Schaum's Outline of Introduction to Mathematical Economics (2000) PDF

 SCHAUM’S OUTLINE OF Theory and Problems of INTRODUCTION TO MATHEMATICAL ECONOMICS Third Edition EDWARD T. DOWLING, Ph.D....

 SCHAUM’S OUTLINE OF Theory and Problems of INTRODUCTION TO MATHEMATICAL ECONOMICS Third Edition EDWARD T. DOWLING, Ph.D. Professor and Former Chairman Department of Economics Fordham University Schaum’s Outline Series McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogotá Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto  To the memory of my parents, Edward T. Dowling, M.D. and Mary H. Dowling EDWARD T. DOWLING is professor of Economics at Fordham University. He was Dean of Fordham College from 1982 to 1986 and Chairman of the Economics Department from 1979 to 1982 and again from 1988 to 1994. His Ph.D. is from Cornell University and his main areas of professional interest are mathematical economics and economic development. In addition to journal articles, he is the author of Schaum’s Outline of Calculus for Business, Economics, and the Social Sciences, and Schaum’s Outline of Mathematical Methods for Business and Economics. A Jesuit priest, he is a member of the Jesuit Community at Fordham. Copyright © 2001, 1992 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-161015-5 MHID: 0-07-161015-4 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-135896-5, MHID: 0-07-135896-X. All trademarks are trademarks of their respective owners. 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This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.  PREFACE The mathematics needed for the study of economics and business continues to grow with each passing year, placing ever more demands on students and faculty alike. Introduction to Mathematical Economics, third edition, introduces three new chapters, one on comparative statics and concave programming, one on simultaneous differential and difference equations, and one on optimal control theory. To keep the book manageable in size, some chapters and sections of the second edition had to be excised. These include three chapters on linear programming and a number of sections dealing with basic elements such as factoring and completing the square. The deleted topics were chosen in part because they can now be found in one of my more recent Schaum books designed as an easier, more detailed introduction to the mathematics needed for economics and business, namely, Mathematical Methods for Business and Economics. The objectives of the book have not changed over the 20 years since the introduction of the first edition, originally called Mathematics for Economists. Introduction to Mathematical Economics, third edition, is designed to present a thorough, easily understood introduction to the wide array of mathematical topics economists, social scientists, and business majors need to know today, such as linear algebra, differential and integral calculus, nonlinear programming, differential and difference equations, the calculus of variations, and optimal control theory. The book also offers a brief review of basic algebra for those who are rusty and provides direct, frequent, and practical applications to everyday economic problems and business situations. The theory-and-solved-problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. The topics and related problems range in difficulty from simpler mathematical operations to sophisticated applications. No mathematical proficiency beyond the high school level is assumed at the start. The learning-by-doing pedagogy will enable students to progress at their own rates and adapt the book to their own needs. Those in need of more time and help in getting started with some of the elementary topics may feel more comfortable beginning with or working in conjunction with my Schaum’s Outline of Mathematical Methods for Business and Economics, which offers a kinder, gentler approach to the discipline. Those who prefer more rigor and theory, on the other hand, might find it enriching to work along with my Schaum’s Outline of Calculus for Business, Economics, and the Social Sciences, which devotes more time to the theoretical and structural underpinnings. Introduction to Mathematical Economics, third edition, can be used by itself or as a supplement to other texts for undergraduate and graduate students in economics, business, and the social sciences. It is largely self-contained. Starting with a basic review of high school algebra in Chapter 1, the book consistently explains all the concepts and techniques needed for the material in subsequent chapters. Since there is no universal agreement on the order in which differential calculus and linear algebra should be presented, the book is designed so that Chapters 10 and 11 on linear algebra can be covered immediately after Chapter 2, if so desired, without loss of continuity. This book contains over 1600 problems, all solved in considerable detail. To get the most from the book, students should strive as soon as possible to work independently of the solutions. This can be done by solving problems on individual sheets of paper with the book closed. If difficulties arise, the solution can then be checked in the book. iii iv PREFACE For best results, students should never be satisfied with passive knowledgeᎏthe capacity merely to follow or comprehend the various steps presented in the book. Mastery of the subject and doing well on exams requires active knowledgeᎏthe ability to solve any problem, in any order, without the aid of the book. Experience has proved that students of very different backgrounds and abilities can be successful in handling the subject matter presented in this text if they apply themselves and work consistently through the problems and examples. In closing, I would like to thank my friend and colleague at Fordham, Dr. Dominick Salvatore, for his unfailing encouragement and support over the past 25 years, and an exceptionally fine graduate student, Robert Derrell, for proofreading the manuscript and checking the accuracy of the solutions. I am also grateful to the entire staff at McGraw-Hill, especially Barbara Gilson, Tina Cameron, Maureen B. Walker, and Deborah Aaronson. EDWARD T. DOWLING  CONTENTS CHAPTER 1 Review 1 1.1 Exponents. 1.2 Polynomials. 1.3 Equations: Linear and Quadratic. 1.4 Simultaneous Equations. 1.5 Functions. 1.6 Graphs, Slopes, and Intercepts. CHAPTER 2 Economic Applications of Graphs and Equations 14 2.1 Isocost Lines. 2.2 Supply and Demand Analysis. 2.3 Income Determination Models. 2.4 IS-LM Analysis. CHAPTER 3 The Derivative and the Rules of Differentiation 32 3.1 Limits. 3.2 Continuity. 3.3 The Slope of a Curvilinear Function. 3.4 The Derivative. 3.5 Differentiability and Continuity. 3.6 Derivative Notation. 3.7 Rules of Differentiation. 3.8 Higher-Order Derivatives. 3.9 Implicit Differentiation. CHAPTER 4 Uses of the Derivative in Mathematics and Economics 58 4.1 Increasing and Decreasing Functions. 4.2 Concavity and Convexity. 4.3 Relative Extrema. 4.4 Inflection Points. 4.5 Optimization of Functions. 4.6 Successive-Derivative Test for Optimization. 4.7 Marginal Concepts. 4.8 Optimizing Economic Functions. 4.9 Relationship among Total, Marginal, and Average Concepts. CHAPTER 5 Calculus of Multivariable Functions 82 5.1 Functions of Several Variables and Partial Derivatives. 5.2 Rules of Partial Differentiation. 5.3 Second-Order Partial Derivatives. 5.4 Optimization of Multivariable Functions. 5.5 Constrained Optimization with Lagrange Multipliers. 5.6 Significance of the Lagrange Multiplier. 5.7 Differentials. 5.8 Total and Partial Differentials. 5.9 Total Derivatives. 5.10 Implicit and Inverse Function Rules. CHAPTER 6 Calculus of Multivariable Functions in Economics 110 6.1 Marginal Productivity. 6.2 Income Determination Multipliers and Comparative Statics. 6.3 Income and Cross Price Elasticities of Demand. 6.4 Differentials and Incremental Changes. 6.5 Optimization of Multivariable Functions in Economics. 6.6 Constrained Optimization of Multivariable v vi CONTENTS Functions in Economics. 6.7 Homogeneous Production Functions. 6.8 Returns to Scale. 6.9 Optimization of Cobb-Douglas Production Functions. 6.10 Optimization of Constant Elasticity of Substitution Production Functions. CHAPTER 7 Exponential and Logarithmic Functions 146 7.1 Exponential Functions. 7.2 Logarithmic Functions. 7.3 Properties of Exponents and Logarithms. 7.4 Natural Exponential and Logarithmic Functions. 7.5 Solving Natural Exponential and Logarithmic Functions. 7.6 Logarithmic Transformation of Nonlinear Functions. CHAPTER 8 Exponential and Logarithmic Functions in Economics 160 8.1 Interest Compounding. 8.2 Effective vs. Nominal Rates of Interest. 8.3 Discounting. 8.4 Converting Exponential to Natural Exponential Functions. 8.5 Estimating Growth Rates from Data Points. CHAPTER 9 Differentiation of Exponential and Logarithmic Functions 173 9.1 Rules of Differentiation. 9.2 Higher-Order Derivatives. 9.3 Partial Derivatives. 9.4 Optimization of Exponential and Logarithmic Functions. 9.5 Logarithmic Differentiation. 9.6 Alternative Measures of Growth. 9.7 Optimal Timing. 9.8 Derivation of a Cobb-Douglas Demand Function Using a Logarithmic Transformation. CHAPTER 10 The Fundamentals of Linear (or Matrix) Algebra 199 10.1 The Role of Linear Algebra. 10.2 Definitions and Terms. 10.3 Addition and Subtraction of Matrices. 10.4 Scalar Multiplication. 10.5 Vector Multiplication. 10.6 Multiplication of Matrices. 10.7 Commutative, Associative, and Distributive Laws in Matrix Algebra. 10.8 Identity and Null Matrices. 10.9 Matrix Expression of a System of Linear Equations. CHAPTER 11 Matrix Inversion 224 11.1 Determinants and Nonsingularity. 11.2 Third-Order Determinants. 11.3 Minors and Cofactors. 11.4 Laplace Expansion and Higher-Order Determinants. 11.5 Properties of a Determinant. 11.6 Cofactor and Adjoint Matrices. 11.7 Inverse Matrices. 11.8 Solving Linear Equations with the Inverse. 11.9 Cramer’s Rule for Matrix Solutions.  CONTENTS vii CHAPTER 12 Special Determinants and Matrices and Their Use in Economics 254 12.1 The Jacobian. 12.2 The Hessian. 12.3 The Discriminant. 12.4 Higher-Order Hessians. 12.5 The Bordered Hessian for Constrained Optimization. 12.6 Input-Output Analysis. 12.7 Characteristic Roots and Vectors (Eigenvalues, Eigenvectors). CHAPTER 13 Comparative Statics and Concave Programming 284 13.1 Introduction to Comparative Statics. 13.2 Comparative Statics with One Endogenous Variable. 13.3 Comparative Statics with More Than One Endogenous Variable. 13.4 Comparative Statics for Optimization Problems. 13.5 Comparative Statics Used in Constrained Optimization. 13.6 The Envelope Theorem. 13.7 Concave Programming and Inequality Constraints. CHAPTER 14 Integral Calculus: The Indefinite Integral 326 14.1 Integration. 14.2 Rules of Integration. 14.3 Initial Conditions and Boundary Conditions. 14.4 Integration by Substitution. 14.5 Integration by Parts. 14.6 Economic Applications. CHAPTER 15 Integral Calculus: The Definite Integral 342 15.1 Area Under a Curve. 15.2 The Definite Integral. 15.3 The Fundamental Theorem of Calculus. 15.4 Properties of Definite Integrals. 15.5 Area Between Curves. 15.6 Improper Integrals. 15.7 L’Hôpital’s Rule. 15.8 Consumers’ and Producers’ Surplus. 15.9 The Definite Integral and Probability. CHAPTER 16 First-Order Differential Equations 362 16.1 Definitions and Concepts. 16.2 General Formula for First-Order Linear Differential Equations. 16.3 Exact Differential Equations and Partial Integration. 16.4 Integrating Factors. 16.5 Rules for the Integrating Factor. 16.6 Separation of Variables. 16.7 Economic Applications. 16.8 Phase Diagrams for Differential Equations. CHAPTER 17 First-Order Difference Equations 391 17.1 Definitions and Concepts. 17.2 General Formula for First-Order Linear Difference Equations. 17.3 Stability Conditions. 17.4 Lagged Income Determination Model. 17.5 The Cobweb Model. 17.6 The Harrod Model. 17.7 Phase Diagrams for Difference Equations. viii CONTENTS CHAPTER 18 Second-Order Differential Equations and Difference Equations 408 18.1 Second-Order Differential Equations. 18.2 Second-Order Difference Equations. 18.3 Characteristic Roots. 18.4 Conjugate Complex Numbers. 18.5 Trigonometric Functions. 18.6 Derivatives of Trigonometric Functions. 18.7 Transformation of Imaginary and Complex Numbers. 18.8 Stability Conditions. CHAPTER 19 Simultaneous Differential and Difference Equations 428 19.1 Matrix Solution of Simultaneous Differential Equations, Part 1. 19.2 Matrix Solution of Simultaneous Differential Equations, Part 2. 19.3 Matrix Solution of Simultaneous Difference Equations, Part 1. 19.4 Matrix Solution of Simultaneous Difference Equations, Part 2. 19.5 Stability and Phase Diagrams for Simultaneous Differential Equations. CHAPTER 20 The Calculus of Variations 460 20.1 Dynamic Optimization. 20.2 Distance Between Two Points on a Plane. 20.3 Euler’s Equation and the Necessary Condition for Dynamic Optimization. 20.4 Finding Candidates for Extremals. 20.5 The Sufficiency Conditions for the Calculus of Variations. 20.6 Dynamic Optimization Subject to Functional Constraints. 20.7 Variational Notation. 20.8 Applications to Economics. CHAPTER 21 Optimal Control Theory 493 21.1 Terminology. 21.2 The Hamiltonian and the Necessary Conditions for Maximization in Optimal Control Theory. 21.3 Sufficiency Conditions for Maximization in Optimal Control. 21.4 Optimal Control Theory with a Free Endpoint. 21.5 Inequality Constraints in the Endpoints. 21.6 The Current-Valued Hamiltonian. Index 515  CHAPTER 1 Review 1.1 EXPONENTS Given n a positive integer, xn signifies that x is multiplied by itself n times. Here x is referred to as the base and n is termed an exponent. By convention an exponent of 1 is not expressed: x1 ⫽ x, 81 ⫽ 8. By definition, any nonzero number or variable raised to the zero power is equal to 1: x0 ⫽ 1, 30 ⫽ 1. And 00 is undefined. Assuming a and b are positive integers and x and y are real numbers for which the following exist, the rules of exponents are outlined below and illustrated in Examples 1 and 2 and Problem 1.1. 1 1. xa(xb) ⫽ xa⫹b 6. ⫽ x⫺a xa xa 2. ⫽ xa⫺b 7. 兹x ⫽ x1/2 xb a 3. (xa)b ⫽ xab 8. 兹x ⫽ x1/a b 4. (xy)a ⫽ xa ya 9. 兹xa ⫽ xa/b ⫽ (x1/b)a 冢冣 a x xa 1 5. ⫽ 10. x⫺(a/b) ⫽ y ya xa/b EXAMPLE 1. From Rule 2, it can easily be seen why any variable or nonzero number raised to the zero power equals 1. For example, x3/x3 ⫽ x3⫺3 ⫽ x0 ⫽ 1; 85/85 ⫽ 85⫺5 ⫽ 80 ⫽ 1. EXAMPLE 2. In multiplication, exponents of the same variable are added; in division, the exponents are subtracted; when raised to a power, the exponents are multiplied, as indicated by the rules above and shown in the examples below followed by illustrations in brackets. a) x2(x3) ⫽ x2⫹3 ⫽ x5 ⫽ x6 Rule 1 [x (x ) ⫽ (x · x)(x · x · x) ⫽ x · x · x · x · x ⫽ x ] 2 3 5 6 x b) ⫽ x6⫺3 ⫽ x3 ⫽ x2 Rule 2 x3 6 冤x 冥 x x·x·x·x·x·x 3 ⫽ ⫽ x · x · x ⫽ x3 x·x·x c) (x4)2 ⫽ x4 · 2 ⫽ x8 ⫽ x16 or x6 Rule 3 [(x ) ⫽ (x · x · x · x)(x · x · x · x) ⫽ x ] 4 2 8 1 2 REVIEW [CHAP. 1 d) (xy)4 ⫽ x4 y4 ⫽ xy4 Rule 4 [(xy) ⫽ (xy)(xy)(xy)(xy) ⫽ (x · x · x · x)(y · y · y · y) ⫽ x y ] 4 4 4 5 x5 x5 冢y冣 x x e) ⫽ 5 ⫽ or 5 Rule 5 y y y 5 (x) (x) (x) (x) (x) x5 冤 冢y冣 冥 x ⫽ ⫽ (y) (y) (y) (y) (y) y5 x3 1 f) ⫽ x3⫺4 ⫽ x⫺1 ⫽ ⫽ x3/4 Rules 2 and 6 x4 x x3 冤x 冥 x·x·x 1 4 ⫽ ⫽ x·x·x·x x g) 兹x ⫽ x1/2 Rule 7 Since 兹x · 兹x ⫽ x and from Rule 1 exponents of a common base are added in multiplication, the exponent of 兹x, when added to itself, must equal 1. With 1–2 ⫹ 1–2 ⫽ 1, the exponent of 兹x is 1–2. Thus, 兹x · 兹x ⫽ x1/2 · x1/2 ⫽ x1/2⫹1/2 ⫽ x1 ⫽ x. h) 兹x ⫽ x1/3 3 Rule 8 Just as 兹x · 兹x · 兹x ⫽ x, so x1/3 · x1/3 · x1/3 ⫽ x1/3⫹1/3⫹1/3 ⫽ x1 ⫽ x. 3 3 3 i) x3/2 ⫽ (x1/2)3 or (x3)1/2 Rule 9 [43/2 ⫽ (41/2)3 ⫽ (兹4)3 ⫽ (⫾2)3 ⫽ ⫾8, or equally valid, 43/2 ⫽ (43)1/2 ⫽ (64)1/2 ⫽ 兹64 ⫽ ⫾8] 1 1 1 j) x⫺2/3 ⫽ ⫽ or 2 1/3 Rule 10 x2/3 (x1/3)2 (x ) 冤 27 冥 ⫺2/3 1 1 1 1 1 1 ⫽ ⫽ ⫽ , or equally valid, 27⫺2/3 ⫽ ⫽ ⫽ (271/3)2 (3)2 9 (272)1/3 (729)1/3 9 See Problem 1.1. 1.2 POLYNOMIALS Given an expression such as 5x3, x is called a variable because it can assume any number of different values, and 5 is referred to as the coefficient of x. Expressions consisting simply of a real number or of a coefficient times one or more variables raised to the power of a positive integer are called monomials. Monomials can be added or subtracted to form polynomials. Each of the monomials comprising a polynomial is called a term. Terms that have the same variables and exponents are called like terms. Rules for adding, subtracting, multiplying, and dividing polynomials are explained in Examples 3 through 5 and treated in Problems 1.2 to 1.4. EXAMPLE 3. Like terms in polynomials can be added or subtracted by adding their coefficients. Unlike terms cannot be so added or subtracted. See Problems 1.2 and 1.3. a) 4x5 ⫹ 9x5 ⫽ 13x5 b) 12xy ⫺ 3xy ⫽ 9xy c) (7x3 ⫹ 5x2 ⫺ 8x) ⫹ (11x3 ⫺ 9x2 ⫺ 2x) ⫽ 18x3 ⫺ 4x2 ⫺ 10x d) (24x ⫺ 17y) ⫹ (6x ⫹ 5z) ⫽ 30x ⫺ 17y ⫹ 5z EXAMPLE 4. Like and unlike terms can be multiplied or divided by multiplying or dividing both the coefficients and variables. a) (5x)(13y2) ⫽ 65xy2 b) (7x3 y5)(4x2 y4) ⫽ 28x5 y9 15x4 y3 z6 c) (2x3 y)(17y4 z2) ⫽ 34x3 y5 z2 d) ⫽ 5x2 yz3 3x2 y2 z3 CHAP. 1] REVIEW 3 4x2 y5 z3 y2 e) 5 3 4 ⫽ 3 8x y z 2x z EXAMPLE 5. In multiplying two polynomials, each term in the first polynomial must be multiplied by each term in the second and their products added. See Problem 1.4. (6x ⫹ 7y)(4x ⫹ 9y) ⫽ 24x2 ⫹ 54xy ⫹ 28xy ⫹ 63y2 ⫽ 24x2 ⫹ 82xy ⫹ 63y2 (2x ⫹ 3y)(8x ⫺ 5y ⫺ 7z) ⫽ 16x2 ⫺ 10xy ⫺ 14xz ⫹ 24xy ⫺ 15y2 ⫺ 21yz ⫽ 16x2 ⫹ 14xy ⫺ 14xz ⫺ 21yz ⫺ 15y2 1.3 EQUATIONS: LINEAR AND QUADRATIC A mathematical statement setting two algebraic expressions equal to each other is called an equation. An equation in which all variables are raised to the first power is known as a linear equation. A linear equation can be solved by moving the unknown variable to the left-hand side of the equal sign and all the other terms to the right-hand side, as is illustrated in Example 6. A quadratic equation of the form ax2 ⫹ bx ⫹ c ⫽ 0, where a, b, and c are constants and a ⫽ 0, can be solved by factoring or using the quadratic formula: ⫺b ⫾ 兹b2 ⫺ 4ac x⫽ (1.1) 2a Solving quadratic equations by factoring is explained in Example 7 and by the quadratic formula in Example 8 and Problem 1.6. EXAMPLE 6. The linear equation given below is solved in three easy steps. x x ⫺3 ⫽ ⫹1 4 5 1. Move all terms with the unknown variable x to the left, here by subtracting x/5 from both sides of the equation. x x ⫺3⫺ ⫽ 1 4 5 2. Move any term without the unknown variable to the right, here by adding 3 to both sides of the equation. x x ⫺ ⫽ 1⫹3 ⫽ 4 4 5 3. Simplify both sides of the equation until the unknown variable is by itself on the left and the solution is on the right, here by multiplying both sides of the equation by 20 and subtracting. 冢 4 ⫺ 5 冣 ⫽ 4 · 20 x x 20 · 5x ⫺ 4x ⫽ 80 x ⫽ 80 EXAMPLE 7. Factoring is the easiest way to solve a quadratic equation, provided the factors are easily recognized integers. Given x2 ⫹ 13x ⫹ 30 ⫽ 0 by factoring, we have (x ⫹ 3)(x ⫹ 10) ⫽ 0 4 REVIEW [CHAP. 1 For (x ⫹ 3)(x ⫹ 10) to equal 0, x ⫹ 3 or x ⫹ 10 must equal 0. Setting each in turn equal to 0 and solving for x, we have x⫹3 ⫽ 0 x ⫹ 10 ⫽ 0 x ⫽ ⫺3 x ⫽ ⫺10 Those wishing a thorough review of factoring and other basic mathematical techniques should consult another of the author’s books, Schaum’s Outline of Mathematical Methods for Business and Economics, for a gentler, more gradual approach to the discipline. EXAMPLE 8. The quadratic formula is used below to solve the quadratic equation 5x2 ⫺ 55x ⫹ 140 ⫽ 0 Substituting a ⫽ 5, b ⫽ ⫺55, c ⫽ 140 from the given equation in (1.1) gives ⫺(⫺55) ⫾ 兹(⫺55)2 ⫺ 4(5)(140) x⫽ 2(5) 55 ⫾ 兹3025 ⫺ 2800 55 ⫾ 兹225 55 ⫾ 15 ⫽ ⫽ ⫽ 10 10 10 Adding ⫹15 and then ⫺15 to find each of the two solutions, we get 55 ⫹ 15 55 ⫺ 15 x⫽ ⫽7 x⫽ ⫽4 10 10 See Problem 1.6. 1.4 SIMULTANEOUS EQUATIONS To solve a system of two or more equations simultaneously, (1) the equations must be consistent (noncontradictory), (2) they must be independent (not multiples of each other), and (3) there must be as many consistent and independent equations as variables. A system of simultaneous linear equations can be solved by either the substitution or elimination method, explained in Example 9 and Problems 2.11 to 2.16, as well as by methods developed later in linear algebra in Sections 11.8 and 11.9. EXAMPLE 9. The equilibrium conditions for two markets, butter and margarine, where Pb and Pm are the prices of butter and margarine, respectively, are given in (1.2) and (1.3): 8Pb ⫺ 3Pm ⫽ 7 (1.2) ⫺Pb ⫹ 7Pm ⫽ 19 (1.3) The prices that will bring equilibrium to the model are found below by using the substitution and elimination methods. Substitution Method 1. Solve one of the equations for one variable in terms of the other. Solving (1.3) for Pb gives Pb ⫽ 7Pm ⫺ 19 2. Substitute the value of that term in the other equation, here (1.2), and solve for Pm. 8Pb ⫺ 3Pm ⫽ 7 8(7Pm ⫺ 19) ⫺ 3Pm ⫽ 7 56Pm ⫺ 152 ⫺ 3Pm ⫽ 7 53Pm ⫽ 159 Pm ⫽ 3 CHAP. 1] REVIEW 5 3. Then substitute Pm ⫽ 3 in either (1.2) or (1.3) to find Pb. 8Pb ⫺ 3(3) ⫽ 7 8Pb ⫽ 16 Pb ⫽ 2 Elimination Method 1. Multiply (1.2) by the coefficient of Pb (or Pm) in (1.3) and (1.3) by the coefficient of Pb (or Pm) in (1.2). Picking Pm, we get 7(8Pb ⫺ 3Pm ⫽ 7) 56Pb ⫺ 21Pm ⫽ 49 (1.4) ⫺3(⫺Pb ⫹ 7Pm ⫽ 19) 3Pb ⫺ 21Pm ⫽ ⫺57 (1.5) 2. Subtract (1.5) from (1.4) to eliminate the selected variable. 53Pb ⫽ 106 Pb ⫽ 2 3. Substitute Pb ⫽ 2 in (1.4) or (1.5) to find Pm as in step 3 of the substitution method. 1.5 FUNCTIONS A function f is a rule which assigns to each value of a variable (x), called the argument of the function, one and only one value [ f(x)], referred to as the value of the function at x. The domain of a function refers to the set of all possible values of x; the range is the set of all possible values for f(x). Functions are generally defined by algebraic formulas, as illustrated in Example 10. Other letters, such as g, h, or the Greek letter ␾, are also used to express functions. Functions encountered frequently in economics are listed below. Linear function: f(x) ⫽ mx ⫹ b Quadratic function: f(x) ⫽ ax2 ⫹ bx ⫹ c (a ⫽ 0) Polynomial function of degree n: f(x) ⫽ an xn ⫹ an⫺1 xn⫺1 ⫹ · · · ⫹ a0 (n ⫽ nonnegative integer; an ⫽ 0) Rational function: g(x) f(x) ⫽ h(x) where g(x) and h(x) are both polynomials and h(x) ⫽ 0. (Note: Rational comes from ratio.) Power function: f(x) ⫽ axn (n ⫽ any real number) EXAMPLE 10. The function f(x) ⫽ 8x ⫺ 5 is the rule that takes a number, multiplies it by 8, and then subtracts 5 from the product. If a value is given for x, the value is substituted for x in the formula, and the equation solved for f(x). For example, if x ⫽ 3, f(x) ⫽ 8(3) ⫺ 5 ⫽ 19 If x ⫽ 4, f(x) ⫽ 8(4) ⫺ 5 ⫽ 27 See Problems 1.7 to 1.9. 6 REVIEW [CHAP. 1 EXAMPLE 11. Given below are examples of different functions: Linear: f(x) ⫽ 7x ⫺ 4 g(x) ⫽ ⫺3x h(x) ⫽ 9 Quadratic: f(x) ⫽ 5x ⫹ 8x ⫺ 2 2 g(x) ⫽ x ⫺ 6x 2 h(x) ⫽ 6x2 Polynomial: f(x) ⫽ 4x3 ⫹ 2x2 ⫺ 9x ⫹ 5 g(x) ⫽ 2x5 ⫺ x3 ⫹ 7 x2 ⫺ 9 5x Rational: f(x) ⫽ (x ⫽ ⫺4) g(x) ⫽ (x ⫽ 2) x⫹4 x⫺2 Power: f(x) ⫽ 2x6 g(x) ⫽ x1/2 h(x) ⫽ 4x⫺3 1.6 GRAPHS, SLOPES, AND INTERCEPTS In graphing a function such as y ⫽ f (x), x is placed on the horizontal axis and is known as the independent variable; y is placed on the vertical axis and is called the dependent variable. The graph of a linear function is a straight line. The slope of a line measures the change in y (⌬y) divided by a change in x (⌬x). The slope indicates the steepness and direction of a line. The greater the absolute value of the slope, the steeper the line. A positively sloped line moves up from left to right; a negatively sloped line moves down. The slope of a horizontal line, for which ⌬y ⫽ 0, is zero. The slope of a vertical line, for which ⌬x ⫽ 0, is undefined, i.e., does not exist because division by zero is impossible. The y intercept is the point where the graph crosses the y axis; it occurs when x ⫽ 0. The x intercept is the point where the line intersects the x axis; it occurs when y ⫽ 0. See Problem 1.10. EXAMPLE 12. To graph a linear equation such as y ⫽ ⫺1–4x ⫹ 3 one need only find two points which satisfy the equation and connect them by a straight line. Since the graph of a linear function is a straight line, all the points satisfying the equation must lie on the line. To find the y intercept, set x ⫽ 0 and solve for y, getting y ⫽ ⫺1–4(0) ⫹ 3, y ⫽ 3. The y intercept is the point (x, y) ⫽ (0, 3). To find the x intercept, set y ⫽ 0 and solve for x. Thus, 0 ⫽ ⫺1–4x ⫹ 3, 1–4x ⫽ 3, x ⫽ 12. The x intercept is the point (x, y) ⫽ (12, 0). Then plot the points (0, 3) and (12, 0) and connect them by a straight line, as in Fig. 1-1, to complete the graph of y ⫽ ⫺1–4x ⫹ 3. See Examples 13 and 14 and Problems 1.10 to 1.12. Fig. 1-1 EXAMPLE 13. For a line passing through points (x1, y1) and (x2, y2), the slope m is calculated as follows: ⌬y y2 ⫺ y1 m⫽ ⫽ x1 ⫽ x2 ⌬x x2 ⫺ x1 For the line in Fig. 1-1 passing through (0, 3) and (12, 0), ⌬y 0⫺3 1 m⫽ ⫽ ⫽⫺ ⌬x 12 ⫺ 0 4 and the vertical intercept can be seen to be the point (0, 3). CHAP. 1] REVIEW 7 EXAMPLE 14. For a linear equation in the slope-intercept form y ⫽ mx ⫹ b m, b ⫽ constants the slope and intercepts of the line can be read directly from the equation. For such an equation, m is the slope of the line; (0, b) is the y intercept; and, as seen in Problem 1.10, (⫺b/m, 0) is the x intercept. One can tell immediately from the equation in Example 12, therefore, that the slope of the line is ⫺1–4, the y intercept is (0, 3), and the x intercept is (12, 0). Solved Problems EXPONENTS 1.1. Simplify the following, using the rules of exponents: a) x4 · x 5 x4 · x5 ⫽ x4⫹5 ⫽ x9 b) x7 · x⫺3 x7 · x⫺3 ⫽ x7⫹(⫺3) ⫽ x4 冤x · x 冥 ⫺3 1 1 7 ⫽ x7 · ⫽x·x·x·x·x·x·x· ⫽ x4 x3 x·x·x c) x⫺2 · x⫺4 1 x⫺2 · x⫺4 ⫽ x⫺2⫹(⫺4) ⫽ x⫺6 ⫽ x6 冤x 冥 ⫺2 1 1 1 · x⫺4 ⫽ · ⫽ x · x x · x · x · x x6 d) x2 · x1/2 x2 · x1/2 ⫽ x2⫹(1/2) ⫽ x5/2 ⫽ 兹x5 [x2 · x1/2 ⫽ (x · x)(兹x) ⫽ (兹x · 兹x · 兹x · 兹x)(兹x) ⫽ (x1/2)5 ⫽ x5/2] x9 e) x3 x9 ⫽ x9⫺3 ⫽ x6 x3 x4 f) x7 x4 1 ⫽ x4⫺7 ⫽ x⫺3 ⫽ 3 x7 x x4 冤x 冥 x·x·x·x 1 7 ⫽ ⫽ x · x · x · x · x · x · x x3 8 REVIEW [CHAP. 1 x3 g) x⫺4 x3 ⫽ x3⫺(⫺4) ⫽ x3⫹4 ⫽ x7 x⫺4 x3 x3 冤x ⫺4 ⫽ 1/x4 ⫽ x3 · x4 ⫽ x7 冥 x3 h) 兹x x3 x3 ⫽ ⫽ x3⫺(1/2) ⫽ x5/2 ⫽ 兹x5 兹x x1/2 i) (x2)5 (x2)5 ⫽ x2 · 5 ⫽ x10 j) (x4)⫺2 1 (x4)⫺2 ⫽ x4 · (⫺2) ⫽ x⫺8 ⫽ x8 1 1 k) · x5 y5 1 1 1 · ⫽ x⫺5 · y⫺5 ⫽ (xy)⫺5 ⫽ x5 y 5 (xy)5 x3 l) y3 x3 3 冢冣 x ⫽ y3 y POLYNOMIALS 1.2. Perform the indicated arithmetic operations on the following polynomials: a) 3xy ⫹ 5xy b) 13yz2 ⫺ 28yz2 c) 36x2 y3 ⫺ 25x2 y3 d ) 26x1 x2 ⫹ 58x1 x2 e) 16x2 y3 z5 ⫺ 37x2 y3 z5 a) 8xy, b) ⫺15yz2, c) 11x2 y3, d ) 84x1 x2, e) ⫺21x2 y3 z5 1.3. Add or subtract the following polynomials as indicated. Note that in subtraction the sign of every term within the parentheses must be changed before corresponding elements are added. a) (34x ⫺ 8y) ⫹ (13x ⫹ 12y) (34x ⫺ 8y) ⫹ (13x ⫹ 12y) ⫽ 47x ⫹ 4y b) (26x ⫺ 19y) ⫺ (17x ⫺ 50y) (26x ⫺ 19y) ⫺ (17x ⫺ 50y) ⫽ 9x ⫹ 31y c) (5x ⫺ 8x ⫺ 23) ⫺ (2x ⫹ 7x) 2 2 (5x2 ⫺ 8x ⫺ 23) ⫺ (2x2 ⫹ 7x) ⫽ 3x2 ⫺ 15x ⫺ 23 d) (13x2 ⫹ 35x) ⫺ (4x2 ⫹ 17x ⫺ 49) (13x2 ⫹ 35x) ⫺ (4x2 ⫹ 17x ⫺ 49) ⫽ 9x2 ⫹ 18x ⫹ 49 CHAP. 1] REVIEW 9 1.4. Perform the indicated operations, recalling that each term in the first polynomial must be multiplied by each term in the second and their products summed. a) (2x ⫹ 9)(3x ⫺ 8) (2x ⫹ 9)(3x ⫺ 8) ⫽ 6x2 ⫺ 16x ⫹ 27x ⫺ 72 ⫽ 6x2 ⫹ 11x ⫺ 72 b) (6x ⫺ 4y)(3x ⫺ 5y) (6x ⫺ 4y)(3x ⫺ 5y) ⫽ 18x2 ⫺ 30xy ⫺ 12xy ⫹ 20y2 ⫽ 18x2 ⫺ 42xy ⫹ 20y2 c) (3x ⫺ 7)2 (3x ⫺ 7)2 ⫽ (3x ⫺ 7)(3x ⫺ 7) ⫽ 9x2 ⫺ 21x ⫺ 21x ⫹ 49 ⫽ 9x2 ⫺ 42x ⫹ 49 d) (x ⫹ y)(x ⫺ y) (x ⫹ y)(x ⫺ y) ⫽ x2 ⫺ xy ⫹ xy ⫺ y2 ⫽ x2 ⫺ y2 SOLVING EQUATIONS 1.5. Solve each of the following linear equations by moving all terms with the unknown variable to the left, moving all other terms to the right, and then simplifying. a) 5x ⫹ 6 ⫽ 9x ⫺ 10 b) 26 ⫺ 2x ⫽ 8x ⫺ 44 5x ⫹ 6 ⫽ 9x ⫺ 10 26 ⫺ 2x ⫽ 8x ⫺ 44 5x ⫺ 9x ⫽ ⫺10 ⫺ 6 ⫺2x ⫺ 8x ⫽ ⫺44 ⫺ 26 ⫺4x ⫽ ⫺16 ⫺10x ⫽ ⫺70 x⫽4 x⫽7 x x c) 9(3x ⫹ 4) ⫺ 2x ⫽ 11 ⫹ 5(4x ⫺ 1) d) ⫺ 16 ⫽ ⫹ 14 3 12 9(3x ⫹ 4) ⫺ 2x ⫽ 11 ⫹ 5(4x ⫺ 1) x x 27x ⫹ 36 ⫺ 2x ⫽ 11 ⫹ 20x ⫺ 5 ⫺ 16 ⫽ ⫹ 14 3 12 27x ⫺ 2x ⫺ 20x ⫽ 11 ⫺ 5 ⫺ 36 x x 5x ⫽ ⫺30 ⫺ ⫽ 14 ⫹ 16 3 12 x ⫽ ⫺6 Multiplying both sides of the equation by the least common denominator (LCD), here 12, gives 冢 3 ⫺ 12 冣 ⫽ 30 · 12 x x 12 · 4x ⫺ x ⫽ 360 x ⫽ 120 5 3 7 e) ⫹ ⫽ [x ⫽ 0, ⫺4] x x⫹4 x 5 3 7 ⫹ ⫽ x x⫹4 x Multiplying both sides by the LCD, we get 冢 x ⫹ x ⫹ 4 冣 ⫽ x · x(x ⫹ 4) 5 3 7 x(x ⫹ 4) · 5(x ⫹ 4) ⫹ 3x ⫽ 7(x ⫹ 4) 8x ⫹ 20 ⫽ 7x ⫹ 28 x⫽8 10 REVIEW [CHAP. 1 1.6. Solve the following quadratic equations, using the quadratic formula: a) 5x2 ⫹ 23x ⫹ 12 ⫽ 0 Using (1.1) and substituting a ⫽ 5, b ⫽ 23, and c ⫽ 12, we get ⫺b ⫾ 兹b2 ⫺ 4ac x⫽ 2a ⫺23 ⫾ 兹(23)2 ⫺ 4(5)(12) ⫺23 ⫾ 兹529 ⫺ 240 ⫽ ⫽ 2(5) 10 ⫺23 ⫾ 兹289 ⫺23 ⫾ 17 ⫽ ⫽ 10 10 ⫺23 ⫹ 17 ⫺23 ⫺ 17 x⫽ ⫽ ⫺0.6 x⫽ ⫽ ⫺4 10 10 b) 3x2 ⫺ 41x ⫹ 26 ⫽ 0 ⫺(⫺41) ⫾ 兹(⫺41)2 ⫺ 4(3)(26) 41 ⫾ 兹1681 ⫺ 312 x⫽ ⫽ 2(3) 6 41 ⫾ 兹1369 41 ⫾ 37 ⫽ ⫽ 6 6 41 ⫹ 37 41 ⫺ 37 2 x⫽ ⫽ 13 x⫽ ⫽ 6 6 3 FUNCTIONS 1.7. a) Given f (x) ⫽ x2 ⫹ 4x ⫺ 5, find f (2) and f (⫺3). Substituting 2 for each occurrence of x in the function gives f (2) ⫽ (2)2 ⫹ 4(2) ⫺ 5 ⫽ 7 Now substituting ⫺3 for each occurrence of x, we get f (⫺3) ⫽ (⫺3)2 ⫹ 4(⫺3) ⫺ 5 ⫽ ⫺8 b) Given f (x) ⫽ 2x3 ⫺ 5x2 ⫹ 8x ⫺ 20, find f (5) and f (⫺4). f (5) ⫽ 2(5)3 ⫺ 5(5)2 ⫹ 8(5) ⫺ 20 ⫽ 145 f (⫺4) ⫽ 2(⫺4)3 ⫺ 5(⫺4)2 ⫹ 8(⫺4) ⫺ 20 ⫽ ⫺260 1.8. In the following graphs (Fig. 1-2), where y replaces f (x) as the dependent variable in functions, indicate which graphs are graphs of functions and which are not. For a graph to be the graph of a function, for each value of x, there can be one and only one value of y. If a vertical line can be drawn which intersects the graph at more than one point, then the graph is not the graph of a function. Applying this criterion, which is known as the vertical-line test, we see that (a), (b), and (d ) are functions; (c), (e), and ( f ) are not. 1.9. Which of the following equations are functions and why? a) y ⫽ ⫺2x ⫹ 7 y ⫽ ⫺2x ⫹ 7 is a function because for each value of the independent variable x there is one and only one value of the dependent variable y. For example, if x ⫽ 1, y ⫽ ⫺2(1) ⫹ 7 ⫽ 5. The graph would be similar to (a) in Fig. 1-2. CHAP. 1] REVIEW 11 Fig. 1-2 b) y2 ⫽ x y2 ⫽ x, which is equivalent to y ⫽ ⫾兹x, is not a function because for each positive value of x, there are two values of y. For example, if y2 ⫽ 9, y ⫽ ⫾3. The graph would be similar to that of (c) in Fig. 1-2, illustrating that a parabola whose axis is parallel to the x axis cannot be a function. c) y ⫽ x2 y ⫽ x2 is a function. For each value of x there is only one value of y. For instance, if x ⫽ ⫺5, y ⫽ 25. While it is also true that y ⫽ 25 when x ⫽ 5, it is irrelevant. The definition of a function simply demands that for each value of x there be one value of y, not that for each value of y there be only one value of x. The graph would be like (d ) in Fig. 1-2, demonstrating that a parabola with axis parallel to the y axis is a function. d) y ⫽ ⫺x2 ⫹ 6x ⫹ 15 y ⫽ ⫺x2 ⫹ 6x ⫹ 15 is a function. For each value of x there is a unique value of y. The graph would be like (b) in Fig. 1-2. e) x2 ⫹ y2 ⫽ 64 x2 ⫹ y2 ⫽ 64 is not a function. If x ⫽ 0, y2 ⫽ 64, and y ⫽ ⫾8. The graph would be a circle, similar to (e) in Fig. 1-2. A circle does not pass the vertical-line test. f) x⫽4 x ⫽ 4 is not a function. The graph of x ⫽ 4 is a vertical line. This means that at x ⫽ 4, y has many values. The graph would look like ( f ) in Fig. 1-2. GRAPHS, SLOPES, AND INTERCEPTS 1.10. Find the x intercept in terms of the parameters of the slope-intercept form of a linear equation y ⫽ mx ⫹ b. Setting y ⫽ 0, 0 ⫽ mx ⫹ b mx ⫽ ⫺b b x⫽⫺ m Thus, the x intercept of the slope-intercept form is (⫺b/m, 0). 12 REVIEW [CHAP. 1 1.11. Graph the following equations and indicate their respective slopes and intercepts: a) 3y ⫹ 15x ⫽ 30 b) 2y ⫺ 6x ⫽ 12 c) 8y ⫺ 2x ⫹ 16 ⫽ 0 d) 6y ⫹ 3x ⫺ 18 ⫽ 0 To graph an equation, first set it in slope-intercept form by solving it for y in terms of x. From Example 14, the slope and two intercepts can then be read directly from the equation, providing three pieces of information, whereas only two are needed to graph a straight line. See Fig. 1-3 and Problems 2.1 to 2.10. Fig. 1-3 a) 3y ⫹ 15x ⫽ 30 b) 2y ⫺ 6x ⫽ 12 3y ⫽ ⫺15x ⫹ 30 2y ⫽ 6x ⫹ 12 y ⫽ ⫺5x ⫹ 10 y ⫽ 3x ⫹ 6 Slope m ⫽ ⫺5 Slope m ⫽ 3 y intercept: (0, 10) y intercept: (0, 6) x intercept: (2, 0) x intercept: (⫺2, 0) c) 8y ⫺ 2x ⫹ 16 ⫽ 0 d ) 6y ⫹ 3x ⫺ 18 ⫽ 0 8y ⫽ 2x ⫺ 16 6y ⫽ ⫺3x ⫹ 18 y ⫽ 1–4 x ⫺ 2 y ⫽ ⫺1–2 x ⫹ 3 Slope m ⫽ 1–4 Slope m ⫽ ⫺1–2 y intercept: (0, ⫺2) y intercept: (0, 3) x intercept: (8, 0) x intercept: (6, 0) CHAP. 1] REVIEW 13 1.12. Find the slope m of the linear function passing through: a) (4, 12), (8, 2); b) (⫺1, 15), (3, 6); c) (2, ⫺3), (5, 18). a) Substituting in the formula from Example 13, we get y2 ⫺ y1 2 ⫺ 12 ⫺10 1 m⫽ ⫽ ⫽ ⫽ ⫺2 x2 ⫺ x1 8⫺4 4 2 6 ⫺ 15 ⫺9 1 b) m⫽ ⫽ ⫽ ⫺2 3 ⫺ (⫺1) 4 4 18 ⫺ (⫺3) 21 c) m⫽ ⫽ ⫽7 5⫺2 3 1.13. Graph (a) the quadratic function y ⫽ 2x2 and (b) the rational function y ⫽ 2/x. Fig. 1-3  CHAPTER 2 Economic Applications of Graphs and Equations 2.1 ISOCOST LINES An isocost line represents the different combinations of two inputs or factors of production that can be purchased with a given sum of money. The general formula is PK K ⫹ PL L ⫽ E, where K and L are capital and labor, PK and PL their respective prices, and E the amount allotted to expenditures. In isocost analysis the individual prices and the expenditure are initially held constant; only the different combinations of inputs are allowed to change. The function can then be graphed by expressing one variable in terms of the other, as seen in Example 1 and Problems 2.5 and 2.6. EXAMPLE 1. Given: PK K ⫹ P L L ⫽ E P K K ⫽ E ⫺ PL L E ⫺ PL L K⫽ PK 冢 冣 E PL K⫽ ⫺ L PK PK This is the familiar linear function of the form y ⫽ mx ⫹ b, where b ⫽ E/PK ⫽ the vertical intercept and m ⫽ ⫺PL/PK ⫽ the slope. The graph is given by the solid line in Fig. 2-1. From the equation and graph, the effects of a change in any one of the parameters are easily discernible. An increase in the expenditure from E to E⬘ will increase the vertical intercept and cause the isocost line to shift out to the right (dashed line) parallel to the old line. The slope is unaffected because the slope depends on the relative prices (⫺PL/PK ) and prices are not affected by expenditure changes. A change in PL will alter the slope of the line but leave the vertical intercept unchanged. A change in PK will alter the slope and the vertical intercept. See Problems 2.5 and 2.6. 14 CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 15 Fig. 2-1 2.2 SUPPLY AND DEMAND ANALYSIS Equilibrium in supply and demand analysis occurs when Qs ⫽ Qd. By equating the supply and demand functions, the equilibrium price and quantity can be determined. See Example 2 and Problems 2.1 to 2.4 and 2.11 to 2.16. EXAMPLE 2. Given: Qs ⫽ ⫺5 ⫹ 3P Qd ⫽ 10 ⫺ 2P In equilibrium, Qs ⫽ Qd Solving for P, ⫺5 ⫹ 3P ⫽ 10 ⫺ 2P 5P ⫽ 15 P⫽3 Substituting P ⫽ 3 in either of the equations, Qs ⫽ ⫺5 ⫹ 3P ⫽ ⫺5 ⫹ 3(3) ⫽ 4 ⫽ Qd 2.3 INCOME DETERMINATION MODELS Income determination models generally express the equilibrium level of income in a four-sector economy as Y ⫽ C ⫹ I ⫹ G ⫹ (X ⫺ Z) where Y ⫽ income, C ⫽ consumption, I ⫽ investment, G ⫽ government expenditures, X ⫽ exports, and Z ⫽ imports. By substituting the information supplied in the problem, it is an easy matter to solve for the equilibrium level of income. Aggregating (summing) the variables on the right allows the equation of be graphed in two-dimensional space. See Example 3 and Problems 2.7 to 2.10 and 2.17 to 2.22. EXAMPLE 3. Assume a simple two-sector economy where Y ⫽ C ⫹ I, C ⫽ C0 ⫹ bY, and I ⫽ I0. Assume further that C0 ⫽ 85, b ⫽ 0.9, and I0 ⫽ 55. The equilibrium level of income can be calculated in terms of (1) the general parameters and (2) the specific values assigned to these parameters. 1. The equilibrium equation is Y ⫽ C⫹I Substituting for C and I, Y ⫽ C0 ⫹ bY ⫹ I0 Solving for Y, Y ⫺ bY ⫽ C0 ⫹ I0 (1 ⫺ b) Y ⫽ C0 ⫹ I0 C0 ⫹ I0 Y⫽ 1⫺b 16 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 The solution in this form is called the reduced form. The reduced form (or solution equation) expresses the endogenous variable (here Y) as an explicit function of the exogenous variables (C0, I0) and the parameters (b). 2. The specific equilibrium level of income can be calculated by substituting the numerical values for the parameters in either the original equation (a) or the reduced form (b): C0 ⫹ I0 85 ⫹ 55 a) Y ⫽ C0 ⫹ bY ⫹ I0 ⫽ 85 ⫹ 0.9Y ⫹ 55 b) Y⫽ ⫽ 1⫺b 1 ⫺ 0.9 Y ⫺ 0.9Y ⫽ 140 140 0.1Y ⫽ 140 ⫽ ⫽ 1400 0.1 Y ⫽ 1400 The term 1/(1 ⫺ b) is called the autonomous expenditure multiplier in economics. It measures the multiple effect each dollar of autonomous spending has on the equilibrium level of income. Since b ⫽ MPC in the income determination model, the multiplier ⫽ 1/(1 ⫺ MPC). Note: Decimals may be converted to fractions for ease in working with the income determination model. For example, 0.1 ⫽ –– 10, 0.9 ⫽ 10, 0.5 ⫽ 2, 0.2 ⫽ 5, etc. 1 9 –– 1 – 1 – 2.4 IS-LM ANALYSIS The IS schedule is a locus of points representing all the different combinations of interest rates and income levels consistent with equilibrium in the goods (commodity) market. The LM schedule is a locus of points representing all the different combinations of interest rates and income levels consistent with equilibrium in the money market. IS-LM analysis seeks to find the level of income and the rate of interest at which both the commodity market and the money market will be in equilibrium. This can be accomplished with the techniques used for solving simultaneous equations. Unlike the simple income determination model in Section 2.3, IS-LM analysis deals explicitly with the interest rate and incorporates its effect into the model. See Example 4 and Problems 2.23 and 2.24. EXAMPLE 4. The commodity market for a simple two-sector economy is in equilibrium when Y ⫽ C ⫹ I. The money market is in equilibrium when the supply of money (Ms) equals the demand for money (Md ), which in turn is composed of the transaction-precautionary demand for money (Mt) and the speculative demand for money (Mz). Assume a two-sector economy where C ⫽ 48 ⫹ 0.8Y, I ⫽ 98 ⫺ 75i, Ms ⫽ 250, Mt ⫽ 0.3Y, and Mz ⫽ 52 ⫺ 150i. Commodity equilibrium (IS) exists when Y ⫽ C ⫹ I. Substituting into the equation, Y ⫽ 48 ⫹ 0.8Y ⫹ 98 ⫺ 75i Y ⫺ 0.8Y ⫽ 146 ⫺ 75i 0.2Y ⫹ 75i ⫺ 146 ⫽ 0 (2.1) Monetary equilibrium (LM) exists when Ms ⫽ Mt ⫹ Mz. Substituting into the equation, 250 ⫽ 0.3Y ⫹ 52 ⫺ 150i 0.3Y ⫺ 150i ⫺ 198 ⫽ 0 (2.2) A condition of simultaneous equilibrium in both markets can be found, then, by solving (2.1) and (2.2) simultaneously: 0.2Y ⫹ 75i ⫺ 146 ⫽ 0 (2.1) 0.3Y ⫺ 150i ⫺ 198 ⫽ 0 (2.2) CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 17 Multiply (2.1) by 2, add the result (2.3) to (2.2) to eliminate i, and solve for Y 0.4Y ⫹ 150i ⫺ 292 ⫽ 0 (2.3) 0.3Y ⫺ 150i ⫺ 198 ⫽ 0 0.7Y ⫺ 490 ⫽ 0 Y ⫽ 700 Substitute Y ⫽ 700 in (2.1) or (2.2) to find i. 0.2Y ⫹ 75i ⫺ 146 ⫽ 0 0.2(700) ⫹ 75i ⫺ 146 ⫽ 0 140 ⫹ 75i ⫺ 146 ⫽ 0 75i ⫽ 6 75 ⫽ 0.08 i ⫽ –– 6 The commodity and money markets will be in simultaneous equilibrium when Y ⫽ 700 and i ⫽ 0.08. At that point C ⫽ 48 ⫹ 0.8(700) ⫽ 608, I ⫽ 98 ⫺ 75(0.08) ⫽ 92, Mt ⫽ 0.3(700) ⫽ 210, and Mz ⫽ 52 ⫺ 150(0.08) ⫽ 40. C ⫹ I ⫽ 608 ⫹ 92 ⫽ 700 and Mt ⫹ Mz ⫽ 210 ⫹ 40 ⫽ 250 ⫽ Ms. Solved Problems GRAPHS 2.1. A complete demand function is given by the equation Qd ⫽ ⫺30P ⫹ 0.05Y ⫹ 2Pr ⫹ 4T where P is the price of the good, Y is income, Pr is the price of a related good (here a substitute), and T is taste. Can the function be graphed? Since the complete function contains five different variables, it cannot be graphed as is. In ordinary demand analysis, however, it is assumed that all the independent variables except price are held constant so that the effect of a change in price on the quantity demanded can be measured independently of the influence of other factors, or ceteris paribus. If the other variables (Y, Pr, T ) are held constant, the function can be graphed. 2.2 (a) Draw the graph for the demand function in Problem 2.1, assuming Y ⫽ 5000, Pr ⫽ 25, and T ⫽ 30. (b) What does the typical demand function drawn in part (a) show? (c) What happens to the graph if the price of the good changes from 5 to 6? (d) What happens if any of the other variables change? For example, if income increases to 7400? a) By adding the new data to the equation in Problem 2.1, the function is easily graphable. See Fig. 2-2. Qd ⫽ ⫺30P ⫹ 0.05Y ⫹ 2Pr ⫹ 4T ⫽ ⫺30P ⫹ 0.05(5000) ⫹ 2(25) ⫹ 4(30) ⫽ ⫺30P ⫹ 420 b) The demand function graphed in part (a) shows all the diferent quantities of the good that will be demanded at different prices, assuming a given level of income, taste, and prices of substitutes (here 5000, 30, 25) which are not allowed to change. c) If nothing changes but the price of the good, the graph remains exactly the same since the graph indicates the different quantities that will be demanded at all the possible prices. A simple change in the price of the good occasions a movement along the curve which is called a change in quantity demanded. When the price goes from 5 to 6, the quantity demanded falls from 270 [420 ⫺ 30(5)] to 240 [420 ⫺ 30(6)], a movement from A to B on the curve. 18 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 Fig. 2-2 Fig. 2-3 d) If any of the other variables change, there will be a shift in the curve. This is called a change in demand because it results in a totally new demand function (and curve) in response to the changed conditions. If income increases to 7400, the new demand function becomes Qd ⫽ ⫺30P ⫹ 0.05(7400) ⫹ 2(25) ⫹ 4(30) ⫽ ⫺30P ⫹ 540 This is graphed as a dashed line in Fig. 2-2. 2.3. In economics the independent variable (price) has traditionally been graphed on the vertical axis in supply and demand analysis, and the dependent variable (quantity) has been graphed on the horizontal. (a) Graph the demand function in Problem 2.2 according to the traditional method. (b) Show what happens if the price goes from 5 to 6 and income increases to 7400. a) The function Qd ⫽ 420 ⫺ 30P is graphed according to traditional economic practice by means of the inverse function, which is obtained by solving the original function for the independent variable in terms of the dependent variable. Solving algebraically for P in terms of Qd, therefore, the inverse function of Qd ⫽ 420 ⫺ 30P is P ⫽ 14 ⫺ –– 1 30 Qd. The graph appears as a solid line in Fig. 2-3. b) If P goes from 5 to 6, Qd falls from 270 to 240. P ⫽ 14 ⫺ –– 1 30 Qd P ⫽ 14 ⫺ –– 1 30 Qd 5 ⫽ 14 ⫺ –– 1 30 Qd 6 ⫽ 14 ⫺ –– 1 30 Qd –– Qd ⫽ 9 1 –– Qd ⫽ 8 1 30 30 Qd ⫽ 270 Qd ⫽ 240 The change is represented by a movement from A to B in Fig. 2-3. If Y ⫽ 7400, as in Problems 2.2(d), Qd ⫽ 540 ⫺ 30P. Solving algebraically for P in terms of Q, the inverse function is P ⫽ 18 ⫺ –– 1 30 Qd. It is graphed as a dashed line in Fig. 2-3. 2.4. Graph the demand function Qd ⫽ ⫺4P ⫹ 0.01Y ⫺ 5Pr ⫹ 10T when Y ⫽ 8000, Pr ⫽ 8, and T ⫽ 4. (b) What type of good is the related good? (c) What happens if T increases to 8, indicating greater preference for the good? (d) Construct the graph along the traditional economic lines with P on the vertical axis and Q on the horizontal axis. a) Qd ⫽ ⫺4P ⫹ 0.01(8000) ⫺ 5(8) ⫹ 10(4) ⫽ ⫺4P ⫹ 80 This is graphed as a solid line in Fig. 2-4(a). b) The related good has a negative coefficient. This means that a rise in the price of the related good will lead to a decrease in demand for the original good. The related good is, by definition, a complementary good. CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 19 Fig. 2-4 c) If T ⫽ 8, indicating greater preference, there will be a totally new demand. Qd ⫽ ⫺4P ⫹ 0.01(8000) ⫺ 5(8) ⫹ 10(8) ⫽ ⫺4P ⫹ 120 See the dashed line in Fig. 2-4(a). d) Graphing P on the vertical calls for the inverse function. Solving for P in terms of Qd, the inverse of Qd ⫽ 80 ⫺ 4P is P ⫽ 20 ⫺ 1–4 Qd and is graphed as a solid line in Fig. 2-4(b). The inverse of Qd ⫽ 120 ⫺ 4P is P ⫽ 30 ⫺ 1–4 Qd. It is the dashed line in Fig. 2-4(b). 2.5 A person has $120 to spend on two goods (X, Y ) whose respective prices are $3 and $5. (a) Draw a budget line showing all the different combinations of the two goods that can be bought with the given budget (B). What happens to the original budget line (b) if the budget falls by 25 percent, (c) if the price of X doubles, (d) if the price of Y falls to 4? a) The general function for a budget line is Px X ⫹ PY Y ⫽ B If Px ⫽ 3, PY ⫽ 5, and B ⫽ 120, 3X ⫹ 5Y ⫽ 120 Solving for Y in terms of X in order to graph the function, Y ⫽ 24 ⫺ 3–5X The graph is given as a solid line in Fig. 2-5(a). Fig. 2-5 20 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 b) If the budget falls by 25 percent, the new budget is 90 [120 ⫺ 1–4(120) ⫽ 90]. The equation for the new budget line is 3X ⫹ 5Y ⫽ 90 Y ⫽ 18 ⫺ 3–5X The graph is a dashed line in Fig. 2-5(a). Lowering the budget causes the budget line to shift parallel to the left. c) If PX doubles, the original equation becomes 6X ⫹ 5Y ⫽ 120 Y ⫽ 24 ⫺ 6–5X The vertical intercept remains the same, but the slope changes and becomes steeper. See the dashed line in Fig. 2-5(b). With a higher price for X, less X can be bought with the given budget. d) If PY now equals 4, 3X ⫹ 4Y ⫽ 120 Y ⫽ 30 ⫺ 3–4X With a change in PY, both the vertical intercept and the slope change. This is shown in Fig. 2-5(c) by the dashed line. 2.6. Either coal (C ) or gas (G) can be used in the production of steel. The cost of coal is 100, the cost of gas 500. Draw an isocost curve showing the different combinations of gas and coal that can be purchased (a) with an initial expenditure (E ) of 10,000, (b) if expenditures increase by 50 percent, (c) if the price of gas is reduced by 20 percent, (d ) if the price of coal rises by 25 percent. Always start from the original equation. a) PC C ⫹ PG G ⫽ E 100C ⫹ 500G ⫽ 10,000 C ⫽ 100 ⫺ 5G The graph is a solid line in Fig. 2-6(a). b) A 50 percent increase in expenditures makes the new outlay 15,000 [10,000 ⫹ 0.5(10,000)]. The new equation is 100C ⫹ 500G ⫽ 15,000 C ⫽ 150 ⫺ 5G The graph is the dashed line in Fig. 2-6(a). (a) Increase in budget (b) Reduction in PG (c) Increase in PC Fig. 2-6 CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 21 c) If the price of gas is reduced by 20 percent, the new price is 400 [500 ⫺ 0.2(500)], and the new equation is 100C ⫹ 400G ⫽ 10,000 C ⫽ 100 ⫺ 4G The graph is the dashed line in Fig. 2-6(b). d) A 25 percent rise in the price of coal makes the new price 125 [100 ⫹ 0.25(100)]. 125C ⫹ 500G ⫽ 10,000 C ⫽ 80 ⫺ 4G The graph appears as a dashed line in Fig. 2-6(c). GRAPHS IN THE INCOME DETERMINATION MODEL 2.7. Given: Y ⫽ C ⫹ I, C ⫽ 50 ⫹ 0.8Y, and I0 ⫽ 50. (a) Graph the consumption function. (b) Graph the aggregate demand function, C ⫹ I0. (c) Find the equilibrium level of income from the graph. a) Since consumption is a function of income, it is graphed on the vertical axis; income is graphed on the horizontal. See Fig. 2-7. When other components of aggregate demand such as I, G, and X ⫺ Z are added to the model, they are also graphed on the vertical axis. It is easily determined from the linear form of the consumption function that the vertical intercept is 50 and the slope of the line (the MPC or ⌬C/⌬Y ) is 0.8. b) Investment in the model is autonomous investment. This means investment is independent of income and does not change in response to changes in income. When considered by itself, the graph of a constant is a horizontal line; when added to a linear function, it causes a parallel shift in the original function by an amount equal to its value. In Fig. 2-7, autonomous investment causes the aggregate demand function to shift up by 50 parallel to the initial consumption function. c) To obtain the equilibrium level of income from a graph, a 45⬚ dashed line is drawn from the origin. If the same scale of measurement is used on both axes, a 45⬚ line has a slope of 1, meaning that as the line moves away from the origin, it moves up vertically (⌬Y ) by one unit for every unit it moves across horizontally (⌬X ). Every point on the 45⬚ line, therefore, has a horizontal coordinate (abscissa) Fig. 2-7 22 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 exactly equal to its vertical coordinate (ordinate). Consequently, when the aggregate demand function intersects the 45⬚ line, aggregate demand (as graphed on the vertical) will equal national income (as graphed on the horizontal). From Fig. 2-7 it is clear that the equilibrium level of income is 500, since the aggregate demand function (C ⫹ I ) intersects the 45⬚ line at 500. 2.8. Given: Y ⫽ C ⫹ I ⫹ G, C ⫽ 25 ⫹ 0.75Y, I ⫽ I0 ⫽ 50, and G ⫽ G0 ⫽ 25. (a) Graph the aggregate demand function and show its individual components. (b) Find the equilibrium level of income. (c) How can the aggregate demand function be graphed directly, without having to graph each of the component parts? a) See Fig. 2-8. b) Equilibrium income ⫽ 400. c) To graph the aggregate demand function directly, sum up the individual components, Agg. D ⫽ C ⫹ I ⫹ G ⫽ 25 ⫹ 0.75Y ⫹ 50 ⫹ 25 ⫽ 100 ⫹ 0.75Y The direct graphing of the aggregate demand function coincides exactly with the graph of the summation of the individual graphs of C, I, and G above. Fig. 2-8 2.9. Use a graph to show how the addition of a lump-sum tax (a tax independent of income) influences the parameters of the income determination model. Graph the two systems indi- vidually, using a solid line for (1) and a dashed line for (2). 1) Y ⫽ C⫹I 2) Y ⫽ C⫹I Yd ⫽ Y ⫺ T C ⫽ 100 ⫹ 0.6Y C ⫽ 100 ⫹ 0.6Yd T ⫽ 50 I0 ⫽ 40 I0 ⫽ 40 The first system of equations presents no problems; the second requires that C first be converted from a function of Yd to a function of Y. 1) Agg. D ⫽ C ⫹ I 2) Agg. D ⫽ C ⫹ I ⫽ 100 ⫹ 0.6Y ⫹ 40 ⫽ 100 ⫹ 0.6Yd ⫹ 40 ⫽ 140 ⫹ 0.6(Y ⫺ T ) ⫽ 140 ⫹ 0.6Y ⫽ 140 ⫹ 0.6(Y ⫺ 50) ⫽ 110 ⫹ 0.6Y CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 23 Fig. 2-9 A lump-sum tax has a negative effect on the vertical intercept of the aggregate demand function equal to ⫺MPC(T ). Here ⫺0.6(50) ⫽ ⫺30. The slope is not affected (note the parallel lines for the two graphs in Fig. 2-9). Income falls from 350 to 275 as a result of the tax. See Fig. 2-9. 2.10. Explain with the aid of a graph how the incorporation of a proportional tax (a tax depending on income) influences the parameters of the income determination model. Graph the model without the tax as a solid line and the model with the tax as a dashed line. 1) Y ⫽ C⫹I 2) Y ⫽ C⫹I Yd ⫽ Y ⫺ T C ⫽ 85 ⫹ 0.75Y C ⫽ 85 ⫹ 0.75Yd T ⫽ 20 ⫹ 0.2Y I0 ⫽ 30 I0 ⫽ 30 1) Agg. D ⫽ C ⫹ I 2) Agg. D ⫽ C ⫹ I ⫽ 85 ⫹ 0.75Y ⫹ 30 ⫽ 85 ⫹ 0.75Yd ⫹ 30 ⫽ 115 ⫹ 0.75(Y ⫺ T ) ⫽ 115 ⫹ 0.75Y ⫽ 115 ⫹ 0.75(Y ⫺ 20 ⫺ 0.2Y ) ⫽ 115 ⫹ 0.75Y ⫺ 15 ⫺ 0.15Y ⫽ 100 ⫹ 0.6Y Incorporation of a proportional income tax into the model affects the slope of the line, or the MPC. In this case it lowers it from 0.75 to 0.6. The vertical intercept is also lowered because the tax structure includes a lump-sum tax of 20. Because of the tax structure, the equilibrium level of income falls from 460 to 250. See Fig. 2-10. EQUATIONS IN SUPPLY AND DEMAND ANALYSIS 2.11. Find the equilibrium price and quantity for the following markets: a) Qs ⫽ ⫺20 ⫹ 3P b) Qs ⫽ ⫺45 ⫹ 8P Qd ⫽ 220 ⫺ 5P Qd ⫽ 125 ⫺ 2P c) Qs ⫹ 32 ⫺ 7P ⫽ 0 d) 13P ⫺ Qs ⫽ 27 Qd ⫺ 128 ⫹ 9P ⫽ 0 Qd ⫹ 4P ⫺ 24 ⫽ 0 24 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 Fig. 2-10 Each of the markets will be in equilibrium when Qs ⫽ Qd. a) Qs ⫽ Qd b) Qs ⫽ Qd ⫺20 ⫹ 3P ⫽ 220 ⫺ 5P ⫺45 ⫹ 8P ⫽ 125 ⫺ 2P 8P ⫽ 240 10P ⫽ 170 P ⫽ 30 P ⫽ 17 Qs ⫽ ⫺20 ⫹ 3P ⫽ ⫺20 ⫹ 3(30) Qd ⫽ 125 ⫺ 2P ⫽ 125 ⫺ 2(17) Qs ⫽ 70 ⫽ Qd Qd ⫽ 91 ⫽ Qs c) Qs ⫽ 7P ⫺ 32 d) Qs ⫽ ⫺27 ⫹ 13P Qd ⫽ 128 ⫺ 9P Qd ⫽ 24 ⫺ 4P 7P ⫺ 32 ⫽ 128 ⫺ 9P ⫺27 ⫹ 13P ⫽ 24 ⫺ 4P 16P ⫽ 160 17P ⫽ 51 P ⫽ 10 P⫽3 Qs ⫽ 7P ⫺ 32 ⫽ 7(10) ⫺ 32 Qd ⫽ 24 ⫺ 4P ⫽ 24 ⫺ 4(3) Qs ⫽ 38 ⫽ Qd Qd ⫽ 12 ⫽ Qs 2.12. Given the following set of simultaneous equations for two related markets, beef (B) and pork (P), find the equilibrium conditions for each market, using the substitution method. 1) QdB ⫽ 82 ⫺ 3PB ⫹ PP 2) QdP ⫽ 92 ⫹ 2PB ⫺ 4PP QsB ⫽ ⫺5 ⫹ 15PB QsP ⫽ ⫺6 ⫹ 32PP Equilibrium requires that Qs ⫽ Qd in each market. 1) QsB ⫽ QdB 2) QsP ⫽ QdP ⫺5 ⫹ 15PB ⫽ 82 ⫺ 3PB ⫹ PP ⫺6 ⫹ 32PP ⫽ 92 ⫹ 2PB ⫺ 4PP 18PB ⫺ PP ⫽ 87 36PP ⫺ 2PB ⫽ 98 This reduces the problem to two equations and two unknowns: 18PB ⫺ PP ⫽ 87 (2.4) ⫺2PB ⫹ 36PP ⫽ 98 (2.5) CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 25 Solving for PP in (2.4) gives PP ⫽ 18PB ⫺ 87 Substituting the value of this term in (2.5) gives ⫺2PB ⫹ 36(18PB ⫺ 87) ⫽ 98 ⫺2PB ⫹ 648PB ⫺ 3132 ⫽ 98 646PB ⫽ 3230 PB ⫽ 5 Substituting PB ⫽ 5 in (2.5), or (2.4), ⫺2(5) ⫹ 36PP ⫽ 98 36PP ⫽ 108 PP ⫽ 3 Finally, substituting the values for PB and PP in either the supply or the demand function for each market, 1) QdB ⫽ 82 ⫺ 3PB ⫹ PP ⫽ 82 ⫺ 3(5) ⫹ (3) 2) QdP ⫽ 92 ⫹ 2PB ⫺ 4PP ⫽ 92 ⫹ 2(5) ⫺ 4(3) QdB ⫽ 70 ⫽ QsB QdP ⫽ 90 ⫽ QsP 2.13. Find the equilibrium price and quantity for two complementary goods, slacks (S) and jackets (J), using the elimination method. 1) QdS ⫽ 410 ⫺ 5PS ⫺ 2PJ 2) QdJ ⫽ 295 ⫺ PS ⫺ 3PJ QsS ⫽ ⫺60 ⫹ 3PS QsJ ⫽ ⫺120 ⫹ 2PJ In equilibrium, 1) QdS ⫽ QsS 2) QdJ ⫽ QsJ 410 ⫺ 5PS ⫺ 2PJ ⫽ ⫺60 ⫹ 3PS 295 ⫺ PS ⫺ 3PJ ⫽ ⫺120 ⫹ 2PJ 470 ⫺ 8PS ⫺ 2PJ ⫽ 0 415 ⫺ PS ⫺ 5PJ ⫽ 0 This leaves two equations 470 ⫺ 8PS ⫺ 2PJ ⫽ 0 (2.6) 415 ⫺ PS ⫺ 5PJ ⫽ 0 (2.7) Multiplying (2.7) by 8 gives (2.8). Subtract (2.6) from (2.8) to eliminate PS, and solve for PJ. 3320 ⫺ 8PS ⫺ 40PJ ⫽ 0 (2.8) ⫺(⫹470 ⫺ 8PS ⫺ 2PJ ⫽ 0) 2850 ⫺ 38PJ ⫽ 0 PJ ⫽ 75 Substituting PJ ⫽ 75 in (2.6), 470 ⫺ 8PS ⫺ 2(75) ⫽ 0 320 ⫽ 8PS PS ⫽ 40 Finally, substituting PJ ⫽ 75 and PS ⫽ 40 into Qd or Qs for each market, 1) QdS ⫽ 410 ⫺ 5PS ⫺ 2PJ ⫽ 410 ⫺ 5(40) ⫺ 2(75) 2) QdJ ⫽ 295 ⫺ PS ⫺ 3PJ ⫽ 295 ⫺ 40 ⫺ 3(75) Qds ⫽ 60 ⫽ QsS QdJ ⫽ 30 ⫽ QsJ 2.14. Supply and demand conditions can also be expressed in quadratic form. Find the equilibrium price and quantity, given the demand function P ⫹ Q2 ⫹ 3Q ⫺ 20 ⫽ 0 (2.9) and the supply function P ⫺ 3Q2 ⫹ 10Q ⫽ 5 (2.10) 26 ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS [CHAP. 2 Either the substitution method or the elimination method can be used, since this problem involves two equations and two unknowns. Using the substitution method, (2.10) is solved for P in terms of Q. P ⫺ 3Q2 ⫹ 10Q ⫽ 5 P ⫽ 3Q2 ⫺ 10Q ⫹ 5 Substituting P ⫽ 3Q2 ⫺ 10Q ⫹ 5 in (2.9), (3Q2 ⫺ 10Q ⫹ 5) ⫹ Q2 ⫹ 3Q ⫺ 20 ⫽ 0 4Q2 ⫺ 7Q ⫺ 15 ⫽ 0 Using the quadratic formula Q1, Q2 ⫽ (⫺b ⫾ 兹b2 ⫺ 4ac)/(2a), where a ⫽ 4, b ⫽ ⫺7, and c ⫽ ⫺15, Q1 ⫽ 3, and Q2 ⫽ ⫺1.25. Since neither price nor quantity can be negative, Q ⫽ 3. Substitute Q ⫽ 3 in (2.9) or (2.10) to find P. P ⫹ (3)2 ⫹ 3(3) ⫺ 20 ⫽ 0 P⫽2 2.15. Use the elimination method to find the equilibrium price and quantity when the demand function is 3P ⫹ Q2 ⫹ 5Q ⫺ 102 ⫽ 0 (2.11) and the supply function is P ⫺ 2Q2 ⫹ 3Q ⫹ 71 ⫽ 0 (2.12) Multiply (2.12) by 3 to get (2.13) and subtract it from (2.11) to eliminate P. 3P ⫹ Q2 ⫹ 5Q ⫺ 102 ⫽ 0 ⫺(3P ⫺ 6Q2 ⫹ 9Q ⫹ 213 ⫽ 0) (2.13) 7Q2 ⫺ 4Q ⫺ 315 ⫽ 0 Use the quadratic formula (see Problem 2.14) to solve for Q, and substitute the result, Q ⫽ 7, in (2.12) or (2.11) to solve for P. P ⫺ 2(7)2 ⫹ 3(7) ⫹ 71 ⫽ 0 P⫽6 2.16. Supply and demand analysis can also involve more than two markets. Find the equilibrium price and quantity for the three substitute goods below. Qd1 ⫽ 23 ⫺ 5P1 ⫹ P2 ⫹ P3 Qs1 ⫽ ⫺8 ⫹ 6P1 Qd2 ⫽ 15 ⫹ P1 ⫺ 3P2 ⫹ 2P3 Qs2 ⫽ ⫺11 ⫹ 3P2 Qd3 ⫽ 19 ⫹ P1 ⫹ 2P2 ⫺ 4P3 Qs3 ⫽ ⫺5 ⫹ 3P3 For equilibrium in each market, Qd1 ⫽ Qs1 Qd2 ⫽ Qs2 Qd3 ⫽ Qs3 23 ⫺ 5P1 ⫹ P2 ⫹ P3 ⫽ ⫺8 ⫹ 6P1 15 ⫹ P1 ⫺ 3P2 ⫹ 2P3 ⫽ ⫺11 ⫹ 3P2 19 ⫹ P1 ⫹ 2P2 ⫺ 4P3 ⫽ ⫺5 ⫹ 3P3 31 ⫺ 11P1 ⫹ P2 ⫹ P3 ⫽ 0 26 ⫹ P1 ⫺ 6P2 ⫹ 2P3 ⫽ 0 24 ⫹ P1 ⫹ 2P2 ⫺ 7P3 ⫽ 0 This leaves three equations with three unknowns: 31 ⫺ 11P1 ⫹ P2 ⫹ P3 ⫽ 0 (2.14) 26 ⫹ P1 ⫺ 6P2 ⫹ 2P3 ⫽ 0 (2.15) 24 ⫹ P1 ⫹ 2P2 ⫺ 7P3 ⫽ 0 (2.16) Start by eliminating one of the variables (here P2). Multiply (2.14) by 2 to get 62 ⫺ 22P1 ⫹ 2P2 ⫹ 2P3 ⫽ 0 CHAP. 2] ECONOMIC APPLICATIONS OF GRAPHS AND EQUATIONS 27 From this subtract (2.16). 62 ⫺ 22P1 ⫹ 2P2 ⫹ 2P3 ⫽ 0 ⫺(24 ⫹ P1 ⫹ 2P2 ⫺ 7P3) ⫽ 0 38 ⫺ 23P1 ⫹ 9P3 ⫽ 0 (2.17) Multiply (2.16) by 3. 72 ⫹ 3P1 ⫹ 6P2 ⫺ 21P3 ⫽ 0 Add the result to (2.15). 26 ⫹ P1 ⫺ 6P2 ⫹ 2P3 ⫽ 0 72 ⫹ 3P1 ⫹ 6P2 ⫺ 21P3 ⫽ 0 98 ⫹ 4P1 ⫺ 19P3 ⫽ 0 (2.18) Now there are two equations, (2.17) and (2.18), and two unknowns. Multiply (2.17) by 19 and (2.18) by 9; then add to eliminate P3. 722 ⫺ 437P1 ⫹ 171P3 ⫽ 0 882 ⫹ 36P1 ⫺ 171P3 ⫽ 0 1604 ⫺ 401P1 ⫽0