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QABEBook_2011 - ECON 1202 ECON 2291 Quantitative Analysis...

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ECON 1202 & ECON 2291 Quantitative Analysis for Business and Economics Lecture Book 2011 u1D465 u1D466 u1D467 u1D453 u1D465 ( u1D465 0 , u1D466 0 ) School of Economics The University of NSW
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ii Based on Quantitative Methods A: Lecture Book , first edition written and produced by Simon Angus, School of Economics, University of NSW, February–May 2006. Revisions and additional materials prepared by Christopher Bidner, Loretti Isabella Dobrescu, Kevin Fox and Judith Watson. Typeset in L A T E X2 u1D700 , a document preparation language, using typeface Computer Modern. Significant packages supporting this document in- clude beamer and pstricks . c 2010 by the School of Economics, UNSW; Sydney, Australia. All rights reserved. No part of this document may be reproduced, stored in a re- trieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without written per- mission from the School.
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Contents 1 Introducing QABE 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Learning in QABE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 What is a Function? . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.8 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Time Value of Money 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 A Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Option 1: Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Option 2: Compound Interest . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Option 3: Continuously Compounded Interest . . . . . . . . . . . . . 19 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 Present Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Evaluating Time-Money Choices 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Equations of value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Net Present Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Internal Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Geometric Progressions and Annuities 31 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Geometric Progressions . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Annuities Due . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Matrices I: Maths by Arrangement 39 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.3 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 iii
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iv CONTENTS 5.4 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6 Matrices II: The Inverse & Determinant in Small Matrices 47 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 The Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.3 Determinant Excursus . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.4 The Inverse Really . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.5 On Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7 Matrices III: Matrix Algebra & Automatic-Matrices! 55 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2 More on Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.3 Matrices on Computers . . . . . . . . . . . . . . . . . . . . . . . . . 58 8 Probability I: Permutations and Combinations 61 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8.2 Why Counting? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.3 Basic Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.4 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 9 Probability II: Probability in action 69 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9.2 Probability Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9.3 Rules of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 9.4 Bayes’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 10 Markov Chains 77 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 10.2 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 10.3 Transitions, Regularity and States . . . . . . . . . . . . . . . . . . . 79 10.4 Markov Chains in Game Theory . . . . . . . . . . . . . . . . . . . . 84 11 Linear Programming I: Solving problems in a world of constraints 87 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 11.2 The Business Headache . . . . . . . . . . . . . . . . . . . . . . . . . 88 11.3 Introduction to Linear programming . . . . . . . . . . . . . . . . . . 88 11.4 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 12 Linear Programming II: Dealing with a Changing World 97 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 12.2 Linear programming usual suspects . . . . . . . . . . . . . . . . . . . 97 12.3 Variations in the LP problem . . . . . . . . . . . . . . . . . . . . . . 98 13 Linear Programming III: Using Solver 105 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 13.2 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 13.3 Using Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 13.4 Changing the Objective Function: Multiple Solutions . . . . . . . . . 113 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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CONTENTS 14 Differentiation: Responding to Change 117 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 14.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 14.3 Rates of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 14.4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 15 Differentiation II: Tricks and Extensions 127 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 15.2 Implicit Differentiation
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