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Microeconomics A

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Microeconomics A. Daripa EC2066 2016 Undergraduate study in Economics, Management, Finance and the Social Sciences This subject guide is for a 200 course offered as part of the University of London International Programmes in Economics, Management, Finance and the Social Sciences. This is equivalent to Level 5 within the Framework for Higher Education Qualifications in England, Wales and Northern Ireland (FHEQ). For more information about the University of London International Programmes undergraduate study in Economics, Management, Finance and the Social Sciences, see: www.londoninternational.ac.uk This guide was prepared for the University of London International Programmes by: Dr Arup Daripa, Lecturer in Financial Economics, Department of Economics, Mathematics and Statistics, Birkbeck, University of London. This is one of a series of subject guides published by the University. We regret that due to pressure of work the author is unable to enter into any correspondence relating to, or arising from, the guide. If you have any comments on this subject guide, favourable or unfavourable, please use the form at the back of this guide. University of London International Programmes Publications Office Stewart House 32 Russell Square London WC1B 5DN United Kingdom www.londoninternational.ac.uk Published by: University of London © University of London 2016 The University of London asserts copyright over all material in this subject guide except where otherwise indicated. All rights reserved. No part of this work may be reproduced in any form, or by any means, without permission in writing from the publisher. We make every effort to respect copyright. If you think we have inadvertently used your copyright material, please let us know. Contents Contents 1 Introduction 1 1.1 Routemap to the subject guide . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Introduction to the subject area and prior knowledge . . . . . . . . . . . 2 1.3 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Aims of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Learning outcomes for the course . . . . . . . . . . . . . . . . . . . . . . 3 1.6 Overview of learning resources . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.1 The subject guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6.4 Online study resources . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6.5 The VLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6.6 Making use of the Online Library . . . . . . . . . . . . . . . . . . 7 Examination advice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.1 Format of the examination . . . . . . . . . . . . . . . . . . . . . . 7 1.7.2 Types of questions . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.3 Specific advice on approaching the questions . . . . . . . . . . . . 8 1.7 2 Consumer theory 2.1 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Preferences, utility and choice . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Preferences and utility . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Indifference curves . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.4 Utility maximisation . . . . . . . . . . . . . . . . . . . . . . . . . 16 Demand curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 i Contents 2.4.1 The impact of income and price changes . . . . . . . . . . . . . . 20 2.4.2 Elasticities of demand . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.3 The compensated demand curve . . . . . . . . . . . . . . . . . . . 24 2.4.4 Welfare measures: ?CS, CV and EV . . . . . . . . . . . . . . . . 25 2.5 Labour supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Saving and borrowing: intertemporal choice . . . . . . . . . . . . . . . . 30 2.7 Present value calculation with many periods . . . . . . . . . . . . . . . . 33 2.7.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.8 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 34 2.9 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 35 2.9.1 35 Sample examination questions . . . . . . . . . . . . . . . . . . . . 3 Choice under uncertainty 3.1 37 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Expected utility theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Risk aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 Risk aversion and demand for insurance . . . . . . . . . . . . . . . . . . 40 3.6.1 Insurance premium for full insurance . . . . . . . . . . . . . . . . 40 3.6.2 How much insurance? . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Risk-neutral and risk-loving preferences . . . . . . . . . . . . . . . . . . . 43 3.8 The Arrow–Pratt measure of risk aversion . . . . . . . . . . . . . . . . . 44 3.9 Reducing risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.10 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 46 3.11 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 46 3.11.1 Sample examination questions . . . . . . . . . . . . . . . . . . . . 46 4 Game theory 4.1 ii 49 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Contents 4.1.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Simultaneous-move or normal-form games . . . . . . . . . . . . . . . . . 50 4.3.1 Dominant and dominated strategies . . . . . . . . . . . . . . . . . 51 4.3.2 Dominated strategies and iterated elimination . . . . . . . . . . . 52 4.3.3 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3.4 Mixed strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.5 Existence of Nash equilibrium . . . . . . . . . . . . . . . . . . . . 58 4.3.6 Games with continuous strategy sets . . . . . . . . . . . . . . . . 59 Sequential-move or extensive-form games . . . . . . . . . . . . . . . . . . 59 4.4.1 Actions and strategies . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.2 Finding Nash equilibria using the normal form . . . . . . . . . . . 61 4.4.3 Imperfect information: information sets . . . . . . . . . . . . . . . 61 Incredible threats in Nash equilibria and subgame perfection . . . . . . . 63 4.5.1 Subgame perfection: refinement of Nash equilibrium . . . . . . . . 64 4.5.2 Perfect information: backward induction . . . . . . . . . . . . . . 65 4.5.3 Subgame perfection under imperfect information . . . . . . . . . . 66 Repeated Prisoners’ Dilemma . . . . . . . . . . . . . . . . . . . . . . . . 68 4.6.1 Cooperation through trigger strategies . . . . . . . . . . . . . . . 69 4.6.2 Folk theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.7 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 74 4.8 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 74 4.8.1 74 4.4 4.5 4.6 Sample examination questions . . . . . . . . . . . . . . . . . . . . 5 Production, costs and profit maximisation 5.1 79 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 A general note on costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Production and factor demand . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4.1 Marginal and average product . . . . . . . . . . . . . . . . . . . . 80 5.5 The short run: one variable factor . . . . . . . . . . . . . . . . . . . . . . 81 5.6 The long run: both factors are variable . . . . . . . . . . . . . . . . . . . 83 5.6.1 83 Isoquants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Contents 5.6.2 Diminishing MRTS . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.6.3 Returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6.4 Optimal long-run input choice . . . . . . . . . . . . . . . . . . . . 84 Cost curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.7.1 Marginal cost and average cost . . . . . . . . . . . . . . . . . . . 84 5.7.2 Fixed costs and sunk costs . . . . . . . . . . . . . . . . . . . . . . 85 5.7.3 Short run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.7.4 Long run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.8 Profit maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.9 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 91 5.10 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 91 5.10.1 Sample examination questions . . . . . . . . . . . . . . . . . . . . 91 5.7 6 Perfect competition in a single market 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 A general comment on zero profit . . . . . . . . . . . . . . . . . . . . . . 94 6.4 Supply decision by a price-taking firm . . . . . . . . . . . . . . . . . . . . 95 6.4.1 Which types of firms have a supply curve? . . . . . . . . . . . . . 95 6.4.2 Short-run supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4.3 Long-run supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Market supply and market equilibrium . . . . . . . . . . . . . . . . . . . 97 6.5.1 Short run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.5.2 Long run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.6 Producer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.7 Applications of the supply-demand model: partial equilibrium analysis . . 99 6.7.1 Tax: deadweight loss and incidence . . . . . . . . . . . . . . . . . 99 6.7.2 Price ceiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.7.3 Price floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.7.4 Quota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.7.5 Price support policy . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.7.6 Tariffs and quotas . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 108 6.5 6.8 iv 93 Contents 6.9 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 109 6.9.1 109 Sample examination questions . . . . . . . . . . . . . . . . . . . . 7 General equilibrium and welfare 7.1 111 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.3 General equilibrium in an exchange economy . . . . . . . . . . . . . . . . 112 7.4 Existence of equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.5 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.6 Welfare theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.6.1 The first theorem of welfare economics . . . . . . . . . . . . . . . 120 7.6.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.6.3 The second theorem of welfare economics . . . . . . . . . . . . . . 122 7.6.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.7 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.8 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 126 7.9 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 127 7.9.1 127 Sample examination questions . . . . . . . . . . . . . . . . . . . . 8 Monopoly 8.1 129 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.3 Properties of marginal revenue . . . . . . . . . . . . . . . . . . . . . . . . 130 8.4 Profit maximisation and deadweight loss . . . . . . . . . . . . . . . . . . 130 8.5 Price discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.6 Natural monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.7 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 135 8.8 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 135 8.8.1 135 Sample examination questions . . . . . . . . . . . . . . . . . . . . v Contents 9 Oligopoly 9.1 137 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.3 Cournot competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.3.1 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.3.2 Cournot with n > 2 firms . . . . . . . . . . . . . . . . . . . . . . 141 9.3.3 Stackelberg leadership . . . . . . . . . . . . . . . . . . . . . . . . 142 Bertrand competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.4.1 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Bertrand competition with product differentiation . . . . . . . . . . . . . 143 9.5.1 Sequential pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.6 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 146 9.7 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 146 9.7.1 146 9.4 9.5 Sample examination questions . . . . . . . . . . . . . . . . . . . . 10 Asymmetric information: adverse selection vi 149 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 150 10.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 150 10.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 10.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 10.3 The scope of economic theory: a general comment . . . . . . . . . . . . . 151 10.4 Akerlof’s (1970) model of the market for lemons . . . . . . . . . . . . . . 152 10.4.1 The market for lemons: an example with two qualities . . . . . . . 152 10.5 A model of price discrimination . . . . . . . . . . . . . . . . . . . . . . . 153 10.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10.5.2 The full information benchmark . . . . . . . . . . . . . . . . . . . 155 10.5.3 Contracts under asymmetric information . . . . . . . . . . . . . . 155 10.6 Spence’s (1973) model of job market signalling . . . . . . . . . . . . . . . 158 10.7 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 159 10.8 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 160 10.8.1 Sample examination questions . . . . . . . . . . . . . . . . . . . . 160 Contents 11 Asymmetric information: moral hazard 161 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 11.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 161 11.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 161 11.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.3 Effort choice and incentive contracts: a formal model . . . . . . . . . . . 162 11.4 Full information: observable effort . . . . . . . . . . . . . . . . . . . . . . 164 11.4.1 Implementing high effort eH . . . . . . . . . . . . . . . . . . . . . 164 11.4.2 Implementing low effort eL . . . . . . . . . . . . . . . . . . . . . . 165 11.4.3 Which effort is optimal for the principal? . . . . . . . . . . . . . . 165 11.5 Asymmetric information: unobservable effort . . . . . . . . . . . . . . . . 166 11.5.1 Implementing low effort eL . . . . . . . . . . . . . . . . . . . . . . 166 11.5.2 Implementing high effort eH . . . . . . . . . . . . . . . . . . . . . 166 11.6 Risk-neutral agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 11.7 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 170 11.8 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 170 11.8.1 Sample examination questions . . . . . . . . . . . . . . . . . . . . 170 12 Externalities and public goods 173 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 12.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.1.3 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.1.4 References cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.2 Overview of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 12.3 Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 12.3.1 Tax and quota policies . . . . . . . . . . . . . . . . . . . . . . . . 177 12.3.2 Coase theorem: the property rights solution . . . . . . . . . . . . 178 12.4 Public goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 12.4.1 Pareto optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 12.4.2 Private provision . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 12.5 The commons problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 12.5.1 Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 12.5.2 A simple model of resource extraction . . . . . . . . . . . . . . . . 186 12.6 A reminder of your learning outcomes . . . . . . . . . . . . . . . . . . . . 187 vii Contents 12.7 Test your knowledge and understanding . . . . . . . . . . . . . . . . . . . 188 12.7.1 Sample examination questions . . . . . . . . . . . . . . . . . . . . 188 viii Chapter 1 Introduction 1.1 Routemap to the subject guide Welcome to this course in Microeconomics. In this introductory chapter, we will look at the overall structure of the subject guide (in the form of a Routemap); we will introduce you to the subject area; to the aims and learning outcomes for the course; and to the learning resources available to you. Finally, we will offer you some Examination advice. We hope that you enjoy this course and we wish you every success in your studies. We start by analysing individual choice. In Chapter 2, we analyse consumer choice. We specify properties of preferences, how to go from preferences to utility and optimal choice by maximising utility under a budget constraint. We show how to obtain demand functions from optimal solutions to the consumer choice problem. We discuss what happens to the consumer’s welfare when prices and/or income change, and appropriate measures to evaluate these changes. We also consider the labour supply decision of individuals, and the intertemporal choice problem faced by savers and borrowers. We provide a brief exposition of some basic algebra of intertemporal choice problems with more than two periods. In Chapter 3, we model the agent’s behaviour in situations involving risk, and analyse insurance problems. We then introduce strategic interaction in Chapter 4 and provide an exposition of the basic tools of game theory. Next, we turn to the supply side of the economy. In Chapter 5, we describe the production technology, structure of costs and principles of profit maximisation by firms. Chapter 5 of the subject guide is essentially a toolbox for analysis in subsequent chapters. Using these tools, we analyse the problem of competitive firms as well as the equilibrium in a competitive market in Chapter 6. In Chapter 7, we then introduce the general equilibrium across all competitive markets and study the two welfare theorems that are fundamental to our understanding of market economies. We also consider some applications of the supply-demand model, and analyse the impact of policies on welfare. Next, we consider markets that are imperfectly competitive. In Chapter 8, we analyse the problem of monopoly, the associated inefficiencies and policy prescriptions aimed at restoring efficiency. This is followed by an analysis of the problem of oligopolistic competition in Chapter 9. Next, we turn to the problem of asymmetric information and analyse the problems of adverse selection (in Chapter 10) and moral hazard (in Chapter 11). Finally, in Chapter 12, we consider the problem of externalities and public goods and consider policy prescriptions arising from associated market failures. Given the emphasis of the EC2066 Microeconomics syllabus on using analytical methods to solve economic problems, you are encouraged to spend a considerable amount of time doing the problems or questions given in the textbooks. Learning by doing is likely to be more profitable than simply reading and re-reading textbooks. 1 1. Introduction Nevertheless, a thorough reading of, and careful note-taking from, the recommended textbook and the subject guide is a prerequisite for successful problem solving. The subject guide aims to indicate as clearly as possible the key elements in microeconomic analysis that you need to learn. The subject guide also presents detailed algebraic derivations for a variety of topics. For each topic, you should consult both the subject guide and the suggested parts of the textbook to understand fully the economic principles involved. 1.2 Introduction to the subject area and prior knowledge The syllabus and subject guide assume that you are competent in basic economic analysis up to the level of the prerequisite courses EC1002 Introduction to economics, ST104a Statistics 1, MT105a Mathematics 1 or MT1174 Calculus. They build on the foundations provided in these courses by specifying how your understanding of the microeconomic principles developed so far should be deepened and extended. Like EC1002 Introduction to economics, EC2066 Microeconomics is designed to equip you with the economic principles necessary to analyse a whole range of economic problems. To maximise your benefit from the subject, you should continue to think carefully about: the assumptions, internal logic and predictions of economic models how economic principles can be applied to solve particular economic problems. The appropriate analysis will depend on the specific facts of a problem. However, you are not expected to know the detailed facts about specific economic issues and policies mentioned in textbook examples. Rather, you should use these examples (and the end-of-chapter Sample examination questions) to aid your understanding of how economic principles can be applied creatively to the analysis of economic problems. If you are taking this course as part of a BSc degree you will also have passed ST104a Statistics 1 and MT105a Mathematics 1 or MT1174 Calculus before beginning this course. Every part of the syllabus can be mastered with the aid of diagrams and relatively simple algebra. The subject guide indicates the minimum level of mathematical knowledge that is required. Knowledge (and use in the examination) of sophisticated mathematical techniques is not required. However, if you are mathematically competent you are encouraged to use mathematical techniques when these are appropriate, as long as you recognise that some verbal explanation is always necessary. 1.3 Syllabus The course examines how economic decisions are made by households and firms, and how they interact to determine the quantities and prices of goods and factors of production and the allocation of resources. Further, it examines the nature of strategic 2 1.4. Aims of the course interaction and interaction under asymmetric information. Finally, it investigates the role of policy as well as economic contracts in improving welfare. The topics covered are: Consumer choice and demand, labour supply. Choice under uncertainty: the expected utility model. Producer theory: production and cost functions, firm and industry supply. Game theory: normal-form and extensive form games, Nash equilibrium and subgame perfect equilibrium, repeated games and cooperative equilibria. Market structure: competition, monopoly and oligopoly. General equilibrium and welfare: competitive equilibrium and efficiency. Pricing in input markets. Intertemporal choice: savings and investment choices. The economics of information: moral hazard and adverse selection, resulting market failures and the role of contracts and institutions. Market failures arising from monopoly, externalities and public goods. The role of policy. 1.4 Aims of the course This subject guide enables you to fully interpret the published syllabus for EC2066 Microeconomics. It identifies what you are expected to know within each area of the syllabus by emphasising the relevant concepts and models and by stating where in specific textbooks that material can be found. This subject guide aims to help you make the best use of textbooks to secure a firm understanding of the microeconomic analysis covered by the syllabus. The subject guide also complements your textbook in certain areas where the coverage in the textbook is deemed inadequate. 1.5 Learning outcomes for the course At the end of this course, and having completed the Essential reading and activities, you should: be able to define and describe: • the determinants of consumer choice, including inter-temporal choice and choice under uncertainty • the behaviour of firms under different market structures • how firms and households determine factor prices • behaviour of agents in static as well as dynamic strategic situations 3 1. Introduction • the nature of economic interaction under asymmetric information be able to analyse and assess: • efficiency and welfare optimality of perfectly and imperfectly competitive markets • the effects of externalities and public goods on efficiency • the effects of strategic behaviour and asymmetric information on efficiency • the nature of policies and contracts aimed at improving welfare be prepared for further courses which require a knowledge of microeconomics. Each chapter includes a list of the learning outcomes that are specific to it. However, you also need to go beyond the learning outcomes of each single chapter by developing the ability of linking the concepts introduced in different chapters, in order to approach the examination well. 1.6 1.6.1 Overview of learning resources The subject guide Each chapter of the subject guide has the following format. The Essential reading lists the relevant textbook chapters and sections of chapters, even though a more detailed indication of the required reading is listed throughout the chapter. The sections that follow specify in detail what you are expected to know about each topic. The relevant sections of the recommended textbooks are referred to. Wherever necessary, the sections integrate the textbook with additional material and explanations. Finally, they draw attention to any problems that occur in textbook expositions and explain how these can be overcome. The boxes that appear in some of the sections give you exercises based on the material discussed. The learning outcomes show you what you should be able to do by the end of the chapter. A final section gives you questions to test your knowledge and understanding. 1.6.2 Essential reading This subject guide is specifically designed to be used in conjunction with the textbook: Nicholson, W. and C. Snyder Intermediate Microeconomics and its Application. (Cengage Learning, 2015) 12th edition [ISBN 9781133189039]. Henceforth in this subject guide this textbook is referred to as ‘N&S.’ 4 1.6. Overview of learning resources This is available as an e-book at a discounted price via the VLE. Please visit the course page for details. Students may use the previous edition instead: Nicholson, W. and C. Snyder Theory and Applications of Intermediate Microeconomics. (Cengage Learning, 2010) 11th edition, international edition [ISBN 9780324599497]. Note that the title of the twelfth edition differs slightly from that of the eleventh. If using this edition, students should refer to the reading supplement on the VLE for customised references. N&S is more adequate for some parts of the syllabus while less so for others; as a consequence we integrate some topics more with the extra material provided in the subject guide. The textbook employs verbal reasoning as the main method of presentation, supplemented by diagrammatic analyses. The textbook’s use of algebra is not uniformly satisfactory. The subject guide supplements the textbook in many cases in this regard. There are also some references to the following textbook: Perloff, J.M. Microeconomics with Calculus. (Pearson Education, 2014) 3rd edition [ISBN 9780273789987]. Detailed reading references in this subject guide refer to the editions of the textbooks listed above. New editions of these textbooks may have been published by the time you study this course. You can use a more recent edition of any of the textbooks; use the detailed chapter and section headings and the index to identify relevant readings. Also, check the virtual learning environment (VLE) regularly for updated guidance on readings. Unless otherwise stated, all websites in this subject guide were accessed in February 2016. We cannot guarantee, however, that they will stay current and you may need to perform an internet search to find the relevant pages. 1.6.3 Further reading Please note that as long as you read the Essential reading you are then free to read around the subject area in any textbook, paper or online resource. You will need to support your learning by reading as widely as possible and by thinking about how these principles apply in the real world. To help you read extensively, you have free access to the virtual learning environment (VLE) and University of London Online Library (see below). Other useful textbooks for this course include: Besanko, D. and R. Braeutigam Microeconomics. (John Wiley & Sons, 2014) 5th edition, international student version [ISBN 9781118716380]. Varian, H.R. Intermediate Microeconomics, a Modern Approach. (W.W. Norton, 2014) 9th edition [ISBN 9780393920772]. Pindyck, R.S. and D.L. Rubinfeld Microeconomics. (Pearson, 2014) 8th edition [ISBN 9781292081977]. 5 1. Introduction 1.6.4 Online study resources In addition to the subject guide and the Essential reading, it is crucial that you take advantage of the study resources that are available online for this course, including the VLE and the Online Library. You can access the VLE, the Online Library and your University of London email account via the Student Portal at: http://my.londoninternational.ac.uk You should have received your login details for the Student Portal with your official offer, which was emailed to the address that you gave on your application form. You have probably already logged in to the Student Portal in order to register! As soon as you registered, you will automatically have been granted access to the VLE, Online Library and your fully functional University of London email account. If you have forgotten these login details, please click on the ‘Forgotten your password’ link on the login page. 1.6.5 The VLE The VLE, which complements this subject guide, has been designed to enhance your learning experience, providing additional support and a sense of community. It forms an important part of your study experience with the University of London and you should access it regularly. The VLE provides a range of resources for EMFSS courses: Electronic study materials: All of the printed materials which you receive from the University of London are available to download, to give you flexibility in how and where you study. Discussion forums: An open space for you to discuss interests and seek support from your peers, working collaboratively to solve problems and discuss subject material. Some forums are moderated by an LSE academic. Videos: Recorded academic introductions to many subjects; interviews and debates with academics who have designed the courses and teach similar ones at LSE. Recorded lectures: For a few subjects, where appropriate, various teaching sessions of the course have been recorded and made available online via the VLE. Audio-visual tutorials and solutions: For some of the first year and larger later courses such as Introduction to Economics, Statistics, Mathematics and Principles of Banking and Accounting, audio-visual tutorials are available to help you work through key concepts and to show the standard expected in examinations. Self-testing activities: Allowing you to test your own understanding of subject material. Study skills: Expert advice on getting started with your studies, preparing for examinations and developing your digital literacy skills. 6 1.7. Examination advice Note: Students registered for Laws courses also receive access to the dedicated Laws VLE. Some of these resources are available for certain courses only, but we are expanding our provision all the time and you should check the VLE regularly for updates. This subject guide is reproduced in colour on the VLE and you may find it easier to understand if you access the online PDF. 1.6.6 Making use of the Online Library The Online Library (http://onlinelibrary.london.ac.uk) contains a huge array of journal articles and other resources to help you read widely and extensively. To access the majority of resources via the Online Library you will either need to use your University of London Student Portal login details, or you will be required to register and use an Athens login. The easiest way to locate relevant content and journal articles in the Online Library is to use the Summon search engine. If you are having trouble finding an article listed in a reading list, try removing any punctuation from the title, such as single quotation marks, question marks and colons. For further advice, please use the online help pages (http://onlinelibrary.london.ac.uk/resources/summon) or contact the Online Library team: onlinelibrary@shl.london.ac.uk 1.7 1.7.1 Examination advice Format of the examination Important: the information and advice given here are based on the examination structure used at the time this subject guide was written. Please note that subject guides may be used for several years. Because of this we strongly advise you to always check both the current Regulations for relevant information about the examination, and the VLE where you should be advised of any forthcoming changes. You should also carefully check the rubric/instructions on the paper you actually sit and follow those instructions. In this examination you should answer eleven of fourteen questions: all eight questions from Section A (5 marks each) and three out of six from Section B (20 marks each). 1.7.2 Types of questions Examples of the types of questions which will appear on the examination paper appear not only in the Sample examination paper on the VLE, but also at the end of chapters. However, in the examination you should not be surprised to see some questions which are not necessarily specific to one particular topic. For example, a question may require knowledge about markets which are oligopolistic as well as those which are monopolistic or competitive. 7 1. Introduction Numerical questions will sometimes require the use of a calculator. A calculator may be used when answering questions on the examination paper for this course and it must comply in all respects with the specification given in the Regulations. Questions will not require knowledge of empirical studies or institutional material. However, you will be awarded some marks for supplementary empirical or institutional material which is directly relevant to the question. 1.7.3 Specific advice on approaching the questions You should follow all the excellent advice to candidates which is published in the annual Examiners’ commentaries. For this course, the following advice is also worth noting: Prepare thoroughly for the examination by attempting the problems/questions in the textbooks and in this subject guide and, in particular, past examination papers (for which there are Examiners’ commentaries where you can check examiners’ responses). Occasionally, you may be unsure exactly what a question is asking. If there is some element of doubt about the interpretation of a question, state at the beginning of your answer how you interpret the question. If you are very uncertain of what is required, and the question is in Section B, do another question. Explain briefly what you are doing: an answer that is simply a list of equations or numbers will not be credited with full marks even if it gets to the correct solution. Moreover, by explaining what you are doing, you will be awarded some marks for correct reasoning even if there are mistakes in some part of the procedure. It is essential to attempt eight questions in Section A. Even if you think you do not know the answer, at least define any terms or concepts which you think may be relevant (including those in the question!) and, if possible, present the question in diagrammatic or algebraic form. The same applies to a specific part of a multi-part question in Section B. The examiners can give no marks for an unattempted question, but they can award marks for relevant points. A single mark may make the difference between passing and failing the examination. Although you should attempt all the questions and parts of questions that are required, to avoid wasting time you should make sure that you do no more than is required. For example, if only two parts of a three-part question need to be answered, only answer two parts. Note the importance of key words. In some of the ‘True or false?’ type questions, the words ‘must’, ‘always’, ‘never’ or ‘necessarily’ usually invite you to explain why the statement is ‘false’. Notice that this is simply a way in which you can start approaching the problem, but there is no way to know in advance the correct answer without analysing every specific question. It is worth noting that in this type of question, simply writing ‘true’ or ‘false’ will not earn you any marks, even if it happens to be the right answer. The examiners are looking for reasoning, not blind guesses. 8 1.7. Examination advice Good answers to most questions require relevant assumptions to be stated and terms to be defined. Also, do use the term ceteris paribus (meaning ‘other things being equal’), where appropriate. If you are asked to examine the effects of a change in a particular exogenous variable, you should not complicate your answers unnecessarily by positing simultaneous changes in other exogenous variables. For many questions, good answers will require diagrammatic and/or algebraic analysis to complement verbal reasoning. Good diagrams can often save much explanation but free-standing diagrams, however well-drawn and labelled, do not portray sufficient information to the examiners. Diagrams need to be explained in the text of the answer. Similarly, symbols in algebraic expressions should be defined and the final line of an algebraic presentation should be explained in words. The examiners are primarily looking for analytical explanations, not descriptions. On reading a question, your first thought should be: ‘what is the appropriate hypothesis, theory, concept or model to use?’ Remember, it is important to check the VLE for: up-to-date information on examination and assessment arrangements for this course where available, past examination papers and Examiners’ commentaries for the course which give advice on how each question might best be answered. 9 1. Introduction 10 Chapter 2 Consumer theory 2.1 Introduction How do people choose what bundle of goods to consume? We cannot observe this process directly, but can we come up with a model to capture the decision-making process so that the predictions from the model match the way people behave? If we can build a sensible model, we should be able to use the model to understand how choice varies when the economic environment changes. This should also help us design appropriate policy. This is the task in this chapter – and as we go through the various steps, you should keep this overarching goal in your mind and try to see how each piece of analysis fits in the overall scheme. Once we build such a model, we use it to analyse how optimal consumption choice responds to price and income variations. We also extend the analysis to cover labour-leisure choice as well as intertemporal consumption choice. 2.1.1 Aims of the chapter This chapter introduces you to the theory of consumer choice. You should be familiar with many of the ideas here from EC1002 Introduction to economics, but we aim to investigate certain aspects at relatively greater depth. The chapter also aims to encourage you to ask questions about the meaning of concepts and their usefulness in understanding the world. For example, you have come across utility functions before. But surely no-one has a utility function – so where do these functions come from? Why is this concept useful? You should not accept such concepts just because they appear in textbooks and are taught in classes. To convey this message is an important aim here. 2.1.2 Learning outcomes By the end of this chapter, the Essential reading and activities, you should be able to: explain the implications of the assumptions on the consumer’s preferences describe the concept of modelling preferences using a utility function draw indifference curve diagrams starting from the utility function of a consumer draw budget lines for different prices and income levels solve the consumer’s utility maximisation problem and derive the demand for a consumer with a given utility function and budget constraint 11 2. Consumer theory analyse the effect of price and income changes on demand explain the notion of a compensated demand function explain measures of the welfare impact of a price change: change in consumer surplus, equivalent variation and compensating variation, and use these measures to analyse the welfare impact of a price change in specific cases construct the market demand curve from individual demand curves explain the notion of elasticity of demand analyse the decision to supply labour analyse the problem of savers and borrowers derive the present discounted value of payment streams and explain bond pricing. 2.1.3 Essential reading N&S Chapters 2 and 3, the Appendix to Chapter 13, from Chapter 14: Sections 14.1, 14.2, 14.5 and from Appendix 14A: Sections A14–3, A14–4. In addition, Chapter 1 and Appendix 1A provide a review of the basics of economic models and some basic techniques. You should be familiar with this material from earlier courses. Nevertheless, you should read this chapter and the appendix and make sure that you understand the content. Throughout this subject guide, we will assume that you are familiar with this material. 2.2 Overview of the chapter We start by analysing preferences, utility and choice. Next, we learn how to construct demand curves, and analyse their properties. We also explain various welfare measures. We then analyse the labour supply decision of an individual before moving on to saving and borrowing with two periods. Finally, we learn to carry out present value calculations with many periods and analyse bond pricing. 2.3 Preferences, utility and choice See N&S Chapter 2. See also Perloff Sections 3.1 and 3.2 for a good discussion on the connection between preferences and utility. 2.3.1 Preferences and utility The theory of choice starts with rational preferences. Generally, preferences are primitives in economics – you take these as given and proceed from there. The task of explaining why certain preferences exist in certain societies falls largely under the domain of subjects such as anthropology or sociology. However, to be able to create a 12 2.3. Preferences, utility and choice model of choice that has some predictive power, we do need to put some restrictions on preferences to rule out irrational behaviour. Just a few relatively sensible restrictions allow us to build a model of choice that has great analytical power. Indeed, this idea of creating an analytical structure that can be manipulated to understand how changes in the economic environment affect economic outcomes underlies the success of economics as a tool for investigating the world around us. Section 2.2 of N&S sets out three restrictions on preferences: completeness, transitivity and non-satiation (‘more is better’). These restrictions allow us to do something very useful. Once we add some further technical requirements (a full specification must await Masters level courses but the main extra condition we need is that preferences have certain continuity properties), these restrictions allow us to represent preferences by a continuous function. This is known as a utility function. Note that a utility function is an artificial concept – no-one actually has a utility function (you knew that of course, since you surely do not have one). But because we can represent preferences using such a function, it is as if agents have a utility function. All subsequent analysis using utility functions and indifference curves has this ‘as if’ property. Ordinal versus cardinal utility Preferences generally give us rankings among bundles rather than some absolute measure of satisfaction derived from bundles. You might prefer apple juice to orange juice but would have difficulty saying exactly how much more satisfaction you derive from the former compared to the latter. Preferences, therefore, typically give us an ‘ordinal’ ranking among bundles of goods. Since utility is simply a representation of preferences, it is also an ordinal measure. This means that if your preferences can be represented by a utility function, then a positive transformation of this function which preserves the ordering among bundles is another function that is also a valid utility function. In other words, there are many possible utility functions that can represent a given set of preferences equally well. However, there are some instances where we use cardinal utility and make absolute comparisons among bundles. Money, for example, is a cardinal measure – you know that 20 pounds is twice as good as 10 pounds. In general, though, you should understand utility as an ordinal concept. 2.3.2 Indifference curves Once we can represent preferences using a continuous utility function, we can draw indifference curves. An indifference curve is the locus of different bundles of goods that yield the same level of utility. In other words, an indifference curve for utility function u(x, y) is given by u(x, y) = k, where k is some constant. As we vary k, we can trace out the indifference map. Note that an indifference curve is simply a level curve of a utility function. Just as you draw contours on a map to represent, say, a mountain, so indifference curves drawn for two goods are contour plots of a utility function over these two goods. You can see Figure 3.3 in Perloff for a pictorial representation. You should read carefully the discussion in N&S (Sections 2.3 to 2.5) on indifference curves. You 13 2. Consumer theory should know how different types of preferences generate different types of indifference curves. Activity 2.1 For each of the following utility functions, write down the equation for an indifference curve and then draw some indifference curves. (a) u(x, y) = xy. (b) u(x, y) = x + y. (c) u(x, y) = min{x, y}. Previously, we put some restrictions on preferences. What do these restrictions imply for indifference curves? We have the following properties: 1. If an indifference curve is further from the origin compared to another indifference curve, any point on the former is preferred to any point on the latter (implied by the assumption that more is better). 2. Indifference curves cannot slope upwards (implied by more is better). 3. Indifference curves cannot be thick (again, implied by more is better). 4. Indifference curves cannot cross (implied by transitivity). 5. Every bundle of goods lies on some indifference curve (follows from completeness). The marginal rate of substitution A further important property concerns the rate at which a consumer is willing to substitute one good for another along an indifference curve. The marginal rate of substitution of a consumer between goods x and y is the units of y the consumer is willing to substitute (i.e. willing to give up) to obtain one more unit of x. The slope of an indifference curve (with good y on the y-axis and good x on the x-axis) is given by: MUx dy =− . dx u constant MUy The marginal rate of substitution is the absolute value of the slope: MRSxy = MUx . MUy Typically, preferences have the following property. Consider a point where a lot of y and very little x is being consumed. Starting from any such point, a consumer is willing to give up a lot of y in exchange for another unit of x while retaining the same level of utility as before. As we keep adding units of x and reducing y while keeping utility constant (i.e. we are moving down an indifference curve), consumers are willing to give up less and less of y in return for a further unit of x. One way to interpret this is that 14 2.3. Preferences, utility and choice people typically have a taste for variety and want to avoid extremes (i.e. avoid situations where a lot of one good and very little of the other good is being consumed). This implies that MRS falls along an indifference curve. This property is referred to as indifference curves being ‘convex to the origin’ in some textbooks. This works as a visual description, but you should be aware that in terms of mathematics, this is not a meaningful description – there is no mathematical concept where something is convex relative to something else. The correct idea of convex indifference curves is as follows. Consider a subset S of Rn . S is a convex set if the following property holds: if points s1 and s2 are in S, then a convex combination λs1 + (1 − λ)s2 is also in the set S for any 0 < λ < 1. Now consider any indifference curve yielding utility level u?. Consider the set of all points that yield utility u? or more. This is the set of all points on an indifference curve plus all points above. Call this set B. Diminishing MRS implies that B is a convex set. Figure 2.1 below shows a convex indifference curve. Note that the set B (part of which is shaded) is a convex set. Try taking any two points in B and then making a convex combination. You will find that the combinations are always inside B. Figure 2.1: A convex indifference curve. Next, Figure 2.2 shows an example of non-convex indifference curves. Note that the set B of points on or above the indifference curve is not convex. If you combine points such as a1 and a2 in the diagram, for some values of λ, the convex combinations fall outside the set B. Note that when two goods are perfect substitutes, you get a straight line indifference curve. At the other extreme, the two goods are perfect complements (no substitutability) and the indifference curve is L-shaped. Indifference curves with diminishing MRS lie in between these two extremes. 15 2. Consumer theory Figure 2.2: A non-convex indifference curve. Here is an activity to get you computing the MRS for different utility functions. Activity 2.2 Compute the MRS for the following utility functions. (a) u(x, y) = √ xy. (b) u(x, y) = ln x + ln y. (c) u(x, y) = 20 + 3(x + y)2 . 2.3.3 Budget constraint Once we have specified our model of preferences, we need to know the set of goods that a consumer can afford to buy. This is captured by the budget constraint. Since consumers are generally taken to be price-takers (i.e. what an individual consumer purchases does not affect the market price for any good), the budget line is a straight line. See Section 2.7 of N&S for the construction of budget sets. You should be aware that budget lines would no longer be a straight line if a consumer buys different units at different prices. This could happen if a consumer is a large buyer in a market or if the consumer gets quantity discounts. See Application 2.6 in N&S for an example. 2.3.4 Utility maximisation The consumer chooses the most preferred point in the budget set. If preferences are such that indifference curves have the usual convex shape, the best point is where an indifference curve is tangent to the budget line. This is shown as point A in Figure 2.3. At A the slope of the indifference curve coincides with the slope of the budget 16 2.3. Preferences, utility and choice Figure 2.3: Consumer optimisation. constraint. So we have: − MUx Px =− . MUy Py Multiplying both sides by −1 we can write this as the familiar condition: MRSxy = Px . Py Let us derive this condition formally using a Lagrange multiplier approach. This is the approach you are expected to use when faced with optimisation problems of this sort. Note that the ‘more is better’ assumption ensures that a consumer spends all income (if not, then the consumer could increase utility by buying more of either good). Therefore, the budget constraint is satisfied with equality. It follows that the consumer maximises u(x, y) subject to the budget constraint Px x + Py y = M . Set up the Lagrangian: L = u(x, y) + λ(M − Px x + Py y). The first-order conditions for a constrained maximum are: ∂u ∂L = − λPx = 0 ∂x ∂x ∂L ∂u = − λPy = 0 ∂y ∂y ∂L = M − Px x + Py y = 0. ∂λ From the first two conditions, we get: Px ∂u/∂x = = MRSxy . Py ∂u/∂y 17 2. Consumer theory Second-order condition The first-order conditions above are, by themselves, not sufficient to guarantee a maximum. We also need the second-order condition to hold. It is better to derive this formally once you have learned matrix algebra, which allows a relatively simple exposition of the second-order condition. For our purposes here, note that the diminishing MRS condition is sufficient to guarantee that a maximum occurs at the point satisfying the first-order conditions. This should also be clear to you from the graph. If indifference curves satisfy the usual convexity property, there is an interior tangency point with the budget constraint line at which the maximum utility is attained. Figure 2.4 below demonstrates that if preferences are not convex, the first-order conditions are not sufficient to guarantee optimality. Figure 2.4: Violation of the second-order condition under non-convex preferences. Note that MRS is not always diminishing. Point A satisfies the first-order condition MRS equal to price ratio, but is not optimal. (Source: Schmalensee, R. and R.N. Stavins (2013) ‘The SO2 allowance trading system: the ironic history of a grand policy experiment,’ Journal of Economic Perspectives, Vol. 27, pp.103–21. Reproduced by kind permission of the American Economic Association.) Read Sections 2.7 to 2.9 of N&S carefully and work through all the examples therein. Note that if two goods are perfect substitutes or complements, the tangency condition does not apply. For perfect substitutes, there is either a corner solution or the budget line coincides with the indifference curve. In the latter case, any point on the budget line is optimal. For the case of perfect complements, the optimum occurs where the kink in the indifference curve just touches the budget line. Note that this is not a tangency point – the slope of the indifference curve is undefined at the kink. N&S clarifies these cases with appropriate diagrams. 18 2.3. Preferences, utility and choice Demand functions The maximisation exercise above gives us the demand for goods x and y at given prices and income. As we vary the price of good x, we can trace out the demand curve for good x. See Section 3.6 of N&S for a discussion. The activities below compute demand functions in specific examples. Example 2.1 A consumer has the following Cobb–Douglas utility function: u(x, y) = xα y β where α, β > 0. The price of x is normalised to 1 and the price of y is p. The consumer’s income is M . Derive the demand functions for x and y. The consumer’s problem is as follows: max xα y β subject to x + py = M. x, y Using the Lagrange multipliers method, we get: MUx Px = . MUy Py This implies: αxα−1 y β 1 = . βxα y β−1 p Simplifying: αy 1 = . βx p Using this in the budget constraint and solving, we get the demand functions: x(p, M ) = αM α+β y(p, M ) = βM . p(α + β) Example 2.2 A consumer has the following utility function: u(x, y) = min{αx, βy} where α, β > 0. The price of x is normalised to 1 and the price of y is p. The consumer’s income is M . Derive the demand functions for x and y. The consumer would choose the bundle at which the highest indifference curve is reached while not exceeding the budget. This is point E in Figure 2.5 where the indifference curve just touches the budget constraint (Figure 2.5 is drawn using α/β = 1/2). Note that this is not a tangency point as the slope of the indifference curve is undefined at the kink. 19 2. Consumer theory Since we are at the kink, it must be that αx = βy. Using this in the budget constraint, we get the demand functions: x(p, M ) = βM β + αp y(p, M ) = αM . β + αp Figure 2.5: The optimum occurs at point E. Note that this is not a tangency point. The slope of the indifference curve at the kink is undefined. 2.4 2.4.1 Demand curves The impact of income and price changes See N&S Chapter 3. Now that we have derived demand curves, we can try to understand various properties of demand by varying income and prices. Income changes Section 3.2 of N&S explains the classification of goods according to the response of demand to income changes. Normal goods: a consumer buys more of these when income increases. Inferior goods: a consumer buys less of these when income increases. 20 2.4. Demand curves Note that it is not possible for all goods to be inferior. This would violate the ‘more is better’ assumption. The full argument is left as an exercise. Activity 2.3 ‘It is not possible for all goods to be inferior.’ Provide a careful explanation of this statement. The income-consumption curve The income-consumption curve of a consumer traces out the path of optimal bundles as income varies (keeping all prices constant). Using this exercise, we also plot the relationship between quantity demanded and income directly. The curve that shows this relationship is called the Engel curve. See Perloff Section 4.2 for an exposition of the income-consumption curve and the Engel curve. The slope of the income-consumption curve indicates the sign of the income elasticity of demand (explained below). Price changes See Sections 3.3 to 3.8 of N&S. It is very important to understand fully the decomposition of the total price effect into income and substitution effects. This decomposition is, of course, a purely artificial thought experiment. But this thought experiment is extremely useful in understanding how the demand for different goods responds to a change in price at different levels of income and given different opportunities to substitute out of a good. You should understand how these effects (and, therefore, the total price effect) differ across normal and inferior goods, and understand how the effect known as Giffen’s paradox can arise. The idea of income and substitution effects can help us understand the design of an optimal tax scheme. See Section 3.4 of N&S for a discussion of this issue. Finally, you should also study the impact on the demand for a good by changes in the price of some other good, and how this effect differs depending on whether the other good is a substitute or a complement. Example 2.3 Suppose u(x, y) = x1/2 y 1/2 . Income is M = 72. The price of y is 1 and the price of x changes from 9 to 4. Calculate the income effect (IE) and the substitution effect (SE). Let (px , py ) denote the original prices and let p0x denote the lower price of x. Under the original prices, the Marshallian demand functions (you should be able to calculate these) are: M x∗ = 2px and: y∗ = M . 2py 21 2. Consumer theory The optimised utility is, therefore: M u∗0 = √ 2 px p y where the subscript of u is a reminder that this is the original utility level (before the price change). The total price effect (PE) from a price fall is: PE = M M − . 0 2px 2px Using the values supplied, this is 9 − 4 = 5. In Figure 2.6, the movement from A to C is the total price effect. To isolate the SE, we must change the price of x, but also take away income so that the consumer is on the original indifference curve. In other words, we must keep the utility at u∗0 . In Figure 2.6, the dashed budget line is the one after the compensating reduction in income. The point B is the optimal point on this compensated budget line. The movement from the original point A to B shows the substitution effect. How much income should we take away to compensate for the price change? This can be calculated as follows. Suppose ?M is the amount of income we take away. We need ?M to be such that: M − ?M p = u∗0 0 2 p x py which implies: M − ?M M p = . √ 2 px py 2 p0x py Using the values supplied, ?M = 24 so that M − ?M = 48. Under a reduced income of 48, and given the new price p0x = 4, the demand for x is 6. The original demand for x was 4. Under the compensated price change, the demand is 6. Therefore, the SE is 2. It follows that the rest of the change must be the IE. Since the total price effect is 5, the IE is 3. In terms of algebra: SE = M − ?M M − . 0 2px 2px The IE is the remainder of the price effect, so that: M M M − ?M M IE = − − − . 2p0x 2px 2p0x 2px Simplifying: IE = ?M . 2p0x Looking ahead, the ?M we calculated here is known as the ‘compensating variation’. We will study this concept later in this chapter. 22 2.4. Demand curves Figure 2.6: As the price of x falls, the change from A to C shows the total price effect. The movement from A to B (under a compensated price change) shows the substitution effect, while the movement from B to C shows the income effect. The market demand curve From individual demand curves, we can construct the market demand curve by aggregating across individuals. See N&S Section 3.10 for a discussion. 2.4.2 Elasticities of demand The notion of elasticity of demand captures the responsiveness of demand to variables such as prices and income. The elasticity of market demand for a good can be estimated from data, and these elasticity estimates are important for firms in setting prices and for formulation of policy. Throughout the course, we will come across several such examples. Sections 3.11 to 3.16 of N&S contain a detailed analysis of elasticities, which you must read carefully. Here, let us summarise the main concepts. Price elasticity of demand This is the percentage change in quantity demanded of a good in response to a given percentage change in the price of the good, given by: ε= dQ/Q P dQ = . dP/P Q dP Note that ε < 0 since demand is typically downward-sloping. Demand is said to be elastic if ε < −1, unit elastic if ε = −1, and inelastic if ε > −1. N&S outlines a variety of uses of this concept, which you should read carefully. You should know how to calculate demand elasticity at different points on a demand curve, and how the elasticity varies along a linear demand curve. 23 2. Consumer theory Price elasticity of demand is the most common measure of elasticity and often referred to as just elasticity of demand. Other than price elasticity, we can define income elasticity and cross-price elasticity. Income elasticity Denoting income by M , income elasticity of demand is given by: εM = P dM . M dP This is positive for normal goods, and negative for inferior goods. When εM exceeds 1, we call the good a luxury good. Necessities like food have income elasticities much lower than 1. Cross-price elasticity Let us consider the elasticity of demand for good i with respect to the price of good j. The cross-price elasticity of demand for good i is given by: εij = Pj dQi . Qi dPj This is negative for complements and positive for substitutes. You should take a long look at the elasticity estimates presented in Section 3.16 of N&S. Practical knowledge of elasticities forms an important part of designing and understanding a variety of tax and subsidy policies in different markets. Activity 2.4 Suppose u(x, y) = xα y β , where α + β = 1. Income is M . Calculate the price elasticity, cross-price elasticity and income elasticity of demand for x. 2.4.3 The compensated demand curve We derived the demand function for a good above. To derive the demand function for good x, we vary the price of good x but hold constant the prices of other goods and income. Of course, as the price changes so that the optimal choice changes, the utility of the consumer at the optimal point also changes. This is the usual demand curve, and is also known as the Marshallian demand curve, or the uncompensated demand curve. Indeed, if we simply mention a demand curve without putting a qualifier before it, it refers to the Marshallian, or uncompensated, demand curve. A compensated, or Hicksian, demand curve can be derived as follows. Suppose as the price of a good changes, we keep utility constant while allowing income to vary. In other words, if the price of x, say, falls (so that the new optimal bundle of the consumer would be associated with a higher level of utility if income is left unchanged), we take away enough income to leave the consumer at the original level of utility. It is clear that this process eliminates the income effect and simply captures the substitution effect. Below, we list some properties of compensated demand curves. 24 2.4. Demand curves A compensated demand curve always slopes downwards. For a normal good, the compensated demand curve is less elastic compared to the uncompensated demand curve. For an inferior good, the compensated demand curve is more elastic compared to the uncompensated demand curve. You should understand that all three properties result from the fact that only the substitution effect matters for the change in compensated demand when the price changes. The next example asks you to calculate the compensated demand curve for a Cobb–Douglas utility function. Example 2.4 Suppose u(x, y) = x1/2 y 1/2 . Income is M . Calculate the compensated demand curves for x and y. To do this, we must first calculate the Marshallian demand curves. These are given by (you should do the detailed calculations to show this): x= M 2px and y= M . 2py The optimised value of utility is: M V = √ . 2 px py Holding utility constant at V implies adjusting M to the value M ∗ so that: M ∗ = 2V √ px py . This is the value of income which is compensated to keep utility constant at the level given by the original choice of x and y. It follows that the compensated demand functions are: r M∗ py xc = =V 2px px and: M∗ yc = =V 2py r px . py Note that the Marshallian demand for x does not depend on py , but the Hicksian, or compensated, demand does. This is because changes in py require income adjustments, which generate an income effect on the demand for x. 2.4.4 Welfare measures: ?CS, CV and EV See Section 3.9 of N&S for a discussion of consumer surplus, but this does not cover the other two measures: compensating variation (CV) and equivalent variation (EV). We provide definitions and applications below. 25 2. Consumer theory When drawing demand curves, we typically draw the inverse demand curve (price on the vertical axis, quantity on the horizontal axis). In such a diagram, the consumer surplus (CS) is the area under the (inverse) demand curve and above the market price up to the quantity purchased at the market price. This is the most widely-used measure of welfare. We can measure the welfare effect of a price rise by calculating the change in CS (denoted by ?CS). Much of our discussion of policy will be based on this measure. Any part of ?CS that does not get translated into revenue or profits is a deadweight loss. The extent of deadweight loss generated by any policy is a measure of inefficiency associated with that policy. However, ?CS is not an exact measure because of the presence of an income effect. Ideally, we would use the compensated demand curve to calculate the welfare change. CV and EV give us two such measures. You should use these measures to understand the design of ideal policies, but when measuring welfare change in practice, use ?CS. Compensating variation (CV) CV is the amount of money that must be given to a consumer to offset the harm from a price increase, i.e. to keep the consumer on the original indifference curve before the price increase. Equivalent variation (EV) EV is the amount of money that must be taken away from a consumer to cause as much harm as the price increase. In this case, we keep the price at its original level (before the rise) but take away income to keep the consumer on the indifference curve reached after the price rise. Comparing the three measures Consider welfare changes from a price rise. For a normal good, we have CV > ?CS > EV, and for an inferior good we have CV < ?CS < EV. The measures would coincide for preferences that exhibit no income effect. The example that follows shows an application of these concepts. Example 2.5 The government decides to give a pensioner a heating fuel subsidy of s per unit. This results in an increase in utility from u0 before the subsidy to u1 after the subsidy. Could the government follow an alternative policy that would result in the same increase in utility for the pensioner, but cost the government less? Let us show that an equivalent income boost would be less costly. Essentially, the EV of a price fall is lower than the expenditure on heating after the price fall. The intuition is that a per-unit subsidy distorts choice in favour of consuming more heating, raising the total cost of the subsidy. To put the same idea differently, an equivalent income boost would raise the demand for fuel through the income effect. 26 2.4. Demand curves But a price fall (the fuel subsidy results in a lower effective price) causes an additional substitution effect boosting the demand for heating. To see this, consider Figure 2.7. The initial choice is point A and after the subsidy the pensioner moves to point B. How much money is the government spending on the subsidy? Note that after the subsidy, H1 units of heating fuel are being consumed. At pre-subsidy prices, buying H1 would mean the pensioner would have E 0 of other goods. Since the price of the composite good is 1, M is the same as total income. It follows that the amount of income that would be spent on heating to buy H1 units of heating at pre-subsidy prices is given by M E 0 . Similarly, at the subsidised price, the amount of income being spent on heating fuel is M B 0 . The difference B 0 E 0 is then the amount of the subsidy. This is the same length as segment BE. Once we understand how to show the amount of spending on the subsidy in the diagram, we are ready to compare this spending with an equivalent variation of income. This is added in Figure 2.8 below. The pensioner’s consumption is initially at A, and moves to B after the subsidy. Since the composite good has a price of 1, the vertical distance between the budget lines (segment BE) shows the extent of the expenditure on the subsidy (as explained above). An equivalent variation in income, on the other hand, would move consumption to C. It follows that DE is the equivalent variation in income, which is smaller than the expenditure on the subsidy. Therefore, a direct income transfer policy would be less costly for the government. Figure 2.7: The segment BE shows the extent of the subsidy. 27 2. Consumer theory Figure 2.8: Per-unit subsidy versus an equivalent variation in income. Example 2.6 Suppose that a consumer has the utility function u(x1 , x2 ) = x1/2 y 1/2 . He originally faces prices (1, 1) and has income 100. Then the price of good 1 increases to 2. Calculate the compensating and equivalent variations. Suppose income is M and the prices are p1 and p2 . You should work out that the demand functions are: M M and x2 = . x1 = 2p1 2p2 Therefore, utility is: M u∗ (p1 , p2 , M ) = √ . 2 p1 p2 √ At the initial prices, u∗ = M/2. Once the price of good 1 increases, u∗∗ = M/2 2. The CV is the extra income that restores utility to the original level. Therefore, it is given by: M + CV M √ = . 2 2 2 Solving: √ CV = ( 2 − 1)M. Using the value M = 100, this is 41.42. The EV is the variation in income equivalent to the price change. This is given by: M M − EV √ = . 2 2 2 Solving: √ ( 2 − 1)M √ EV = . 2 Using M = 100, this is 29.29. 28 2.5. Labour supply 2.5 Labour supply See Appendix 13A of N&S. The analysis presented here complements the somewhat basic coverage in the textbook. The tools developed above can also be used to analyse the labour supply decision of an agent. Every economic agent has, in a day, 24 hours. An agent must choose how many of these hours to spend working, and how many hours of leisure to enjoy. Working earns the agent income, which represents all goods the agent can consume. But the agent also enjoys leisure. If the hourly wage is w, this can be seen as the price that the agent must pay to enjoy an hour of leisure. The agent, therefore, faces the following problem. Let Z denote the number of hours worked, N denote the number of leisure hours, M denote income and M? denote unearned income (inheritance, gifts etc). The utility maximisation problem is: max u(N, M ) N, M subject to: Z = 24 − N M = wZ + M? . We can simplify the constraints to M = w(24 − N ) + M? , or M + wN = 24w + M? . Therefore, we have a familiar utility maximisation problem: max u(N, M ) N, M subject to the budget constraint: M + wN = 24w + M? . At the optimum we have the slope of the indifference curve (−MRS) equal to the slope of the budget line (−w). Therefore: MUN = w. MUM How does the optimal choice of labour respond to a change in w? We can analyse this using income and substitution effects. In this case, a change in w also changes income directly (as you can see from the budget constraint), so the exercise is a little different compared to that under standard goods. Suppose w rises. Income effect The rise in w raises income at current levels of labour and leisure. Assuming leisure is a normal good (this should be your default assumption), this raises the demand for leisure. 29 2. Consumer theory Substitution effect The rise in w makes leisure relatively more expensive, causing the agent to substitute away from leisure. This reduces demand for leisure. The two effects go in opposite directions, therefore the direction of the total effect is unclear. In most cases, the substitution effect dominates, giving us an upward-sloping labour supply function. However, it is possible, especially at high levels of income (i.e. when wage levels are high), that the income effect might dominate. In that case we would get a backward-bending labour supply curve which initially slopes upward but then turns back and has a negative slope. If leisure is, on the other hand, an inferior good, the two effects would go in the same direction and labour supply would necessarily slope upwards. Figure 2.9 (a) below shows a backward-bending labour supply curve while Figure 2.9 (b) shows an increasing labour supply curve. Figure 2.9: Labour supply curves. 2.6 Saving and borrowing: intertemporal choice See N&S Sections 14.1 and 14.2. The discussion below complements the somewhat basic discussion in the textbook. We focus on a two-period problem. Suppose the agent’s endowment is Y0 in period 0 and Y1 in period 1. Given a rate of interest r, the present value of income in period 0 is Y0 + Y1 /(1 + r). If the individual consumes all income in period 0, C0 is equal to this present value, and C1 = 0. If all income is saved for period 1, then income at period 1 is (1 + r)Y0 + Y1 . In this case, C1 is this amount and C0 = 0. Therefore, we have 30 2.6. Saving and borrowing: intertemporal choice C1 = (Y0 − C0 )(1 + r) + Y1 , which gives us the intertemporal budget constraint. The problem is then as follows: max u(C0 , C1 ) C0 , C 1 subject to the intertemporal budget constraint: C0 + C1 Y1 = Y0 + . 1+r 1+r This is similar to a standard optimisation problem with two goods, C0 and C1 , where the price of the former is 1 and the price of the latter is 1/(1 + r). Unsurprisingly, the optimum satisfies the property that: MUC0 = 1 + r. MUC1 If the optimal consumption bundle is C0 = Y0 and C1 = Y1 , the agent is neither a saver nor a borrower. If C0 > Y0 the agent is a borrower, and if C0 < Y0 the agent is a saver (lender). How does the intertemporal consumption bundle change when r changes? Again, we can see this by decomposing the effect into income and substitution effects. Let us look at the problem of borrowers and savers separately. Throughout the following analysis, we assume that both C0 and C1 are normal goods. This should be your default assumption. Note that the total income available to consume in period 1 is Y1 + (Y0 − C0 )(1 + r). The problem of a borrower Income effect For a borrower, Y0 < C0 , so that a rise in the interest rate lowers income tomorrow. Given consumption is normal in both periods, the agent should consume less in period 0 (borrow less). Substitution effect A rise in the interest rate makes immediate consumption more costly. Therefore, the substitution effect suggests that the individual should choose to lower C0 and, therefore, borrow less. Since the two effects go in the same direction, the direction of change is unambiguous: a rise in the rate of interest lowers borrowing. 31 2. Consumer theory The problem of a saver (lender) Income effect For a saver, Y0 > C0 , so that a rise in the interest rate raises income tomorrow. Given consumption is normal in both periods, the agent should consume more in period 0 (save less). Substitution effect A rise in the interest rate makes immediate consumption more costly. Therefore, the substitution effect suggests that the individual should choose to lower C0 (save more). Since the two effects go in opposite directions, the total effect on saving is uncertain. Usually, the substitution effect dominates so that agents save less when the interest rate rises, but it could go the other way. Figure 2.10 shows the intertemporal budget constraint. The endowment point is (Y0 , Y1 ). As the rate of interest increases, the budget constraint pivots around the endowment point as shown. Figure 2.10: Intertemporal budget constraint. Note that consumers reaching an optimum in the part of the budget constraint above the endowment point are savers, and those reaching an optimum somewhere in the lower part are borrowers. 32 2.7. Present value calculation with many periods Activity 2.5 Using Figure 2.10 above, explain that a saver cannot become a borrower if the rate of interest rises. 2.7 Present value calculation with many periods The previous section discussed the calculation of present value for a two-period stream of payoffs. We can extend this easily to multiple (or infinite) periods. This calculation is useful in many cases – for example, in calculating the repeated game payoff in game theory. This is also useful in understanding bond pricing. In this course, you need to know only the basics, which we present below. Suppose we have a stream of payoffs y0 , y1 , . . . , yn in periods 0, 1, . . . , n, respectively. Suppose the rate of interest is given by r. The present value in period 0 of this stream of payoffs is given by: PV = y0 + y2 yn y1 + + · · · + . 1 + r (1 + r)2 (1 + r)n If we write δ = 1/(1 + r), we can write this as: PV = y0 + δy1 + δ 2 y2 + · · · + δ n yn . Suppose y0 = y1 = · · · = yn = y. In this case: PV = y(1 + δ + δ 2 + · · · + δ n ). We can sum this as follows. Let: S = 1 + δ + δ2 + · · · + δn. Then: δS = δ + δ 2 + · · · + δ n+1 . We have: S − δS = 1 − δ n+1 . Therefore: S= 1 − δ n+1 . 1−δ So: PV = y 1 − δ n+1 . 1−δ If the payoff stream is infinite, the present value is very simple: 2 PV = y(1 + δ + δ + · · · ) = y 1 1−δ . 33 2. Consumer theory 2.7.1 Bonds A bond typically pays a fixed coupon amount x each period (next period onwards) until a maturity date T , at which point the face value F is paid. The price of the bond, P , is simply the present value given by: P = δx + δ 2 x + · · · + δ T F. Note that the price declines if δ falls, which happens if r rises. Therefore, the price of a bond has an inverse relationship with the rate of interest. A special type of bond is a consol or a perpetuity that never matures. The price of a consol has a particularly simple expression: P = δx + δ 2 x + · · · = x δ . 1−δ Now δ = 1/(1 + r). Therefore: δ 1/(1 + r) 1 = = . 1−δ r/(1 + r) r It follows that: x . r This makes the inverse relationship between P and r clear. P = 2.8 A reminder of your learning outcomes Having completed this chapter, the Essential reading and activities, you should be able to: explain the implications of the assumptions on the consumer’s preferences describe the concept of modelling preferences using a utility function draw indifference curve diagrams starting from the utility function of a consumer draw budget lines for different prices and income levels solve the consumer’s utility maximisation problem and derive the demand for a consumer with a given utility function and budget constraint analyse the effect of price and income changes on demand explain the notion of a compensated demand function explain measures of the welfare impact of a price change: change in consumer surplus, equivalent variation and compensating variation, and use these measures to analyse the welfare impact of a price change in specific cases construct the market demand curve from individual demand curves 34 2.9. Test your knowledge and understanding explain the notion of elasticity of demand analyse the decision to supply labour analyse the problem of savers and borrowers derive the present discounted value of payment streams and explain bond pricing. 2.9 2.9.1 Test your knowledge and understanding Sample examination questions 1. Indifference curves of an agent cannot cross. Is this true or false? Explain. 2. The Hicksian demand curve for a good must be more elastic than the Marshallian demand curve for a good. Is this true or false? Explain. 3. Savers gain more when the rate of interest rises. Is this true or false? Explain. 4. Consider the utility function u(x, y) = x2 + y 2 . (a) Does this satisfy the property of diminishing MRS? Show algebraically, and also show by drawing indifference curves. (b) Show that using the tangency condition (MRS equals price ratio) would not lead to an optimum in this case. (c) Show (in a diagram) the possible optimal bundles. 5. Consider the quasilinear utility function u(x1 , x2 ) = ln x1 + x2 (this is linear in x2 , but not in x1 , hence the name ‘quasilinear’). Let p1 and p2 denote the prices of x1 and x2 , respectively. Let m denote income. (a) Calculate the demand functions. (b) Draw the income-consumption curve. (c) Calculate the price elasticity of demand for each good. (d) Calculate the income elasticity of demand for each good. 35 2. Consumer theory 36 Chapter 3 Choice under uncertainty 3.1 Introduction In the previous chapter, we studied consumer choice in environments that had no element of uncertainty. However, many important economic decisions are made in situations involving some degree of risk. In this chapter, we cover a model of decision-making under uncertainty called the expected utility model. The model introduces the von Neumann–Morgenstern (vN–M) utility function. This is unlike the ordinal utility functions we saw in the previous chapter and has special properties. In particular, the curvature of the vN–M utility function can indicate a consumer’s attitude towards risk. Once we introduce the model, we use it to derive the demand for insurance and also introduce a measure of the degree of risk aversion. 3.1.1 Aims of the chapter This chapter aims to introduce the expected utility model which tells us how a consumer evaluates a risky prospect. We aim to show how this helps us understand attitudes towards risk and analyse the demand for insurance. We also aim to set up a measure of the degree of risk aversion. 3.1.2 Learning outcomes By the end of this chapter, the Essential reading and activities, you should be able to: calculate the expected value of a gamble explain the nature of the vN–M utility function and calculate the expected utility from a gamble explain the different risk attitudes and what they imply for the vN–M utility function analyse the demand for insurance and show the relationship between insurance and premium explain the concept of diversification calculate the Arrow–Pratt measure of risk aversion for different specifications of the vN–M utility function. 37 3. Choice under uncertainty 3.1.3 Essential reading N&S Sections 4.1, 4.2 and 4.3 up to and including the discussion on diversification (up to page 135). N&S does not cover the expected utility model or the Arrow–Pratt measure of risk aversion. We provide details below. These topics are also covered in Perloff Section 16.2 (exclude the last part on willingness to gamble). 3.2 Overview of the chapter This chapter covers expected utility theory and uses the theory to derive the demand for insurance. It also covers the Arrow–Pratt measure of risk aversion. 3.3 Preliminaries You should already be familiar with concepts such as probability and expected value. Do familiarise yourself with these concepts if this is not the case. Section 4.1 of N&S discusses these. Random variable A variable that represents the outcomes from a random event. A random variable has many possible values, and each value occurs with a specified probability. Expected value of a random variable Suppose X is a random variable that has values x1 , . . . , xn . For each i = 1, 2, . . . , n, the value xi occurs with probability pi , where p1 + p2 + · · · + pn = 1. The expected (or ‘average’) value of X is given by: X E(X) = p1 x1 + p2 x2 + · · · + pn xn = p i xi . i 3.4 Expected utility theory Expected utility theory was developed by John von Neumann and Oscar Morgenstern in their book The Theory of Games and Economic Behavior. (Princeton University Press, 1944; expected utility appeared in an appendix in the second edition in 1947). A proper exposition of their theory must await a Masters level course, but let us try to give a rough idea of what is involved. Suppose an agent faces a gamble G that yields an amount x1 with probability p1 , x2 with probability p2 , . . . , and xn with probability pn . How should the agent evaluate this gamble? Von Neumann and Morgenstern specified certain axioms, i.e. restrictions on 38 3.5. Risk aversion choice under uncertainty that might be deemed reasonable. They showed that under their axioms, there exists a function u such that the gamble can be evaluated using the following ‘expected utility’ formulation: E(U (G)) = p1 u(x1 ) + p2 u(x2 ) + · · · + pn u(xn ). The function u is known as the von Neumann–Morgenstern (vN–M) utility function. The vN–M utility function is somewhat special. It is not entirely an ordinal function like the utility functions you saw in the last chapter. Starting from a vN–M u function, we can make transformations of the kind a + bu, with b > 0 (these are called positive affine transformations), without changing the expected utility property but not any other kinds of transformations (for example, u2 is not allowed). The reason is that, as we discuss below, the curvature of the vN–M utility function captures attitude towards risk. Transformations other than positive affine ones change the curvature of this function, and therefore the transformed u function would not represent the same risk-preferences as the original. Thus vN–M utility functions are partly cardinal. Note that the expected utility representation is very convenient. Once we know the vN–M utility function u, we can evaluate any gamble easily by simply taking the expectation over the vN–M utility values. 3.5 Risk aversion We can show that an agent with a concave vN–M utility function over wealth is risk-averse. Let us show this by establishing that an agent with a concave u function would reject a fair gamble. Recall that a function f (W ) is concave if f 00 (W ) < 0, i.e. the second derivative of the function with respect to W is negative. Suppose G is a gamble which yields 20 with probability 1/2, and 10 with probability 1/2. Suppose an agent has wealth 15 and is given the following choice: invest 15 in gamble G, or do nothing. Note that the expected value of the gamble, E(G), is exactly 15, so that this is a fair gamble (the expected wealth is the same whether G is accepted or rejected). An agent who simply cared about expected value, and not about risk, would be indifferent between accepting and rejecting G. However, a risk-averse individual would reject a fair gamble. The expected utility of an agent from G is: 1 1 × u(20) + × u(10). 2 2 As Figure 3.1 shows, given a concave u-function: E(U (G)) = E(U (G)) < u(15) = u(E(G)). Therefore, the agent would not accept a fair gamble. This shows that a concave u-function implies risk aversion. Note that one of the implications of a concave vN–M utility function is that the marginal utility of wealth is declining. The point is noted in N&S Section 4.2. 39 3. Choice under uncertainty Figure 3.1: The vN–M utility function for a risk-averse individual. Note that the function is concave and u(E(G)) > E(U (G)) so that the agent does not accept a fair gamble. 3.6 Risk aversion and demand for insurance A risk-averse individual would pay to obtain insurance. To see that, it is useful to define the certainty equivalent (CE) of a gamble. The CE of a gamble is the certain wealth that would make an agent indifferent between accepting the gamble and accepting the certain wealth. As Figure 3.1 shows, the CE is lower than the expected income of 10. Suppose an agent simply faced gamble G (i.e. did not have the choice between G and 10, but simply faced G). Clearly, since the CE is lower than the expected outcome of G, this agent would be willing to pay a positive amount to buy insurance. How much would the agent be willing to pay? The amount an agent pays for insurance is called the risk premium. We now work through an example to understand how to calculate the risk premium. 3.6.1 Insurance premium for full insurance Kim’s utility depends on wealth W . Kim’s vN–M utility function is given by: √ u(W ) = W. Kim’s wealth is uncertain. With probability 0.5 wealth is 100, and with probability 0.5 a loss occurs so that wealth becomes 64. In what follows, we will assume that Kim can only buy full insurance. In other words, the insurance company offers to pay Kim 36 whenever the loss occurs and in exchange Kim pays them a premium of R in every state (i.e. whether the loss occurs or not). In what follows, we will calculate the maximum and minimum value of R. 40 3.6. Risk aversion and demand for insurance Let us first calculate Kim’s expected utility. This is given by: √ √ E(U ) = 0.5 × 100 + 0.5 × 64 = 9. How do we know Kim would be prepared to pay to buy full insurance? You can draw a diagram as above to show that Kim would be prepared to pay a positive premium if wealth is fully insured. Alternatively, you could point out that the expected utility of the uncertain wealth (which is 9) is lower than the utility of expected wealth since: √ √ u(E(W )) = 0.5 × 100 + 0.5 × 64 = 82 = 9.055. This implies that the premium that Kim is willing to pay is positive. √ You could also point out that W is a concave function (check that the second derivative is negative), implying that Kim is risk-averse. Then draw the CE point as above and point out that since expected wealth exceeds the certainty equivalent, the premium is positive. Let us now calculate the maximum premium that Kim would be willing to pay to buy full insurance. First, calculate the certainty equivalent of the gamble Kim is facing. This is given by: u(CE) = E(U ). Therefore: √ CE = 9 implying that CE = 81. Therefore, the maximum premium Kim is willing to pay is 100 − 81 = 19. There is another way of finding this, which considers the maximum premium in terms of expected wealth (i.e. how much expected wealth would Kim give up in order to fully insure?). You need to understand this, since some textbooks use this way of identifying the premium. For example, this is the approach adopted by Perloff. Unfortunately, textbooks (including Perloff) never make clear exactly what they are doing, which can be very confusing for students. Reading the exposition here should clarify the matter once and for all. The maximum premium in terms of expected wealth is calculated as follows. Note that under full insurance the gross expected wealth Kim would receive is E(W ) = 82. We also know that CE = 81. Therefore, the maximum amount of expected wealth Kim would give up is 82 − 81 = 1. (Note that this should explain why in the diagram on risk aversion in Perloff Section 16.2, and subsequent solved problems, the risk premium is identified as the difference between expected wealth and the CE.) To connect this approach to the one above, consider the actual premium and coverage. Kim loses 36 with probability 0.5. So full insurance means a coverage of 36, which is paid when the loss occurs. In return, Kim pays an actual premium of 19 in each state. Therefore, the change in expected wealth for Kim is: 0.5 × (−19) + 0.5 × (36 − 19) = 18 − 19 = −1. In other words, Kim is giving up 1 unit of expected wealth, as shown above. 41 3. Choice under uncertainty Once again, the purpose of writing this out in detail is to make you aware that textbooks vary in their treatment of this. Some talk about premium in terms of expected wealth, while others calculate the actual premium, but they do not make it clear what it is that they are doing. In answering questions of this sort in an examination, it is easiest (and clearest) to calculate the actual premium. You can follow the other route and define the premium in terms of expected wealth, but in that case you should make that clear in your answer. Next, we calculate the minimum premium. Assuming the insurance company is risk-neutral, it must break even. So the minimum premium (or fair premium) is Rmin such that it equals the expected payout, which is 0.5 × (100 − 64) = 18. (Note that this is simply 100 − E(W ), where E(W ) is expected wealth, which is 82 in this case.) As above, the other way of answering the question is to say that in terms of the expected wealth that Kim needs to give up, the minimum is zero. Think of this as follows. Kim simply hands over her actual wealth to the insurance company, and in return receives the expected wealth in all states. The insurance company is risk-neutral, and in expected wealth terms it is giving and receiving the same amount, and breaks even. 3.6.2 How much insurance? We calculated the premium for full insurance above. But suppose we gave a risk-averse agent a continuous ...