Essential mathematics for economics analysis 4th by sydaeter

Essential mathematics for ecohomics analysis 4th by sydaeter Essential mathematics for ecohomics analysis 4th by sydaeter Essential mathematics for ecohomics analysis 4th by sydaeter Essential mathematics for ecohomics analysis 4th by sydaeter Essential mathematics for ecohomics analysis 4th by sydaeter Essential mathematics for ecohomics analysis 4th by sydaeter

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Learning online with MyMathLab Global  ‘Allows students to work at their own pace, get immediate feedback, and overcome problems by using the step-wise advice This is an excellent tool for all students.’ Jana Vyrastekova, University of Nijmegen, the Netherlands Go to www.mymathlab.com/global – your gateway to all the online resources for this book • MyMathLab Global provides you with the opportunity for unlimited practice, guided solutions with tips and hints to help you solve challenging questions, an interactive eBook, as well as a personalised study plan to help focus your revision efforts on the topics where you need most support • Short answers are available to almost all of the 1,000 problems in the book for students to self check In addition, a Students’ Manual is provided in the online resources, with extended worked answers to selected problems • If you have purchased this text as part of a pack, the book contains a code and full instructions allowing you to register for access to MyMathLab Global If you have purchased this text on its own, you can still purchase access online at www.mymathlab.com/global See the Guided Tour at the front of this text for more details Peter Hammond is currently a Professor of Economics at the University of Warwick, where he moved in 2007 after becoming an Emeritus Professor at Stanford University He has taught mathematics for economists at both universities Arne Strøm has extensive experience in teaching mathematics for economists in the Department of Economics at the University of Oslo www.pearson-books.com Essential Mathematics for Economic Analysis FO U RT H E D I T I O N FOURTH EDITION Sydsæter & Hammond with StrØm Knut Sydsæter is an Emeritus Professor of Mathematics in the Economics Department at the University of Oslo, where he has been teaching mathematics for economists since 1965 Essential Mathematics for Economic Analysis All the mathematical tools an economist needs are provided in this worldwide bestseller Knut Sydsæter & Peter Hammond with Arne StrØm E S S E N T I A L M AT HE M AT I C S F OR E C O NOMI C A N A LY S I S E S S E N T I A L M AT HE M AT I C S F OR E C O NOMI C A N A LY S I S F O UR T H E D I T I O N Knut Sydsæter and Peter Hammond with Arne Strøm 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.pearson.com/uk First published by Prentice-Hall, Inc 1995 Second edition published 2006 Third edition published 2008 Fourth edition published by Pearson Education Limited 2012 © Prentice-Hall, Inc 1995 © Knut Sydsỉter and Peter Hammond 2002, 2006, 2008, 2012 The rights of Knut Sydsæter and Peter Hammond to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 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 Pearson Education is not responsible for the content of third-party internet sites ISBN 978-0-273-76068-9 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 16 15 14 13 12 Typeset in 10/13 pt Times Roman by Matematisk Sats and Arne Strøm, Norway Printed and bound by Ashford Colour Press Ltd, Gosport, UK To the memory of my parents Elsie (1916–2007) and Fred (1916–2008), my first teachers of Mathematics, basic Economics, and many more important things — Peter CONTENTS Preface xi Introductory Topics I: Algebra 1.1 The Real Numbers 1.2 Integer Powers 1.3 Rules of Algebra 1.4 Fractions 1.5 Fractional Powers 1.6 Inequalities 1.7 Intervals and Absolute Values Review Problems for Chapter 1 10 14 19 24 29 32 Introductory Topics II: Equations 35 2.1 2.2 2.3 2.4 2.5 How to Solve Simple Equations Equations with Parameters Quadratic Equations Linear Equations in Two Unknowns Nonlinear Equations Review Problems for Chapter Introductory Topics III: Miscellaneous 3.1 3.2 Summation Notation Rules for Sums Newton’s Binomial Formula 35 38 41 46 48 49 51 51 55 3.3 Double Sums 3.4 A Few Aspects of Logic 3.5 Mathematical Proofs 3.6 Essentials of Set Theory 3.7 Mathematical Induction Review Problems for Chapter 59 61 67 69 75 77 Functions of One Variable 79 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 79 80 86 89 95 99 105 112 114 119 124 Introduction Basic Definitions Graphs of Functions Linear Functions Linear Models Quadratic Functions Polynomials Power Functions Exponential Functions Logarithmic Functions Review Problems for Chapter Properties of Functions 5.1 5.2 5.3 5.4 5.5 Shifting Graphs New Functions from Old Inverse Functions Graphs of Equations Distance in the Plane Circles 127 127 132 136 143 146 viii viii CCOONN T ET N T ST S EN 5.6 General Functions Review Problems for Chapter Differentiation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Slopes of Curves Tangents and Derivatives Increasing and Decreasing Functions Rates of Change A Dash of Limits Simple Rules for Differentiation Sums, Products, and Quotients Chain Rule Higher-Order Derivatives Exponential Functions Logarithmic Functions Review Problems for Chapter Derivatives in Use 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 Implicit Differentiation Economic Examples Differentiating the Inverse Linear Approximations Polynomial Approximations Taylor’s Formula Why Economists Use Elasticities Continuity More on Limits Intermediate Value Theorem Newton’s Method 7.11 Infinite Sequences 7.12 L’Hôpital’s Rule Review Problems for Chapter Single-Variable Optimization 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Introduction Simple Tests for Extreme Points Economic Examples The Extreme Value Theorem Further Economic Examples Local Extreme Points Inflection Points Review Problems for Chapter 150 153 155 155 157 163 165 169 174 178 184 188 194 197 203 205 205 210 214 217 221 225 228 233 237 245 249 251 256 259 259 262 266 270 276 281 287 291 Integration 9.1 Indefinite Integrals 9.2 Area and Definite Integrals 9.3 Properties of Definite Integrals 9.4 Economic Applications 9.5 Integration by Parts 9.6 Integration by Substitution 9.7 Infinite Intervals of Integration 9.8 A Glimpse at Differential Equations 9.9 Separable and Linear Differential Equations Review Problems for Chapter 10 Topics in Financial Economics 10.1 Interest Periods and Effective Rates 10.2 Continuous Compounding 10.3 Present Value 10.4 Geometric Series 10.5 Total Present Value 10.6 Mortgage Repayments 10.7 Internal Rate of Return 10.8 A Glimpse at Difference Equations Review Problems for Chapter 10 11 Functions of Many Variables 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Functions of Two Variables Partial Derivatives with Two Variables Geometric Representation Surfaces and Distance Functions of More Variables Partial Derivatives with More Variables Economic Applications Partial Elasticities Review Problems for Chapter 11 12 Tools for Comparative Statics 12.1 A Simple Chain Rule 12.2 Chain Rules for Many Variables 12.3 Implicit Differentiation along a Level Curve 12.4 More General Cases 293 293 299 305 309 315 319 324 330 336 341 345 345 349 351 353 359 364 369 371 374 377 377 381 387 393 396 400 404 406 408 411 411 416 420 424 12.5 Elasticity of Substitution 12.6 Homogeneous Functions of Two Variables 12.7 Homogeneous and Homothetic Functions 12.8 Linear Approximations 12.9 Differentials 12.10 Systems of Equations 12.11 Differentiating Systems of Equations Review Problems for Chapter 12 13 Multivariable Optimization 13.1 Two Variables: Necessary Conditions 13.2 Two Variables: Sufficient Conditions 13.3 Local Extreme Points 13.4 Linear Models with Quadratic Objectives 13.5 The Extreme Value Theorem 13.6 Three or More Variables 13.7 Comparative Statics and the Envelope Theorem Review Problems for Chapter 13 CO ON NT E N T S C 428 431 435 440 444 449 452 458 461 461 466 470 475 482 487 491 495 14 Constrained Optimization 497 14.1 The Lagrange Multiplier Method 14.2 Interpreting the Lagrange Multiplier 14.3 Several Solution Candidates 14.4 Why the Lagrange Method Works 14.5 Sufficient Conditions 14.6 Additional Variables and Constraints 14.7 Comparative Statics 14.8 Nonlinear Programming: A Simple Case 14.9 Multiple Inequality Constraints 14.10 Nonnegativity Constraints Review Problems for Chapter 14 15 Matrix and Vector Algebra 15.1 15.2 15.3 15.4 Systems of Linear Equations Matrices and Matrix Operations Matrix Multiplication Rules for Matrix Multiplication 497 504 507 509 513 516 522 526 532 537 541 545 545 548 551 556 15.5 The Transpose 15.6 Gaussian Elimination 15.7 Vectors 15.8 Geometric Interpretation of Vectors 15.9 Lines and Planes Review Problems for Chapter 15 16 Determinants and Inverse Matrices 16.1 Determinants of Order 16.2 Determinants of Order 16.3 Determinants of Order n 16.4 Basic Rules for Determinants 16.5 Expansion by Cofactors 16.6 The Inverse of a Matrix 16.7 A General Formula for the Inverse 16.8 Cramer’s Rule 16.9 The Leontief Model Review Problems for Chapter 16 17 Linear Programming 17.1 A Graphical Approach 17.2 Introduction to Duality Theory 17.3 The Duality Theorem 17.4 A General Economic Interpretation 17.5 Complementary Slackness Review Problems for Chapter 17 ix 562 565 570 573 578 582 585 585 589 593 596 601 604 610 613 616 621 623 623 629 633 636 638 643 Appendix: Geometry 645 The Greek Alphabet 647 Answers to the Problems 649 Index 739 www.downloadslide.net 732 ANSWERS TO THE PROBLEMS Expanding each of these determinants by the column (b1 , b2 , b3 ), we find that D1 = −5b1 + b2 + 2b3 , D2 = 5b1 − b2 − 15 b3 , x2 = − 21 b1 + 10 b2 + 35 b3 , x3 = − 21 b1 + 10 b2 + 25 b3 3b2 −6b3 , D3 = 5b1 −7b2 −4b3 Hence, x1 = 21 b1 − 10 Show that the determinant of the coefficient matrix is equal to −(a + b3 + c3 − 3abc), and use Theorem 16.8.2 16.9 x1 = 41 x2 + 100, x2 = 2x3 + 80, x3 = 21 x1 Solution: x1 = 160, x2 = 240, x3 = 80 (a) Let x and y denote total production in industries A and I, respectively Then x = 16 x + 41 y + 60 and y = x + 41 y + 60 So 56 x − 41 y = 60 and − 41 x + 43 y = 60 (b) The solution is x = 320/3 and y = 1040/9 (a) No sector delivers to itself (b) The total amount of good i needed to produce one unit of each good (c) This column vector gives the number of units of each good which are needed to produce one unit of good j (d) No meaningful economic interpretation (The goods are usually measured in different units, so it is meaningless to add them together As the saying goes: “Don’t add apples and oranges!”) 0.8x1 − 0.3x2 = 120 and −0.4x1 + 0.9x2 = 90, with solution x1 = 225 and x2 = 200 The Leontief system for this three-sector model is 0.9x1 − 0.2x2 − 0.1x3 = 85 −0.3x1 + 0.8x2 − 0.2x3 = 95 , −0.2x1 − 0.2x2 + 0.9x3 = 20 which does have the claimed solution β 0 γ The sums of the elements in each column are less than provided α < 1, α 0 β < 1, and γ < 1, respectively Then, in particular, the product αβγ < The input matrix is A = The quantity vector x0 must satisfy (∗) (In − A)x0 = b and the price vector p0 must satisfy (∗∗) p0 (In − A) = v Multiplying (∗∗) from the right by x0 yields v x0 = (p0 (In − A))x0 = p0 ((In − A)x0 ) = p0 b Review Problems for Chapter 16 (a) 5(−2) − (−2)3 = −4 (b) − a (c) 6a b + 2b3 (d) λ2 − 5λ (a) −4 (b) (Subtract row from rows and Then subtract twice row from row The resulting determinant has only one nonzero term in its third row.) (c) (Use exactly the same row operations as in (b).) 1 1 , so A−1 = 2I2 − = 1 0 −1 −2 2 −1/2 −1/2 = = − 41 −2 2 −1/2 Transposing each side yields A−1 − 2I2 = −2 −2 −2 Hence, using (16.6.3), A = − 2 = (a) |At | = t + 1, so At has an inverse if and only if t = −1 (b) Multiplying the given equation from the right 0 −1 by A1 yields BA1 + X = I3 Hence X = I3 − BA1 = 0 −1 −2 −1 Essential Math for Econ Analysis, 4th edn EME4_Z01.TEX, 16 May 2012, 14:24 Page 732 www.downloadslide.net CHAPTER 17 733 |A| = (p + 1)(q − 2), |A + E| = 2(p − 1)(q − 2) A + E has an inverse for p = and q = Obviously, |E| = Hence |BE| = |B||E| = 0, so BE has no inverse −2 −t −3 t = 5t −45t +40 = 5(t −1)(t −8) So by Cramer’s t − −7 rule, there is a unique solution if and only if t = and t = The determinant of the coefficient matrix is We see that (I − A)(I + A + A2 + A3 ) = I + A + A2 + A3 − A − A2 − A3 − A4 = I − A4 = I Then use (16.6.4) (a) (In + aU)(In + bU) = In2 + bU + aU + abU2 = In + (a + b + nab)U, because U2 = nU, as is easily verified −3 −3 (b) A−1 = −3 −3 10 −3 −3 From the first equation, Y = B − AX Inserting this into the second equation and solving for X, yields X = 2A−1 B − C Moreover, Y = AC − B 10 (a) For a = and a = 2, there is a unique solution If a = 1, there is no solution If a = 2, there are infinitely many solutions (b) When a = and b1 − b2 + b3 = 0, or when a = and b1 = b2 , there are infinitely many solutions 11 −6 = A, so A2 + cA = 2I2 if c = −1 18 −10 (b) If B2 = A, then |B|2 = |A| = −2, which is impossible 11 (a) |A| = −2 A2 − 2I2 = 12 Note first that if A A = In , then rule (16.6.5) implies that A = A−1 , so AA = In But then (A B−1 A)(A BA) = A B−1 (AA )BA = A B−1 In BA = A (B−1 B)A = A In A = A A = In By rule (16.6.5) again, it follows that (A BA)−1 = A B−1 A 13 For once we use “unsystematic elimination” Solve the first equation to get y = − ax, then the second to get z = − x, and the fourth to get u = − y Substituting for all these in the third equation gives the result − ax + a(2 − x) + b(1 − + ax) = or a(b − 2)x = −2a + 2b + There is a unique solution provided that a(b − 2) = The solution is: x= 2b − 2a + , a(b − 2) y= 2a + b − , b−2 z= 2ab − 2a − 2b − , a(b − 2) u= − 2a b−2 14 |B3 | = |B|3 Because B is a × 3-matrix, we have |−B| = (−1)3 |B| = −|B| Since B3 = −B, it follows that |B|3 = −|B|, and so |B|(|B|2 + 1) = The last equation implies |B| = 0, and thus B can have no inverse 15 (a) The determinant on the left is equal to (a + x)d − c(b + y) = (ad − bc) + (dx − cy), and this is the sum of the determinants on the right (b) For simplicity look at the case r = 16 For a = b the solutions are x1 = 21 (a + b) and x2 = − 21 (a + b) If a = b, the determinant is for all values of x Chapter 17 17.1 (a) From Fig A17.1.1a we see that the solution is at the intersection of the two lines 3x1 +2x2 = and x1 +4x2 = Solution: max = 36/5 for (x1 , x2 ) = (8/5, 3/5) (b) From Fig.A17.1.1b we see that the solution is at the intersection of the two lines u1 + 3u2 = 11 and 2u1 + 5u2 = 20 Solution: = 104 for (u1 , u2 ) = (5, 2) Essential Math for Econ Analysis, 4th edn EME4_Z01.TEX, 16 May 2012, 14:24 Page 733 www.downloadslide.net 734 ANSWERS TO THE PROBLEMS x2 u2 3x1 + 4x2 = c 10u1 + 27u2 = c P P x1 Figure A17.1.1a 10 u1 Figure A17.1.1b (a) A graph shows that the solution is at the intersection of the lines −2x1 + 3x2 = and x1 + x2 = Hence max = 98/5 for (x1 , x2 ) = (9/5, 16/5) (b) The solution satisfies 2x1 + 3x2 = 13 and x1 + x2 = Hence max = 49 for (x1 , x2 ) = (5, 1) (c) The solution satisfies x1 − 3x2 = and x2 = Hence max = −10/3 for (x1 , x2 ) = (2, 2/3) (a) max = 18/5 for (x1 , x2 ) = (4/5, 18/5) (b) max = for (x1 , x2 ) = (8, 0) (c) max = 24 for (x1 , x2 ) = (8, 0) (d) = −28/5 for (x1 , x2 ) = (4/5, 18/5) (e) max = 16 for all (x1 , x2 ) of the form (x1 , − 21 x1 ) where x1 ∈ [4/5, 8] (f) min= −24 for (x1 , x2 ) = (8, 0) (follows from the answer to (c)) (a) No maximum exists Consider Fig A17.1.4 By increasing c, the dashed level curve x1 + x2 = c moves to the north-east and so this function can take arbitrarily large values and still have points in common with the shaded set (b) Maximum at P = (1, 0) The level curves are the same as in (a), but the direction of increase is reversed x2 −x1 + x2 = −1 x1 + x2 = c −x1 + 3x2 = −1 x1 −1 Figure A17.1.4 The slope of the line 20x1 + tx2 = c must lie between −1/2 (the slope of the flour border) and −1 (the slope of the butter border) For t = 0, the line is vertical and the solution is the point D in Fig in the text For t = 0, the slope of the line is −20/t Thus, −1 ≤ −20/t ≤ −1/2, which implies that t ∈ [20, 40] ⎧ ⎪ ⎨ 3x + 5y ≤ 3900 The LP problem is: max 700x + 1000y subject to x + 3y ≤ 2100 , x ≥ , y ≥ ⎪ ⎩ 2x + 2y ≤ 2200 A figure showing the admissible set and an appropriate level line for the objective function will show that the solution is at the point where the two lines 3x + 5y = 3900 and 2x + 2y = 2200 intersect Solving these equations yields x = 800 and y = 300 The firm should produce 800 sets of type A and 300 of type B Essential Math for Econ Analysis, 4th edn EME4_Z01.TEX, 16 May 2012, 14:24 Page 734 www.downloadslide.net CHAPTER 17 735 17.2 (a) (x1 , x2 ) = (2, 1/2) and u∗1 = 4/5 (b) (x1 , x2 ) = (7/5, 9/10) and u∗2 = 3/5 (c) Multiplying the two ≤ constraints by 4/5 and 3/5, respectively, then adding, we obtain (4/5)(3x1 + 2x2 ) + (3/5)(x1 + 4x2 ) ≤ · (4/5) + · (3/5), which reduces to 3x1 + 4x2 ≤ 36/5 u1 + 2u2 + u3 ≥ , 2u1 + 3u2 + u3 ≥ 8u1 + 13u2 + 6u3 subject to (a) 6u1 + 4u2 subject to 3u1 + u2 ≥ , 2u1 + 4u2 ≥ (b) max 11x1 + 20x2 subject to x1 + 2x2 ≤ 10 3x1 + 5x2 ≤ 27 u1 ≥ 0, u2 ≥ 0, u3 ≥ u1 ≥ 0, u2 ≥ , x1 ≥ 0, x2 ≥ (a) A graph shows that the solution is at the intersection of the lines x1 + 2x2 = 14 and 2x1 + x2 = 13 Hence max = for (x1∗ , x2∗ ) = (4, 5) u1 + 2u2 ≥ (b) The dual is 14u1 + 13u2 subject to , u1 ≥ 0, u2 ≥ A graph shows that the solution 2u1 + u2 ≥ is at the intersection of the lines u1 + 2u2 = and 2u1 + u2 = Hence = for (u∗1 , u∗2 ) = (1/3, 1/3) 17.3 (a) x = and y = gives max = 21 See Fig A17.3.1a, where the optimum is at P 4u1 + 3u2 ≥ (b) The dual problem is 20u1 + 21u2 subject to , u1 ≥ 0, u2 ≥ 5u1 + 7u2 ≥ u1 = and u2 = 1, which gives = 21 See Fig A17.3.1b (c) Yes y It has the solution u2 2x + 7y = c P P 20u1 + 21u2 = c x 1 Figure A17.3.1a u1 Figure A17.3.1b max 300x1 + 500x2 subject to 10x1 + 25x2 ≤ 10 000 , 20x1 + 25x2 ≤ 000 x1 ≥ , x2 ≥ The solution can be found graphically It is x1∗ = 0, x2∗ = 320, and the value of the objective function is 160 000, the same value found in Example 17.1.2 for the optimal value of the primal objective function (a) The profit from selling x1 small and x2 medium television sets is 400x1 +500x2 The first constraint, 2x1 +x2 ≤ 16, says that we cannot use more hours in division than the hours available The other constraints have similar interpretations (b) max = 3800 for x1 = and x2 = (c) Division should have its capacity increased 17.4 According to formula (1), z∗ = u∗1 Essential Math for Econ Analysis, 4th edn b1 + u∗2 b2 = · 0.1 + · (−0.2) = −0.2 EME4_Z01.TEX, 16 May 2012, 14:24 Page 735 www.downloadslide.net 736 ANSWERS TO THE PROBLEMS ⎧ ⎨ 6x1 + 3x2 ≤ 54 (a) max 300x1 + 200x2 subject to 4x1 + 6x2 ≤ 48 , ⎩ 5x1 + 5x2 ≤ 50 x1 ≥ 0, x2 ≥ where x1 and x2 are the number of units produced of A and B, respectively Solution: (x1 , x2 ) = (8, 2) (b) Dual solution: (u1 , u2 , u3 ) = (100/3, 0, 20) (c) Increase in optimal profit: π ∗ = u∗1 · + u∗3 · = 260/3 17.5 4u∗1 + 3u∗2 = > and x ∗ = 0; 5u∗1 + 7u∗2 = and y ∗ = > Also 4x ∗ + 5y ∗ = 15 < 20 and u∗1 = 0; 3x ∗ + 7y ∗ = 21 and u∗2 = > So (1) and (2) are satisfied (a) See Figure A17.5.2 The minimum is attained at (y1∗ , y2∗ ) = (3, 2) (b) The dual is: max 15x1 + 5x2 − 5x3 − 20x4 x1 + x2 − x3 + x4 ≤ , 6x1 + x2 + x3 − 2x4 ≤ s.t xj ≥ 0, j = 1, , The maximum is at (x1∗ , x2∗ , x3∗ , x4∗ ) = (1/5, 4/5, 0, 0) (c) If the first constraint is changed to y1 + 6y2 ≥ 15.1, the solution of the primal is still at the intersection of the lines (1) and (2) in Fig A17.5.2, but with (1) shifted up slightly The solution of the dual is completely unchanged In both problems the optimal value increases by (15.1 − 15) · x1∗ = 0.02 y2 (4) 10 (3) y1 + 2y2 = Z0 (2) (3, 2) (1) 10 15 y1 Figure A17.5.2 (a) 10 000y1 + 000y2 + 11 000y3 subject to (b) The dual is: 10y1 + 20y2 + 20y3 ≥ 300 , 20y1 + 10y2 + 20y3 ≥ 500 ⎧ ⎪ ⎨ 10x1 + 20x2 ≤ 10 000 max 300x1 + 500x2 subject to 20x1 + 10x2 ≤ 000 , ⎪ ⎩ 20x1 + 20x2 ≤ 11 000 y1 ≥ 0, y2 ≥ 0, y3 ≥ x1 ≥ 0, x2 ≥ Solution: max = 255 000 for x1 = 100 and x2 = 450 Solution of the primal: = 255 000 for (y1 , y2 , y3 ) = (20, 0, 5) (c) The minimum cost will increase by 2000 (a) For x3 = 0, the solution is x1 = x2 = 1/3 For x3 = 3, the solution is x1 = and x2 = (b) Let zmax denote the maximum value of the objective function If ≤ x3 ≤ 7/3, then zmax (x3 ) = 2x3 + 5/3 for x1 = 1/3 and x2 = x3 + 1/3 If 7/3 < x3 ≤ 5, then zmax (x3 ) = x3 + for x1 = x3 − and x2 = − x3 If x3 > 5, then zmax (x3 ) = for x1 = and x2 = Because zmax (x3 ) is increasing, the maximum is for x3 ≥ (c) The solution to the original problem is x1 = and x2 = 0, with x3 as an arbitrary number ≥ Essential Math for Econ Analysis, 4th edn EME4_Z01.TEX, 16 May 2012, 14:24 Page 736 www.downloadslide.net 737 CHAPTER 17 Review Problems for Chapter 17 (a) x ∗ = 3/2, y ∗ = 5/2 (A diagram shows that the solution is at the intersection of x + y = and −x + y = 1.) u1 − u2 + 2u3 ≥ , u1 ≥ 0, u2 ≥ 0, u3 ≥ (b) The dual is 4u1 + u2 + 3u3 subject to u1 + u − u ≥ Using complementary slackness, the solution of the dual is: u∗1 = 3/2, u∗2 = 1/2, and u∗3 = ⎧ −x1 + 2x2 ≤ 16 ⎪ ⎪ ⎪ ⎨ x1 − 2x2 ≤ , x1 ≥ 0, x2 ≥ Maximum at (x1 , x2 ) = (0, 8) (a) max −x1 + x2 subject to ⎪ − x2 ≤ −8 −2x ⎪ ⎪ ⎩ − 4x1 − 5x2 ≤ −15 (b) (y1 , y2 , y3 , y4 ) = ( 21 (b + 1), 0, b, 0) for any b satisfying ≤ b ≤ 1/5 (c) The maximand for the dual becomes kx1 + x2 The solution is unchanged provided that k ≤ −1/2 (a) x ∗ = 0, y ∗ = (A diagram shows that the solution is at the intersection of x = and 4x + y = 4.) (b) The dual problem is: max 4u1 + 3u2 + 2u3 − 2u4 4u1 + 2u2 + 3u3 − u4 ≤ u1 + u2 + 2u3 + 2u4 ≤ subject to u1 , u2 , u3 , u4 ≥ By complementary slackness, its solution is: u∗1 = 1, u∗2 = u∗3 = u∗4 = x2 4000 500x1 + 250x2 = c 3000 2000 1000 P 1000 2000 3000 4000 x1 Figure A17.R.4 (a) See Fig A17.R.4 The solution is at P , where (x1 , x2 ) = (2000, 2000/3); (b) See SM (c) a ≤ 1/24 (a) If the numbers of units produced of the three goods are x1 , x2 , and x3 , the profit is 6x1 + 3x2 + 4x3 , and the time spent on the two machines is 3x1 + x2 + 4x3 and 2x1 + 2x2 + x3 , respectively The LP problem is therefore max 6x1 + 3x2 + 4x3 subject to 3x1 + x2 + 4x3 ≤ b1 , 2x1 + 2x2 + x3 ≤ b2 (b) The dual problem is obviously as given Optimum at P = (y1∗ , y2∗ ) = (3/2, 3/4) and (e) see SM Essential Math for Econ Analysis, 4th edn x1 , x2 , x3 ≥ (c) x1∗ = x2∗ = 25 EME4_Z01.TEX, 16 May 2012, 14:24 For (d) Page 737 www.downloadslide.net Essential Math for Econ Analysis, 4th edn EME4_Z01.TEX, 16 May 2012, 14:24 Page 738 www.downloadslide.net INDEX A Absolute extreme point/value (see Global extreme point/value) Absolute risk aversion, 193 Absolute value, 30 Active constraint, 530 Adjoint of matrix, 610 Admissible set LP, 627 NLP, 527, 532 Affine function, 397 Alien cofactor, 602 Annuity due, 366 ordinary, 360, 366 Antiderivative, 294 Approximations linear, 217, 441 quadratic, 222 higher-order, 223 Areas under curves, 299–301 Arithmetic mean, 29, 56, 397 Arithmetic series, 59 Associative law (of matrix multiplication), 556 Asymptote, 238, 240, 244 Asymptotic stability difference equations, 373 differential equations, 338 Augmented coefficient matrix, 568 Essential Math for Econ Analysis, 4th edn Average cost, 133, 182 Average elasticity, 229 Average rate of change, 165 B Bernoulli’s inequality, 78 Binding constraint, 530 Binomial coefficients, 57 Binomial formula, 57 Bordered Hessian, 516 Boundary point, 482, 488 Bounded interval, 29 Bounded set, 483, 488 Budget constraint, 69, 581 Budget plane (set), 69, 94, 394, 483, 581 C Cardinality (of a set) 73 Cartesian coordinate system, 86 Cauchy–Schwarz inequality, 104, 576 CES function, 254, 406, 430, 435, 439 Chain rule one variable, 184, 187 several variables, 412, 417, 418 Change of variables (in integrals), 319–322 Circle area, circumference, 645 equation for, 147 C k function, 403 Closed interval, 29 Closed set, 482, 488 Cobb–Douglas function, 254, 378, 390, 397, 404, 406, 407, 418, 429, 431, 433, 437, 500, 503, 514, 519, 524 Codomain, 151 Coefficient matrix, 549 Cofactor, 601 Cofactor expansion (of a determinant), 589, 601, 603 Column (of a matrix), 548 Column vector, 548, 570 Compact set, 483, 488 Comparison test for convergence of integrals, 327 Complement (of a set), 71 Complementary inequalities, 528 Complementary slackness, LP, 639 NLP, 527, 533, 538 Completing the square, 42 Composite functions, 134, 418 Compound functions, 145 Compound interest, 6, 116, 345 Concave function one variable, 190, 289 EME4_Z02.TEX, 18 May 2012, 22:20 Page 739 www.downloadslide.net 740 INDEX Concave function (continued) two variables, 467 Cone, 435, 646 Consistent system of equations, 450, 546 Constant returns to scale, 437 Consumer demand, 398 Consumer surplus, 313 Consumption function, 96 Continuous compounding, 349 Continuous depreciation, 116 Continuous function one variable, 234, n variables, 398 one-sided, 239 properties of, 235 Continuously differentiable, 403 Convergence of general series, 357 of geometric series, 355 of integrals, 324–328 of sequences, 249 Convex function one variable, 190, 289 two variables, 467 Convex polyhedron, 627 Convex set, 466 Correlation coefficient, 577 Counting rule, 449 Covariance (statistical), 479, 577 Cramer’s rule two unknowns, 586 three unknowns, 590 n unknowns, 613 Critical point (see Stationary point) Cross-partials, 401 Cubic function, 105 Cumulative distribution function, 310 D Decreasing function, 84, 163 Decreasing returns to scale, 437 Deductive reasoning, 68 Definite integral, 302 Degrees of freedom, 450 linear systems, 567 Demand and supply, 97, 129, 130, 211–212, 422, 424, 428 Demand functions, 437, 500, 518 Denominator, 14 Essential Math for Econ Analysis, 4th edn Dependent (endogenous) variable, 81, 378 Depreciation, 8, 98, 116, 350 Derivative (one variable) definition, 158 higher order, 188, 192 left, 243 recipe for computing, 159 right, 243 Descartes’s folium, 256 Determinants × 2, 586 × 3, 589 n × n, 594 by cofactors, 589, 601, 603 geometric interpretations, 587, 591 rules for, 596 Difference equations, 371–373 linear, 372 Difference of sets, 70 Difference-of-squares formula, 11 Differentiable function, 174 Differential equation, 330 for logistic growth, 333 for natural growth, 331 linear, 338 separable, 336 Differentials one variable, 218 two variables, 444 n variables, 447 first (second) order, 448 geometric interpretation, 444 invariance of, 447 partial derivatives from, 445 rules for differentials, 219, 446 Differentiation, 174 Direct partials, 401 Discontinuous function, 234 Discounted value, 352 continuous income stream, 363 Discount factor (rate), 351 Discriminating monopolist, 464, 475 Discriminating monopsonist, 477 Disjoint sets, 71 Distance formula in ‫ޒ‬, 30 in ‫ޒ‬2 , 147 in ‫ޒ‬3 , 395 in ‫ޒ‬n , 487 Distributive laws (of matrix multiplication), 556 Divergence of general series, 357 of geometric series, 356 of integrals, 324–328 of sequences, 249 Dollar cost averaging, 38, 400 Domain (of a function) general case, 151 one variable, 80, 83 two variables, 377, 379 n variables, 396 Dot product (of vectors), 571 Double sums, 59 Doubling time, 115, 121 Duality theorem (LP), 634 Dual problem (LP), 631, 632 Duopoly, 464, 481 E e (=2.7182818284590 ), 117, 201, 250 Economic growth, 337, 340, 341 Effective interest rate, 347, 348, 350 Elastic function, 231 Elasticities, one variable, 230 two variables, 406 n variables, 407 logarithmic derivatives, 231, 406, 407 rules for, 232 Elasticity of substitution, 429, 430, 435 Elementary operations, 566 Elements of a matrix, 548 of a set, 69 Ellipse, 148 Ellipsoid, 394 Empty set, 71 Endogenous variables, 39, 81, 378, 456 Engel elasticities, 232 Entries (of a matrix), 548 Envelope theorems, 492, 523 Equilibrium demand theory, 97 difference equations, 373 differential equations, 338 equation system, 456 Equivalence, 62–63 Equivalence arrow (⇐⇒), 63 EME4_Z02.TEX, 18 May 2012, 22:20 Page 740 www.downloadslide.net INDEX Euclidean n-dimensional space (‫ޒ‬n ), 399, 575 Euler’s theorem for homogeneous functions two variables, 431 n variables, 436 Even function, 135 Exhaustion method, 299 Exogenous variables, 39, 81, 378, 456 Exp, 117 Exponential distribution, 324, 329 Exponential function, 114–117, 194, 196 properties of, 116, 196 Extreme points and values one variable, 260, 281 two variables, 462 n variables, 487 Extreme points (LP), 628 Extreme value theorem one variable, 270 two variables, 484 n variables, 488 F Factorials, 58 Factoring, 12 Feasible set LP, 627 NLP, 537, 532 First-derivative test global extrema, 263 local extrema, 282 First-order conditions (FOC) one variable, 261 two variables, 462, 470 n variables, 488 with equality constraints, 499, 516, 519 with inequality constraints, 527, 533, 538 FMEA, xiii Fractional powers, 19 Freedom, degrees of, 450, 567 Functions one variable, 80 two variables, 377 n variables, 396 composite, 134, 418 compound, 145 concave, 190, 289, 467 Essential Math for Econ Analysis, 4th edn continuous, 234, 398 convex, 190, 289, 467 cubic, 105 decreasing, 84, 163 differentiable, 174 discontinuous, 234 even, 135 exponential, 114–117, 194, 196 general concept, 151 graph of, 87, 387, 399 homogeneous, 431, 435 homothetic, 438 increasing, 84, 163 inverse, 138, 152, 214 linear, 89, 397 logarithmic, 122, 123, 197, 200 log-linear, 216, 231, 397 odd, 136 one-to-one, 137, 152 polynomial, 106 power, 112, 201 quadratic, 99 rational, 110 symmetric, 135, 136 Fundamental theorem of algebra, 106 Future value of continuous income stream, 363 Future value (of an annuity) continuous, 363 discrete, 361 G Gaussian density function, 118, 328, 330 Gaussian elimination, 565–569 Gauss–Jordan method, 568 Generalized power rule, 184 Geometric mean, 29, 397 Geometric series, 353–355 Giffen good, 97 Global extreme point/value one variable, 260, 281 two variables, 462 n variables, 487 Graph of a function one variable, 87 two variables, 387 n variables, 399 Graph of an equation, 143, 393, 424 Greek alphabet, 647 Growth factor, 7, 741 Growth towards a limit, 332 H Hadamard product, 551 Half-open interval, 29 Harmonic mean, 29, 397, 400 Harmonic series, 357 Hessian matrix, 401 bordered, 516 Higher-order derivatives one variable, 188, 192 two variables, 384–385 n variables, 401–402 of composite functions, 414, 416 Higher-order polynomial approximations, 223 Homogeneous functions two variables, 431 n variables, 435 geometric interpretations, 433, 434 Homogeneous systems of linear equations, 614–615 Homothetic functions, 438 Hotelling’s lemma, 493 Hyperbola, 110, 149 Hyperplane, 399, 581 Hypersurface, 399 I Idempotent matrix, 562 Identity matrix, 559 Iff (if and only if), 63 Image (of a function), 152 Implication, 62 Implication arrow ( ⇒), 62 Implicit differentiation, 207, 420, 421, 425, 427 Improper integrals, 324–328 Improper rational function, 110 Inactive constraint, 530 Income distribution, 310–312 Inconsistent (system of equations), 450, 546 Increasing function, 84, 163 Increasing returns to scale, 437 Increment (of a function), 444 Incremental cost, 166 Indefinite integral, 294 Independent (exogenous) variable, 81, 378, 456 EME4_Z02.TEX, 18 May 2012, 22:20 Page 741 www.downloadslide.net 742 INDEX Indeterminate form, 251, 253 Indifference curve, 502 Indirect proof, 67 Indirect utility function, 523 Individual demand functions, 518 Induction proof, 75 Inductive reasoning, 68 Inelastic function, 231 Inequalities, 24–27 Infinite geometric series, 355 Infinite sequence, 249 Infinity (∞), 30, 238 Inflection point, 287 test for, 287 Inner product (of vectors), 571 rules for 572 Input–output model of Leontief, 616–619 Insoluble integrals, 307 Instantaneous rate of change, 165 Integer, Integer roots (of polynomial equations), 107 Integral definite, 302 improper, 324–328 indefinite, 294 infinite limits, 324–326 Newton–Leibniz, 307 Riemann integral, 307 unbounded integrand, 326–328 Integrand, 294 Integrating factor, 339 Integration, by parts, 315–317 by substitution, 319–322 of rational functions, 322 Interest rate, 7, 345 Interior (of a set), 488 Interior point, 482, 488 Intermediate value theorem, 245 Internal rate of return, 369 Intersection (of sets), 70 Interval, 29 Invariance of the differential, 447 Inverse functions formula for the derivative, 214 general definition, 138, 152 geometric characterization, 140 Inverse matrix, 604 by elementary operations, 611 general formula, 610 properties of, 607 Essential Math for Econ Analysis, 4th edn Invertible matrix, 604 Investment projects, 369 Involutive matrix, 600 Irrational numbers, 3, 250 Irremovable discontinuity, 234 IS–LM model, 457 Isoquant, 390, 399 K Kernel (of a composite function), 134 Kink in a graph, 243 Kuhn–Tucker necessary conditions, 528, 533, 538 L Laffer curve, 80 Lagrange multiplier method one constraint, 499, 516 several constraints, 519 economic interpretations, 504, 522 NLP, 527, 533, 538 Lagrange’s form of the remainder, 226, 228, 274 theorem, 511 Lagrangian, 498, 516, 519, 527, 533, 538 Laspeyres’s price index, 54 Law of natural growth, 331–332 LCD (least common denominator), 16 Left continuous, 239 Left derivative, 243 Left limit, 239 Lemniscate, 210 Length (of a vector), 488, 576 Leontief matrix, 618 Leontief model, 616–619 Level curve, 388 Level surface, 399 L’Hôpital’s rule, 252, 253 Limits, 169–173, 237–243 at infinity, 240 ε–δ definition, 243 one-sided, 238–239 rules for, 171 Line in ‫ޒ‬2 , 89 in ‫ޒ‬3 , 578–579 in ‫ޒ‬n , 579 Linear algebra, 545 Linear approximation one variable, 217 two variables, 441 n variables, 441 Linear combination of vectors, 570 Linear difference equation, 372 Linear differential equation, 338 Linear expenditure systems, 398, 506 Linear function, one variable, 89 n variables, 397 Linear inequalities, 93 Linear models, 95 Linear regression, 478–480 Linear systems of equations two variables, 46–47 n variables, 546 in matrix form, 555 Local extreme point/value one variable, 281 two variables, 470 Logarithmic differentiation, 200 Logarithms natural, 119, 122 properties of, 120, 123, 200 with bases other than e, 123 Logical equivalence, 62–63 Logistic differential equation, 333 function, 333 growth, 333 Log-linear relations, 216, 231, 397 Lower triangular matrix, 595 LP (linear programming), 623 Luxury good, 408 M Macroeconomic models, 38–39, 50, 210–213, 220, 256, 451, 454–455, 457–458, 547, 588, 592 Main diagonal (of a matrix), 548 Malthus’s law, 332 Mapping, 152 Marginal cost (MC), 166 product, 167, 404 propensity to consume, 96, 167 propensity to save (MPS), 168 EME4_Z02.TEX, 18 May 2012, 22:20 Page 742 www.downloadslide.net INDEX rate of substitution (MRS), 428, 439 tax rate, 168 utility, 523 Mathematical induction, 75–76 Matrix, 548 adjoint, 610 Hessian, 401 idempotent, 562 identity, 559 inverse, 604, 610 involutive, 600 lower triangular, 595 multiplication, 553 nonsingular, 605 order of, 548 orthogonal, 564 powers of, 558 product, 553 singular, 605 skew-symmetric, 583 square, 548 symmetric, 563 transpose, 562 upper triangular, 566, 594 zero, 550 Maximum and minimum (global) one variable, 259, 281 two variables, 466, 470 n variables, 487 Maximum and minimum (local) one variable, 281 two variables, 462 n variables, 489 Mean arithmetic, 29, 56, 397 geometric, 29, 397 harmonic, 29, 397, 400 Mean income, 311 Mean value theorem, 273 Members (elements) of a set, 69 Minimum (see Maximum and minimum) Minor, 601 Mixed partials, 401 Monopolist (discriminating), 464, 475 Monopoly problem, 101, 268 Monopsonist (discriminating), 477 Mortgage repayment, 366, 373 MRS (marginal rate of substitution), 428, 439 Multiplier–accelerator model, 372 Essential Math for Econ Analysis, 4th edn N Natural exponential function, 117, 194 properties of, 196 Natural logarithm, 119–123 properties of, 120 Natural number, n-ball, 488 Necessary conditions, 63 Nerlove–Ringstad production function, 428 Net investment, 315 Newton–Leibniz integral, 307 Newton quotient, 158 Newton’s binomial formula, 57 Newton’s law of cooling, 335 Newton’s method (approximate roots), 247 convergence, 248 NLP (see Nonlinear programming) Nonnegativity constraints in LP, 627 in NLP, 537–540 Nonsingular matrix, 605 Nontrivial solution, 614 Norm (of a vector), 488, 576 Normal (Gaussian) distribution, 118, 328, 330 n-space (‫ޒ‬n ), 399, 575 nth-order derivative, 192 nth power, nth root, 20 Numerator, 14 n-vector, 396, 548, 570 O Objective function (LP), 627 Odd function, 136 Oil extraction, 183, 309 One-sided continuity, 239 One-sided limits, 238–239 One-to-one function, 137, 152 Open interval, 29 Open set, 482, 488 Optimal value function (see Value function) Order (of a matrix), 548 Ordered pair, 87 Ordinary least-square estimates, 479 Orthogonality in econometrics, 577 Orthogonal matrix, 564 743 Orthogonal projection, 578 Orthogonal vectors, 576 P Paasche’s price index, 54 Parabola, 99 Paraboloid, 389 Parameter, 38 Pareto’s income distribution, 177, 311 Partial derivatives two variables, 381–384 n variables, 400–403 geometric interpretation, 390–391 higher-order, 384–385, 401–402 Partial elasticities two variables, 406 n variables, 407 as logarithmic derivatives, 406, 407 Pascal’s triangle, 58 Peak load pricing, 540 Perfectly competitive firm, 102 Perfectly elastic/inelastic function, 231 Periodic decimal fraction, Periodic rate (of interest), 345 Plane in ‫ޒ‬3 , 393, 580 in ‫ޒ‬n , 581 Point–point formula, 92 Point–slope formula, 91 Pollution and welfare, 405, 413–414 Polynomial, 106 Polynomial division, 108–109 Population growth, 114–115 Postmultiply (a matrix), 554 Power function, 112, 201 Power rule for differentiation, 176 Powers of matrix, 558 Premultiply (a matrix), 554 Present (discounted) value, 352, 360 continuous income stream, 362 of an annuity, 360 Price adjustment mechanism, 339 Price elasticity of demand, 229, 408 Price indices, 53, 54 Primal problem, (LP) 631, 632 Principle of mathematical induction, 76 Producer surplus, 314 EME4_Z02.TEX, 18 May 2012, 22:20 Page 743 www.downloadslide.net 744 INDEX Production functions, 191, 289, 378, 390, 404, 428, 433, 437, 439, 445, 459 Profit function, 133, 493 Profit maximization, 276, 285, 463, 464, 472, 493 Proof by induction, 75 direct, 67 indirect, 67 Proper rational function, 110 Proportional rate of change, 165 Pyramid, 646 Pythagoras’s theorem, 647 Q Quadratic Quadratic Quadratic Quadratic Quadratic approximation, 222 equations, 41 formula, 43 function, 99 identities, 10 R Range (of a function) one variable, 80, 83 two variables, 378 general case, 152 Rate of change, 165 Rate of extraction, 309 Rate of interest, 7, 345 Rate of investment, 166 Rational function, 110 Rational number, Real number, Real wage rate, 182 Recurring decimal fraction, Rectangular distribution, 329 Reduced form (of a system of equations), 39, 456 Relative extreme point/value (see Local extreme point/value) Relative rate of change, 165 Relative risk aversion, 193 Remainder theorem, 106 Removable discontinuity, 234 Revenue function, 133 Riemann integral, 307 Right continuous, 239 Right derivative, 243 Right limit, 239 Risk aversion, 193 Essential Math for Econ Analysis, 4th edn ‫ޒ‬n , 399, 575 Roots of polynomial equations, 106 of quadratic equations, 43 Row (of a matrix), 548 Row vector, 548, 570 Roy’s identity, 524 Rule of 70, 218 S Saddle point, 462, 470 second-order conditions for, 471 Sarrus’s rule, 591 Scalar product (of vectors), 571 Search model, 426 Second-derivative test (global) one variable, 264 two variables, 466 with constraints, 516, 528 Second-derivative test (local) one variable, 283 two variables, 471 with constraint, 515 Second-order conditions (global) one variable, 264 two variables, 466 with constraints, 516, 528 Second-order conditions (local) one variable, 283 two variables, 471 with constraints, 515 Separable differential equations, 336 Sequence (infinite), 249 Series (general), 357 Set difference (minus), 70 Shadow price, 504, 522, 636 Shephard’s lemma, 525 Sign diagram, 25 Singular matrix, 605 Skew-symmetric matrix, 583 Slope of a curve, 156 of a level curve, 420 of a straight line, 89, 90 Sphere equation for, 395 surface area, volume, 646 Square matrix, 548 Square root, 19 Stability difference equations, 373 differential equations, 338 Stationary point one variable, 260 two variables, 462 n variables, 488 Straight line depreciation, 98 Straight line point–point formula of, 92 point–slope formula of, 91 slope of, 89, 90 Strictly concave (convex) function, 290 Strictly increasing (decreasing) function, 84, 163 Strict maximum/minimum point, 260 local, 470 Structural form (of a system of equations), 39, 456 Subset, 70 Substitutes (in consumption), 406 Sufficient conditions, 63 Summation formulas binomial, 57 finite geometric series, 354 infinite geometric series, 355 other sums, 57 Summation notation, 51 Supply and demand (see Demand and supply) Supply curve, 102 Surface, 393, 399 Symmetric function, 100, 135, 136 Symmetric matrix, 563 T Tangent, 157–158 Tangent plane, 442 Target (of a function), 151 Taylor polynomial, 223, 225 Taylor’s formula, 226 Total derivative, 412 Transformation, 152 Translog cost function, 440 Transpose of a matrix, 562 rules for, 563 Triangle, 645 Triangle inequality, 34, 578 Trivial solution, 614 U Uniform distribution, 329 EME4_Z02.TEX, 18 May 2012, 22:20 Page 744 www.downloadslide.net INDEX Union (of sets), 70 Unit elastic function, 231 Universal set, 71 Upper triangular matrix, 566, 594 Utility function, 398 Utility maximization, 501–502, 518–519 V Value function, equality constraints, 504, 522 inequality constraint, 536 unconstrained, 491, 492 Variance (statistical), 56, 479, 577 Essential Math for Econ Analysis, 4th edn Vectors, 548, 570 angle between, 577 column, 548, 570 geometric interpretation, 573–576 inner (scalar) product of, 571 linear combination of, 570 norm (length), 488, 576 orthogonal, 576 row, 548, 570 Venn diagram, 72 Vertex (of a parabola), 99 Vertical asymptote, 238 Vertical-line test, 144 745 W Wicksell’s law, 459 w.r.t (with respect to), 161 Y y-intercept, 90 Young’s theorem, 402 Z Zero (0), division by, Zero matrix, 550 Zero of a polynomial, 106 Zeros of a quadratic function, 43 EME4_Z02.TEX, 18 May 2012, 22:20 Page 745 www.downloadslide.net Essential Math for Econ Analysis, 4th edn EME4_Z02.TEX, 18 May 2012, 22:20 Page 746 ... Furthermore, for many economics students, it may be some years since their last formal mathematics course Accordingly, as mathematics becomes increasingly essential for specialist studies in economics, ... economic theory, and a great deal more in econometrics The purpose of Essential Mathematics for Economic Analysis, therefore, is to help economics students acquire enough mathematical skill to access... divided by zero.” (Steven Wright) Essential Math for Econ Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page www.downloadslide.net CHAPTER / INTRODUCTORY TOPICS I: ALGEBRA PROBLEMS FOR SECTION