organization for Economic Cooperation and Development, 130 OECD, 130 Output, 11, 21-22, 43-44, 59, 64-65, 71, 75, 84, 110, 113, 129-130, 160, 164, 180, 182, 196, 209, 218-220, 229, 233-234, 241, 245, 252, 272, 274, 276, 281-285, 287, 290, 299, 301, 315, 317-318, 331-332, 342, 344, 357-358, 388, 484 equilibrium, 75, 84, 274, 283 potential, 22, 220, 276, 484 relationship between labor and, 290 P Pakistan, 70 Parameter, 8, 14, 16, 36, 41, 83, 87, 90-91, 151, 170, 178, 196, 200, 219, 237, 277, 300, 330-332, 341, 347, 379, 391, 398, 401, 408, 441, 454, 485 parentheses, 15, 53, 118, 128, 134, 150, 166, 174, 185, 272, 303, 327, 331 Participation rate, 158-159, 165 Patent, 277 Patents, 277 Per-capita income, 70 Percentage changes, 178, 189, 217 Perception, 186 Perfect competition, 110, 282 Performance, 5, 251, 278 performance reviews, 251 Permanent income, 347, 363, 426, 441 Permits, 287, 318 Personal income, 33, 162 Place, 3, 219, 293, 410, 426, 452 Plans, 277 Poland, 280 inflation, 280 Policies, 146, 195, 276, 391, 465 Poor countries, 5, 218-220, 422-423 Population, 12, 48-50, 55, 59, 66, 284, 393-395 Portfolio, 272 Model, 272 Poverty, 5, 396, 422-423 Power, 5, 36-39, 43, 47, 118, 173-174, 176, 182-184, 273, 291, 380, 382-383 491 PPP, 429 Premium, 213, 474 Present value, 45, 56-59, 71, 346, 361, 364-366, 372-373, 377-378, 380, 385-386, 388-389, 393, 401-403, 405 Presidential elections, 231 Price, 3-6, 8-9, 12, 23, 26-27, 33-34, 45, 56-57, 59, 62-63, 67-69, 71, 75-82, 85, 102, 110-111, 113, 125, 146, 152, 160, 180, 189-191, 193-195, 197-199, 218, 228-230, 251-252, 271-274, 279, 282-285, 290-294, 299-301, 322-323, 326, 328, 330, 332, 361, 363-365, 377-378, 385, 389-390, 402-403, 407, 416, 425, 427, 429-435, 443, 465-471, 473-474, 484-487 defined, 4, 152, 160, 189, 198, 228, 328, 385, 403, 484 price discrimination, 273 price elasticity, 4, 189, 228, 294 Price discrimination, 273 Price elasticity, 4, 189, 228, 294 Price elasticity of demand, 228, 294 Price level, 12, 62, 67, 69, 75, 79, 180, 189, 199, 218, 279, 430-431, 434-435, 465-471, 473-474, 484-487 money demand and, 180 Price ratio, 322 Prices, 5, 9, 57, 64, 67-69, 81, 110, 113, 130, 189-190, 193, 197-199, 211, 273, 277, 283, 290-291, 304-305, 345, 350, 362, 389, 429-430, 432-433, 435, 446, 465, 467, 469, 473 equilibrium, 81, 193, 198, 283, 429 flexible, 110, 469 input, 113, 198 maximum, 304-305, 362 minimum, 304-305 wages and, 110, 277 Pricing, 57, 273, 291, 432 dynamic, 432 payment, 57 strategy, 291 value, 57, 432 Principal, 50-51, 54-55, 128, 308-313, 315, 337-340, 345 Principles, Probability, 204, 292, 362, 396-401, 403-404 objective, 362 Production, 5-6, 8-9, 11, 22, 32, 43-44, 59, 64, 71, 76-77, 109-110, 113, 128-130, 196-198, 202, 209, 218-221, 224-225, 229, 233-238, 241, 245, 249, 252, 255, 281, 283-285, 290, 293, 299, 307, 317, 324, 328, 332, 342, 344-345, 352, 357, 388, 402-403, 407, 421-422, 459 national, 5, 233, 236-237, 241, 281, 284, 422, 459 Production function, 5, 11, 43-44, 59, 64, 71, 196, 202, 209, 218-221, 224-225, 229, 233, 235-238, 241, 245, 249, 252, 281, 283-285, 290, 299, 332, 342, 344-345, 352, 357, 421-422, 459 Productivity, 33, 49, 59-60, 154, 161, 198, 211, 218-220, 224-225, 229, 350 labor, 59-60, 161, 198, 218, 220, 224-225, 229 Products, 62, 99-100, 116-117, 173, 218, 237, 249, 283, 290, 292, 300, 352 defined, 99-100 levels of, 283 Profit, 7, 110, 113, 179-180, 257, 272-274, 276, 282-285, 291-294, 299-301, 305, 328, 332, 364, 470 Profits, 26, 49, 110, 113, 257, 272-274, 276, 282-283, 285, 290-291, 294, 299-301, 304-305, 307, 315, 332 projection, 320 Property, 25-26, 62, 65, 69, 71, 88, 90, 100, 104-105, 108, 116, 136, 235-236, 238, 249-250, 328, 379, 424, 439, 444, 475, 484 Prosperity, Protection, 109, 277 Psychology, 324 Public choice, Public opinion, 391 Purchasing, 318, 389 purpose, 6-7, 43, 163 general, 7, 163 specific, 7, 43 Q Quality, 268, 294 Quantity demanded, 8, 23, 27, 80-82, 197, 228-229, 283, 403, 425 492 Quantity supplied, 8, 80-82, 195, 274, 402-403, 425 R Race, 212, 271 Rate of return, 218, 220 Rates, 1, 3, 5, 35, 45-47, 50, 52-56, 58-59, 66-67, 70-71, 83, 122, 127, 145-150, 155, 158, 161, 164-167, 171, 178, 180, 182, 188-189, 192, 196, 200, 218, 233, 258, 278, 280, 299, 389, 401-402, 429, 432, 470, 484, 486 definition of, 53, 55 gross, 70, 83 Rational expectations, 431, 467-468, 478 Ratios, 12, 323, 327, 348-349 Reach, 155 readability, Real exchange rates, 429 Real GDP, 189 Real interest rate, 170, 230, 279, 345, 388, 392, 430, 471, 474 Real interest rates, 171 Real wages, 109, 218 Recession, 398 Reform, 388 Relationships, 3, 5, 7, 21, 32, 62, 71, 73, 76, 88, 115, 132, 138-141, 143-144, 154, 180, 185, 196, 211, 240, 277, 281, 451, 458-459, 465, 471 preferences, 277 Relative cost, 347 reports, 15, 149, 394 Representations, 401 research, 3, 5-6, 111, 185, 220, 349 primary, 111, 185 Reserve ratio, 186, 253 Reserves, 185 Resources, 82, 130, 271, 279, 317 Restricted, 16, 43, 61 Restrictions, 209 Retirement, 347-348, 352 Revenue, 7, 63, 146-152, 154-156, 160, 162-165, 173, 179-180, 199, 251, 258-259, 263, 272-273, 282, 284-285, 291, 298-299, 301, 363-364, 378, 388 marginal, 154-156, 160, 162-165, 173, 179-180, 199, 272-273, 282, 284, 291, 363-364, 378 Revenues, 9, 44, 110, 145-149, 155, 163-165, 258-259, 272, 279-280, 283, 293-294, 298-299, 307, 317, 324 Risk, 204-206, 210, 474 asset, 474 interest rate, 474 Rivalry, 276 Role, 4, 45, 66, 71, 100, 276, 292, 294, 361, 407 Rules of thumb, 67 Russia, 280 S Salaries, 110 Salary, 49, 110, 251, 285, 378, 425 Sales, 9, 146, 155, 299, 307 Sales tax, 146 Samples, 396 Saving, 280, 345, 350 increase in, 345 Scarcity, 256 scope, 345, 440, 480 Securities, 57 Security, 22 Selection, 320 Sellers, 80 Sensitivity, 5, 82 Services, 45, 233, 272, 292, 349, 378, 422 differentiation, 378 Ships, 434 SIMPLE, 6, 8, 19, 22, 26, 29, 34, 36, 41, 66, 73, 76, 79, 86, 90, 98, 105-106, 109, 114, 131, 134, 137, 139, 146-147, 162, 170, 176, 180-182, 192, 197, 199, 211-212, 214, 218, 231, 236, 364, 379, 381, 386, 401, 407, 426, 428, 441, 449, 468-469, 471 SIR, 13 Size, 45, 84, 86, 146-147, 149-151, 167, 218, 237, 349, 351 Skilled labor, 109-110, 128, 130 Skills, 17, 44, 219-220 Slope, 16, 28-29, 35, 44, 68-69, 146-149, 154-159, 161-166, 191, 195, 200, 207, 209, 212, 220, 225, 244, 247-250, 252-253, 282, 290, 297, 321-322, 332, 466, 468-469, 471, 483, 485, 487 Smoke, 267-268, 270 Social Security, 22 Societies, 394 Society, 394 Spillover effects, 83 Standard of living, 45 statistics, 218, 225, 272, 362, 396, 399, 401, 441 Steady state, 7, 140, 410-411, 413, 415, 418-419, 421-428, 433-436, 442-443, 448-449, 451-459, 463-466, 468-469, 471, 473, 477-481, 483-487 Steel industry, 130 Stock, 12, 59, 69, 184-186, 218, 224, 234, 253, 281-282, 284, 353, 387, 416, 421-422, 432-433, 435, 458, 465-467, 473 Stock of capital, 387, 458 Strategic trade policies, 276 Strategic trade policy, 276 Strategy, 10, 62, 68, 132, 274-275, 291, 300, 342, 379, 475 global, 275 Stress, 23 Students, 3, 271, 324, 342-343 Subsidies, 276 Substitution, 44, 75, 77, 79, 85, 87, 98, 157, 252, 318-320, 323-325, 327, 344-345, 352, 358, 381-382, 387, 390, 407-409, 414, 441 superscripts, 19 Supply, 6-8, 33-34, 73, 75-78, 80-84, 114, 122, 127, 130, 145, 157, 180, 184-186, 189, 192, 194, 198, 253, 274-276, 279, 323, 330, 349, 402-403, 425, 429-431, 435, 443, 465-474, 484-485, 487 aggregate, 75, 425, 485 currency, 184-186, 253, 466, 470-471, 484 excess, 402-403, 429, 470, 474, 485 inelastic, 274 long-run, 429, 431, 435, 468 of capital, 130, 470 of labor, 83, 402 of money, 180, 184, 279, 429, 465, 468-469 Supply and demand, 73, 76, 78 Supply curve, 33, 75, 78, 402 aggregate, 75 labor, 402 market, 33, 402 rightward shift, 33 Support, 176 Surplus, 9, 188, 362, 363, 389-392, 402-403, 434, 465, 467 consumer, 9, 362, 363, 389-392, 402-403 producer, 402-403 total, 9, 363, 402 surveys, 212 Sustainability, 188 Sweden, 55, 197 system, 55, 73-74, 75, 85, 90-92, 94, 96-98, 102-103, 105-111, 113-114, 115, 121-122, 124-127, 129, 131-132, 135-141, 161-162, 211, 267, 325, 335, 426, 437, 443-448, 474-481, 483-487 T Tables, 46 Tariff, 145, 166-168, 402-403 Tariffs, 402 Tax rates, 145-150, 155, 161, 164-165, 200, 233, 258 Tax revenues, 9, 44, 145-149, 163, 165, 258-259, 279, 283, 299 Tax system, 161-162 Taxes, 146-147, 163-164, 233, 279, 298 consumption, 163 income, 164, 279, 298 sales, 146 Technological advances, 235 Technological progress, 233-235, 284 Technology, 49, 64-65, 82, 110, 129, 196, 233 Tenure, 59-60, 198, 212-214, 216 Terminal value, 408, 433, 452, 462-463 The Economist, 236, 328 Time dimension, Total cost, 7, 20-22, 160, 164, 180, 195, 272-273, 282-283, 291, 300-301, 307, 332, 341, 378 short-run, 332 Total costs, 272, 285 Total revenue, 7, 152, 164, 179-180, 272-273, 282, 284-285, 291, 301, 363-364 Total utility, 346, 352 Trade, 4, 6, 9, 44, 49, 109, 111, 114, 128-129, 141, 196, 240, 272, 276, 280-281, 317, 402, 429, 433-434, 465-467 deficit, 196, 434, 465 domestic, 129, 196, 402, 429, 434, 465-466 strategic trade policy, 276 surplus, 9, 402, 434, 465, 467 Trade balance, 196, 240, 434, 466-467 Trade deficit, 434, 465 Trade policies, 276 Trade surplus, 434, 465 Trade-offs, 9, 317 Training, 3, 211-212 methods for, 3, 211 Transactions, 272 consistent, 272 Treasury bonds, 59 Trends, 109, 235 Web, 363-364 Web site, 363-364 Widgets, 6-9, 76-78, 80, 86, 92, 102, 108, 115-116, 146, 155, 193, 197-198 Women, 157-159, 165, 171, 212-214, 294, 351 Won, 12-13, 22, 280 Work, 94, 157, 165, 171, 206, 217, 235, 271, 274, 280, 323, 343, 350-351, 353, 433 Workers, 22, 82-84, 86, 107, 109-111, 213, 219, 224, 285 skilled, 107, 109-111 unskilled, 107, 109-111 workforce, 217 workplace, 94, 217, 228 World, 46, 49, 192, 211, 267, 281-282, 403, 433, 459, 473-474 World Bank, 46, 192 U Z Unemployed, 12 Unemployment, 82, 184, 231-233, 277-278, 487 natural rate of, 277, 487 Unemployment rate, 231, 487 United States, 49, 70-71, 109, 111, 128, 185, 218-220, 229, 235, 237, 240, 282, 391-392, 394, 473-474 international trade in, 128 Unskilled labor, 109-110, 128, 130 uppercase letters, 122 U.S, 59, 70, 109, 113, 229, 236, 240, 285, 349, 429 U.S., 59, 70, 109, 113, 229, 236, 240, 285, 349, 429 U.S dollar, 240, 429 U.S economy, 429 Utilities, 322, 347, 351 Utility, 7, 29, 36, 59, 70, 143, 173, 198-202, 204-206, 209-210, 242-245, 247-248, 252-253, 257, 291, 304, 317, 320-324, 326, 328, 333-334, 340-341, 343, 346-353, 357-358 Y Yen, 12, 464 ZIP codes, 35 V Value, 4, 8-9, 11-20, 23, 25-28, 31, 33, 35-36, 39-41, 45-62, 65, 69, 71, 76-77, 81, 85, 101, 106, 111, 115, 132, 139-140, 143, 145, 147-155, 158-159, 162-169, 171, 174, 176, 178, 181, 185, 188, 190, 196, 199, 201-202, 205, 207-208, 210, 211, 213-214, 219-220, 226-228, 235-237, 241-243, 251-252, 255-256, 257-263, 265-267, 270, 272, 275, 278-280, 285, 288-289, 292, 294-295, 302-303, 306, 309, 313, 317-320, 324-325, 327-328, 330-332, 337, 346, 349-351, 353-358, 361-362, 363-370, 372-378, 380, 385-386, 388-389, 392-394, 397, 399, 401-405, 407-408, 410-416, 418-419, 421-422, 424-429, 431-437, 441, 443, 446, 448, 451-457, 461-464, 466, 468-472, 474, 479, 484, 487-488 building, 11, 408 defined, 4, 28, 53, 152, 154, 174, 178, 228, 242, 257, 262, 265, 313, 328, 358, 369-370, 372, 375-376, 385, 399, 403, 411, 484 market value, 59 Variables, 1, 4, 6-10, 11-12, 15-17, 27, 33, 43, 45, 47-48, 62-64, 66-68, 71, 73, 75-77, 79, 85, 87, 89-92, 96-97, 102, 105, 108, 112-113, 115, 122-125, 131, 137, 139-141, 143-144, 145, 147, 151, 166, 188-189, 191-192, 211, 214-215, 217, 230-233, 240, 245, 268, 294, 297, 308, 317, 320, 323, 325, 330-333, 336-337, 339-340, 361, 396-397, 407-408, 421, 425, 443, 445-446, 448, 451, 458, 474-479, 483, 488 Variance, 297, 401, 403-404 Venezuela, 197 Volume, 429 W Wages, 45, 82-83, 107, 109-111, 152, 157, 212-214, 218, 277, 284, 294, 342, 351 differences in, 45, 212, 214, 218 efficiency, 284 equilibrium, 82 real, 45, 109, 218 Water, 268 Wealth, 17-18, 347, 349, 353 493 ... One of Mathematical Methods for Economics, Second Edition Michael W Klein Copyright © 2002 by Pearson Education, Inc All rights reserved This page intentionally left blank Chapter The Mathematical. .. mathematics in economics helps develop mathematical comprehension as well as hone economic intuition In this W From Chapter of Mathematical Methods for Economics, Second Edition Michael W Klein Copyright... Chapter The Mathematical Framework of Economic Analysis It is typically the case that the assumptions of a formal mathematical model are explicit and transparent Therefore a formal mathematical