Microeconomics A. Daripa EC2066 2016 Undergraduate study in Economics, Management, Finance and the Social Sciences This is an extract from a subject guide for an undergraduate course offered as part of the University of London International Programmes in Economics, Management, Finance and the Social Sciences. Materials for these programmes are developed by academics at the London School of Economics and Political Science (LSE). For more information, 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. 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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: [email protected] 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 ū. Consider the set of all points that yield utility ū 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 M̄ denote unearned income (inheritance, gifts etc). The utility maximisation problem is: max u(N, M ) N, M subject to: Z = 24 − N M = wZ + M̄ . We can simplify the constraints to M = w(24 − N ) + M̄ , or M + wN = 24w + M̄ . Therefore, we have a familiar utility maximisation problem: max u(N, M ) N, M subject to the budget constraint: M + wN = 24w + M̄ . 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 choice of levels of insurance. Can we say something general about how much insurance an agent would choose? As it turns out, we can. If insurance is actuarially fair (which is another way of saying that the insurance company just breaks even, so that the premium is equal to the expected payment to the agent), we can show that any risk-averse agent would buy full insurance. If the premium is higher than this, less than full insurance would be bought. To relate this to the section above, note that there the agent was given a simple choice between full insurance and no insurance, and in that case the maximum willingness to pay for insurance is 19, even though the fair premium is 18. However, if a more continuous choice of insurance levels was provided to that agent, the agent would optimally buy full insurance only at a premium of 18, and optimally buy less-than-full insurance at a premium of 19. If insurance is fair, it does not matter what the degree of risk-aversion is. Every risk-averse agent would buy full insurance. If, on the other hand, the insurance premium is greater than the fair level, how much insurance an agent buys depends on their degree of risk-aversion. No agent with a finite degree of risk-aversion would buy full insurance anymore, but the extent of insurance purchased increases as the agent’s degree of risk-aversion rises. Let us now show that full insurance is purchased when the premium is fair. Suppose a risk-averse agent has wealth W , but faces the prospect of a loss of L with probability p, where 0 < p < 1. The agent can buy a coverage of X by paying the premium rX. The wealth if loss occurs is given by WL = W − L + X − rX, and the wealth when no loss occurs is given by WN = W − rX. The expected utility of the agent is: E(U ) = pu(W − L + X − rX) + (1 − p)u(W − rX) = pu(WL ) + (1 − p)u(WN ). 42 3.7. Risk-neutral and risk-loving preferences Maximising with respect to X, we get the first-order condition: pu0 (WL )(1 − r) − (1 − p)u0 (WN )r = 0. Note that the second-order condition for a maximum is satisfied since the agent is risk-averse implying that u00 < 0 (u is concave). Therefore, we have: u0 (WL ) (1 − p)r = . 0 u (WN ) (1 − r)p If insurance is fair, that implies the insurance company breaks even, i.e. the expected payout pX equals the expected receipt rX. Since pX = rX, we have: p = r. It follows that: u0 (WL ) = u0 (WN ). Since u0 is a decreasing function (because u00 < 0), it is not possible to have this equality if WL 6= WN . (Note that if u0 was a non-monotonic function and was going up and down, it would be possible to have WL different from WN but have the same value of u0 at these two different points.) It follows that WL = WN , i.e. we have: W − L + X − rX = W − rX implying that X = L. Therefore, the agent would optimally fully insure (cover the entire loss) at the fair premium. Note also what would happen if r > p. Then (1 − p)r > (1 − r)p. Therefore: u0 (WL ) (1 − p)r = > 1. u0 (WN ) (1 − r)p This implies that: u0 (WL ) > u0 (WN ). Again, because u0 is a decreasing function (u00 < 0), this implies that WL < WN , which in turn implies that X < L. Thus if the premium exceeds the fair premium, less than full insurance would be purchased. 3.7 Risk-neutral and risk-loving preferences Just as a concave vN–M utility function represents risk aversion, the opposite – a convex vN–M utility function (so that we have u00 > 0) – represents risk-loving behaviour, and the vN–M utility function is a straight line (u00 = 0) for a risk-neutral agent. A risk-neutral agent does not care about risk and only cares about the expected value of a gamble. In other words, a risk-neutral agent is indifferent between accepting and rejecting a fair gamble. For a risk-neutral agent, we can write the vN–M utility function of wealth simply as: u(W ) = W. 43 3. Choice under uncertainty Figure 3.2: The vN–M utility function for a risk-neutral individual. Note that the function is linear and u(E(G)) = E(U (G)) so that the agent is indifferent between a fair gamble and the safe alternative of 15. Figure 3.2 shows the vN–M utility function for a risk-neutral agent. The figure refers to the gamble introduced in the section on risk aversion (either keep 15 or invest in a fair gamble yielding 20 with probability 0.5, and 10 with probability 0.5). A risk-loving agent, on the other hand, prefers a risky bet to a safe alternative when they have the same expected outcome. In other words, a risk-loving agent would prefer to accept a fair gamble. As Figure 3.3 below shows, for a risk-loving agent, the CE of a gamble is higher than the expected value of the gamble (you would have to pay a risk-loving agent to give up a risky gamble in favour of the safe alternative of getting the expected value of the gamble). Figure 3.3 shows the vN–M utility function for a risk-loving agent. 3.8 The Arrow–Pratt measure of risk aversion We discussed different degrees of risk aversion in the section above. How do we measure the degree of risk aversion? As you might guess, the degree of risk aversion has to do with the curvature of the vN–M utility function u. The more concave it is, the greater the degree of risk aversion. The closer it is to a straight line, the lower the degree of risk aversion. Since the second derivative captures the curvature, a measure of risk aversion might be u00 . However, this would not be ideal for the following reason. We know that a positive affine transformation of u, say u b = a + bu, where a and b are positive constants, does not change attitude towards risk. But such a transformation would change the second derivative and, therefore, change the risk measure. This problem could be avoided if u00 is divided by u0 . Furthermore, since the most common risk attitude is risk aversion, and for this case u00 < 0, putting a negative sign in front of u00 would deliver a 44 3.8. The Arrow–Pratt measure of risk aversion Figure 3.3: The vN–M utility function for a risk-loving individual. Note that the function is convex and u(E(G)) < E(U (G)) so that the agent prefers a fair gamble to the safe wealth of 15. positive risk measure under risk aversion. These help to interpret the Arrow–Pratt measure of risk aversion, which is given by: u00 ρ=− 0. u That is, the Arrow–Pratt measure of risk aversion is −1 times the ratio of the second derivative and the first derivative of the vN–M utility function. This is the most common measure of risk aversion. There are other measures, which you will encounter in Masters level courses. It can be shown that the larger the Arrow–Pratt measure of risk aversion, the more small gambles an individual will take. A derivation of this result must await a Masters level course as well. As noted above, for a risk-averse individual, u00 < 0, so the minus sign in front makes the measure a positive number. For a risk-neutral agent, u00 = 0 so that ρ = 0, and for a risk-loving agent u00 > 0 so that the measure is negative. Example 3.1 Let us calculate the Arrow–Pratt measure for different specifications of the vN–M utility function. (a) Suppose u(W ) = ln W . Then u0 (W ) = 1/W and u00 (W ) = −1/W 2 . It follows that: 1 ρ= . W This agent has risk aversion that is decreasing in wealth. 45 3. Choice under uncertainty (b) Next, suppose u(W ) = W α , where 0 < α < 1. Then: ρ=− (α − 1)αW α−2 1−α . = α−1 αW W Note that as α increases, the degree of risk aversion decreases. As α goes to 1, the risk aversion measure goes to 0, which is right since at α = 1 the agent is risk-neutral. (c) Next, suppose u(W ) = −e−α . Then u0 (W ) = αe−α and u00 (W ) = −α2 e−α . Therefore, ρ = α. In this case the degree of risk aversion does not depend on the wealth level. 3.9 Reducing risk Insurance provides a way to reduce risk. Diversification is also another way to reduce risk. You should read carefully the discussion on this in N&S Section 4.3 (you do not need to study this section beyond diversification). 3.10 A reminder of your learning outcomes Having completed 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. 3.11 Test your knowledge and understanding 3.11.1 Sample examination questions 1. A risk-averse individual is offered a choice between a gamble that pays 1000 with a probability of 1/4, and 100 with a probability of 3/4, or a payment of 325. Which would they choose? What if the payment was 320? 46 3.11. Test your knowledge and understanding 2. Suppose u(W ) = −1/W . What is the risk attitude of this person? Calculate the Arrow–Pratt measure of risk aversion for this preference. 3. Suppose an agent has vN–M utility function u(W ). Under what condition p would a + bu(W ) also be a valid vN–M utility function for this agent? Would u(W ) be a valid vN–M utility function for this agent? 4. Suppose u(W ) = ln W for an agent. The agent faces the following gamble: with probability 0.5 wealth is 100, and with probability 0.5 a loss occurs so that wealth becomes 64. The agent can buy any amount of insurance: a coverage of X can be purchased by paying premium rX. (a) Work out the insurance coverage X that the agent would optimally purchase as a function of r. (b) Plot the optimal X as a function of r. (c) Calculate the value of r for which full insurance is purchased. (d) Calculate the value of r for which no insurance is purchased. 47 3. Choice under uncertainty 48 Chapter 4 Game theory 4.1 Introduction In many economic situations agents must act strategically by taking into account the behaviour of others. Game theory provides a set of tools that enables you to analyse such situations in a logically coherent manner. For each concept introduced, you should try to understand why it makes sense as a tool of analysis and (this is much harder) try to see what its shortcomings might be. This is not the right place to ask questions such as ‘what is the policy-relevance of this concept?’. The concepts you come across here are just tools, and that is the spirit in which you should learn them. As you will see, some of these concepts are used later in this course (as well as in a variety of other economics courses that you might encounter later) to analyse certain types of economic interactions. 4.1.1 Aims of the chapter The chapter aims to familiarise you with a subset of basic game theory tools that are used extensively in modern microeconomic theory. The chapter aims to cover simultaneous-move games as well as sequential-move games, followed by an analysis of the repeated Prisoners’ Dilemma game. 4.1.2 Learning outcomes By the end of this chapter, the Essential reading and activities, you should be able to: analyse simultaneous-move games using dominant strategies or by eliminating dominated strategies either once or in an iterative fashion calculate Nash equilibria in pure strategies as well as Nash equilibria in mixed strategies in simultaneous-move games explain why Nash equilibrium is the central solution concept and explain the importance of proving existence specify strategies in extensive-form games analyse Nash equilibria in extensive-form games explain the idea of refining Nash equilibria in extensive-form games using backward induction and subgame perfection 49 4. Game theory analyse the infinitely-repeated Prisoners’ Dilemma game with discounting and analyse collusive equilibria using trigger strategies explain the multiplicity of equilibria in repeated games and state the folk theorem for the Prisoners’ Dilemma game. 4.1.3 Essential reading N&S Chapter 5. However, the content of this chapter is not sufficient by itself. This chapter of the subject guide fleshes out the basic tools that you are required to know in some detail, and you should also read this and follow the exercises carefully. For repeated games, the interpretation of payoffs presented here is slightly different from the textbook. While the analyses are formally equivalent, the approach in this subject guide follows the standard one in the literature. You are likely to find this fits better with analyses found in other, more advanced, courses or research papers. 4.1.4 Further reading You might find the following textbook useful for further explanations of the concepts covered in this subject guide. Osborne, M.J. An Introduction to Game Theory (Oxford University Press 2009) International edition [ISBN 9780195322484]. 4.2 Overview of the chapter The chapter introduces simultaneous-move games and analyses dominance criteria as well as Nash equilibrium. Next, the chapter introduces sequential-move games and analyses Nash equilibrium as well as subgame-perfect Nash equilibrium. Finally, the chapter introduces repeated games and analyses the infinitely-repeated Prisoners’ Dilemma. 4.3 Simultaneous-move or normal-form games A simultaneous-move game (also known as a normal-form game) requires three elements. First, a set of players. Second, each player must have a set of strategies. Once each player chooses a strategy from their strategy set, we have a strategy profile. For example, suppose there are two players, 1 and 2, and 1 can choose between Up or Down (so 1’s strategy set is {U, D}) and 2 can choose between Left or Right (so 2’s strategy set is {L, R}). Then if 1 chooses Up and 2 chooses Left, we have the strategy profile {U, L}. There are altogether 4 such strategy profiles: the one just mentioned, plus {D, L}, {U, R} and {D, R}. Once you understand what a strategy profile is, we can define the third element of a game: a payoff function for each player. A payoff function for any player is defined over the set of strategy profiles. For each strategy profile, each player gets a payoff. 50 4.3. Simultaneous-move or normal-form games Suppose Si denotes the set of strategies of player i. In the example above, S1 = {U, D} and S2 = {L, R}. Let si denote a strategy of i (that is, si is in the set Si ). If there are n players, a strategy profile is (s1 , s2 , . . . , sn ). For any such strategy profile, there is a payoff for each player. Let ui (s1 , s2 , . . . , sn ) denote the payoff of player i. Continuing the example above, suppose u1 ({U, L}) = u1 ({D, R}) = 1 and u1 ({D, L}) = u1 ({U, R}) = 2. Furthermore, suppose u2 ({U, L}) = u2 ({D, R}) = 2, u2 ({D, L}) = 3 and u2 ({U, R}) = 1. We write this in a convenient matrix form (known as the normal form) as follows. The first number in each cell is the payoff of player 1 and the second number is the payoff of player 2. Note that player 1 chooses rows and player 2 chooses columns. Player 1 U D Player 2 L R 1, 2 2, 1 2, 3 1, 2 It is sometimes useful to write the strategy profile (s1 , s2 , . . . , sn ) as (si , s−i ), where s−i is the profile of strategies of all players other than player i. So: s−i = (s1 , s2 , . . . , si−1 , si+i . . . . , sn ) (the ith element is missing). With this notation, we can write the payoff of player i as ui (si , s−i ). 4.3.1 Dominant and dominated strategies Let us now try to understand how rational players should play a game. In some cases, there is an obvious solution. Suppose a player has a strategy that gives a higher payoff compared to other strategies irrespective of the strategy choices of others. Such a strategy is called a dominant strategy. If a player has a dominant strategy, his problem is simple: he should clearly play that strategy. If each player has a dominant strategy, the equilibrium of the game is obvious: each player plays his own dominant strategy. Consider the following game. Each player has two strategies C (cooperate) and D (defect, which means not to cooperate). There are 4 possible strategy profiles and each profile generates a payoff for each player. The first number in each cell is the payoff of player 1 and the second number is the payoff of player 2. Again, note that player 1 chooses the row that is being played and player 2 chooses the column that is being played. Player 1 C D Player 2 C D 2, 2 0, 3 3, 0 1, 1 Here each player has a dominant strategy, D. This is the well-known Prisoners’ Dilemma game. Rational players, playing in rational self-interest, get locked into a 51 4. Game theory dominant-strategy equilibrium that gives a lower payoff compared to the situation where both players cooperate. However, cooperating cannot be part of any equilibrium, since D is the dominant strategy. Later on we will see that if the game is infinitely-repeated, then under certain conditions cooperation can emerge as an equilibrium. But in a one-shot game (i.e. a game that is played once) the only possible equilibrium is that each player plays their dominant strategy. In the game above, the dominant strategy equilibrium is (D, D). In terms of the notation introduced before, we can define a dominant strategy as follows. Dominant strategy Strategy s∗i of player i is a dominant strategy if: ui (s∗i , s−i ) > ui (si , s−i ) for all si different from s∗i and for all s−i . That is, s∗i performs better than any other strategy of player i no matter what others are playing. 4.3.2 Dominated strategies and iterated elimination Even if a player does not have a dominant strategy, he might have one or more dominated strategies. A dominated strategy for i is a strategy of i (say si ) that yields a lower payoff compared to another strategy (say s0i ) irrespective of what others are playing. In other words, the payoff of i from playing si is always (i.e. under all possible choices of other players) lower than the payoff from playing s0i . Since si is a dominated strategy, i would never play this strategy. Thus we can eliminate dominated strategies. Indeed, we can eliminate such strategies not just once, but in an iterative fashion. If in some game, all strategies except one for each player can be eliminated by iteratively eliminating dominated strategies, the game is said to be dominance solvable. Consider the game in Section 4.3. Note that no strategy is dominant for player 1, but for player 2 L dominates R. So we can eliminate the possibility of player 2 playing R. Once we do this, in the remaining game, for player 1 D dominates U (2 is greater than 1). So we can eliminate U . We are then left with (D, L), which is the equilibrium by iteratively eliminating dominated strategies. The game is dominance solvable. Here is another example of a dominance solvable game. We find the equilibrium of this game by iteratively eliminating dominated strategies. Player 1 Top Middle Bottom Left 4, 3 5, 5 3, 5 Player 2 Middle Right 2, 7 0, 4 5, −1 −4, −2 1, 5 −1, 6 We can eliminate dominated strategies iteratively as follows. 52 4.3. Simultaneous-move or normal-form games 1. For player 1, Bottom is dominated by Top. Eliminate Bottom. 2. In the remaining game, for player 2, Right is dominated by Middle. Eliminate Right. 3. In the remaining game, for player 1, Top is dominated by Middle. Eliminate Top. 4. In the remaining game, for player 2, Middle is dominated by Left. Eliminate Middle. This gives us (Middle, Left) as the unique equilibrium. 4.3.3 Nash equilibrium However, for many games the above criteria of dominance do not allow us to find an equilibrium. Players might not have dominant strategies; moreover none of the strategies of any player might be dominated. The following game provides an example. A1 Player 1 B1 C1 A2 3, 1 1, 0 2, 3 Player B2 1, 3 3, 1 2, 0 2 C2 4, 2 3, 0 3, 2 As noted above, the problem with dominance criteria is that they apply only to some games. For games that do not have dominant or dominated strategies, the idea of deriving an equilibrium using dominance arguments does not work. If we cannot derive an equilibrium by using dominant strategies or by (iteratively) eliminating dominated strategies, how do we proceed? If we want to derive an equilibrium that does not rely on specific features such as dominance, we need a concept of equilibrium that applies generally to all games. As we show below, a Nash equilibrium (named after the mathematician John Nash) is indeed such a solution concept. Pure and mixed strategies Before proceeding further, we need to clarify something about the nature of strategies. In the discussion above, we identified strategies with single actions. For example, in the Prisoners’ Dilemma game, we said each player has the strategies C and D. However, this is not a full description of strategies. A player could also do the following: play C with probability p and play D with probability (1 − p), where p is some number between 0 and 1. Such a strategy is called a ‘mixed’ strategy; while a strategy that just chooses one action (C or D) is called a ‘pure’ strategy. We start by analysing Nash equilibrium under pure strategies. Later we introduce mixed strategies. We then note that one can prove an existence result: all games have at least one Nash equilibrium (in either pure or mixed strategies). This is why Nash equilibrium is the central solution concept in game theory. 53 4. Game theory A strategy profile (s∗1 , s∗2 , . . . , s∗n ) is a Nash equilibrium (in pure strategies) if it is a mutual best response. In other words, for every player i, the strategy s∗i is a best response to s∗−i (as explained above, this is the strategy profile of players other than i). In yet other words, if (s∗1 , s∗2 , . . . , s∗n ) is a Nash equilibrium, it must satisfy the property that given the strategy profile s∗−i of other players, player i cannot improve his payoff by replacing s∗i with any other strategy. A more formal definition is as follows. Nash equilibrium in pure strategies A strategy profile (s∗i , s∗−i ) is a Nash equilibrium if for each player i: ui (s∗i , s∗−i ) ≥ ui (si , s∗−i ) for all strategies si in the set Si . To find out the Nash equilibrium of the game above, we must look for the mutual best responses. Let us check the best response of each player. Player 1’s best response is as follows: Player 2’s strategy A2 B2 C2 Player 1’s best response A1 B1 A1 Player 2’s best response: Player 1’s strategy A1 B1 C1 Player 2’s best response B2 B2 A2 Note from these that the only mutual best response is (B1 , B2 ). This is the only Nash equilibrium in this game. You could also check as follows: If player 1 plays A1 , player 2’s best response is B2 . However, if player 2 plays B2 , player 1 will not play A1 (B1 is a better response than A1 ). Therefore, there is no Nash equilibrium involving A1 . If player 1 plays B1 , player 2’s best response is B2 . If player 2 plays B2 , player 1’s best response is B1 . Therefore, (B1 , B2 ) is a Nash equilibrium. If player 1 plays C1 , player 2’s best response is A2 . However, if player 2 plays A2 , player 1 would not play C1 (A1 is a better response). Therefore, there is no Nash equilibrium involving C1 . From the three steps above, we can conclude that (B1 , B2 ) is the only Nash equilibrium. 54 4.3. Simultaneous-move or normal-form games Let us also do a slightly different exercise. Suppose you want to check if a particular strategy profile is a Nash equilibrium. Suppose you want to check if (A1 , C2 ) is a Nash equilibrium. You should check as follows. If player 2 plays C2 , player 1 cannot do any better by changing strategy from A1 (4 is better than 3 from B1 or 3 from C1 ). However, player 2 would not want to stay at (A1 , C2 ) since player 2 can do better by switching to B2 (3 from B2 is better than 2 from C2 ). We can therefore conclude that (A1 , C2 ) is not a Nash equilibrium. You can similarly check that all boxes other than (B1 , B2 ) have the property that some player has an incentive to switch to another box. However, if the players are playing (B1 , B2 ) no player has an incentive to switch away. Neither player can do better by switching given what the other player is playing. Since player 2 is playing B2 , player 1 gets 3 from B1 which is better than 1 from A1 or 2 from C1 . Since player 1 is playing B1 , player 2 gets 1 from B2 which is better than 0 from A2 or C2 . Therefore (B1 , B2 ) is a Nash equilibrium. Nash equilibrium is not necessarily unique. Consider the following game. A Player 1 B Player 2 A B 2, 1 0, 0 0, 0 1, 2 Note there are multiple pure strategy Nash equilibria. Both (A, A) and (B, B) are Nash equilibria. Dominance criteria and Nash equilibrium Note that while a dominant strategy equilibrium is also a Nash equilibrium, Nash equilibrium does not require dominance. However, the greater scope of Nash equilibrium comes at a cost: it places greater rationality requirements on players. To play (B1 , B2 ), player 1 must correctly anticipate that player 2 is going to play B2 . Such a requirement is even more problematic when there are multiple Nash equilibria. On the other hand, if players have dominant strategies, they do not need to think at all about what others are doing. A player would simply play the dominant strategy since it is a best response no matter what others do. This is why a dominant strategy equilibrium (or one achieved through iterative elimination of dominated strategies) is more convincing than a Nash equilibrium. However, as noted before, many games do not have dominant or dominated strategies, and are therefore not dominance solvable. We need a solution concept that applies generally to all games, and Nash equilibrium is such a concept. 4.3.4 Mixed strategies Consider the following game. A1 Player 1 B1 Player 2 A2 B2 3, 1 2, 3 2, 1 3, 0 55 4. Game theory Notice that this game has no pure strategy Nash equilibrium. However, as you will see below, the game does have a Nash equilibrium in mixed strategies. Let us first define a mixed strategy. Mixed strategy A mixed strategy si is a probability distribution over the set of (pure) strategies. In the game above, A1 and B1 are the pure strategies of player 1. A mixed strategy of player 1 could be A1 with probability 1/3, and B1 with probability 2/3. Notice that a pure strategy is only a special case of a mixed strategy. A Nash equilibrium can then be defined in the usual way: a profile of mixed strategies that constitute a mutual best response. A mutual best response in mixed strategies has an essential property that makes it easy to find mixed strategy Nash equilibria. Let us consider games with two players to understand this property. Suppose we have an equilibrium in which both players play strictly mixed strategies: player 1 plays A1 with probability p and B1 with probability (1 − p) where 0 < p < 1, and player 2 plays A2 with probability q and B2 with probability (1 − q) where 0 < q < 1. In this case, whenever player 1 plays A1 , she gets an expected payoff of: π1 (A1 ) = 3q + 2(1 − q). Whenever player 1 plays B1 , she gets an expected payoff of: π1 (B1 ) = 2q + 3(1 − q). What must be true of these expected payoffs that player 1 gets from playing A1 and B1 ? Suppose π1 (A1 ) > π1 (B1 ). Then clearly player 1 should simply play A1 , rather than any strictly mixed strategy, to maximise her payoff. In other words, player 1’s best response in this case would be to choose p = 1, rather than a strictly mixed strategy p < 1. But then we do not have a mixed strategy Nash equilibrium. Similarly, if π1 (A1 ) < π1 (B1 ), player 1’s best response would be to choose p = 0 (i.e. just to play B1 ), and again we cannot have a mixed strategy Nash equilibrium. So if player 1 is going to play a mixed strategy in equilibrium it must be that she is indifferent between the two strategies. How does such indifference come about? This is down to player 2’s strategy choice. Player 2’s choice of q must be such that player 1 is indifferent between playing A1 or B1 . In other words, in equilibrium q must be such that π1 (A1 ) = π1 (B1 ), i.e. we have: 3q + 2(1 − q) = 2q + 3(1 − q) which implies q = 1/2. But if player 2 is going to choose q = 1/2 it must be that he is indifferent between A2 and B2 (otherwise player 2 would not want to mix, but would play a pure strategy). How can such indifference come about? Well, player 1 must choose p in such a way so as 56 4.3. Simultaneous-move or normal-form games to make player 2 indifferent between A2 and B2 . In other words, player 1’s choice of p is such that π2 (A2 ) = π2 (B2 ), i.e. we have: 1 = 3p which implies p = 1/3. Therefore, the mixed strategy Nash equilibrium is as follows. Player 1 plays A1 with probability 1/3 and B1 with probability 2/3, while player 2 plays A2 with probability 1/2 and B2 with probability 1/2. We can also show this in a diagram. Let us first write down the best response function of each player. Player 1’s best response function is given by: q > 1/2 =⇒ p = 1 is the best response. q = 1/2 =⇒ any p in [0, 1] is a best response. q < 1/2 =⇒ p = 0 is the best response. Player 2’s best response function is given by: p > 1/3 =⇒ q = 0 is the best response. p = 1/3 =⇒ any q in [0, 1] is a best response. p < 1/3 =⇒ q = 1 is the best response. Figure 4.1 below shows these best response functions and shows the equilibrium point (where the two best response functions intersect). Figure 4.1: The best-response functions. They cross only at E, which is the mixed strategy Nash equilibrium. There are no pure strategy Nash equilibria in this case. 57 4. Game theory To recap, the essential property of a mixed strategy Nash equilibrium in a two-player game is that each player’s chosen probability distribution must make the other player indifferent between the strategies he is randomising over. In a k-player game, the joint distribution implied by the choices of each player in every combination of (k − 1) players must be such that the kth player receives the same expected payoff from each of the strategies he plays with positive probability. The game above has no Nash equilibrium in pure strategies, but has a mixed strategy Nash equilibrium. However, in other games that do have pure strategy Nash equilibria, there might be yet more equilibria in mixed strategies. For instance, we could find a mixed strategy Nash equilibrium in the Battle of the sexes. Activity 4.1 Find the mixed strategy Nash equilibrium in the following game. Also show all Nash equilibria of the game in a diagram by drawing the best response functions. A Player 1 B 4.3.5 Player 2 A B 2, 1 0, 0 0, 0 1, 2 Existence of Nash equilibrium Once we include mixed strategies in the set of strategies, we have the following existence theorem, proved by John Nash in 1951. Existence theorem Every game with a finite number of players and finite strategy sets has at least one Nash equilibrium. Nash proved the existence theorem for his equilibrium concept using a mathematical result called a ‘fixed-point theorem’. Take any strategy profile and compute the best response to it for every player. So the best response is another strategy profile. Suppose we do this for every strategy profile. So long as certain conditions are satisfied, a fixed-point theorem says that there is going to be at least one strategy profile which is a best response to itself. This is, of course, a Nash equilibrium. Thus upon setting up the strategy sets and the best response functions properly, a fixed-point theorem can be used to prove existence. You will see a formal proof along these lines if you study game theory at more advanced levels. Here, let us point out the importance of this result. The importance of proving existence The existence theorem is indeed very important. It tells us that no matter what game we look at, we will always be able to derive at least one Nash equilibrium. If existence did not hold for some equilibrium concept (for example, games do not necessarily have a dominant strategy equilibrium), we could derive wonderful properties of that concept, but we could not be sure such derivations would be of any use. The particular game 58 4.4. Sequential-move or extensive-form games that we confront might not have an equilibrium at all. But being able to prove existence for Nash equilibrium removes such problems. Indeed, as noted before as well, this is precisely what makes Nash equilibrium the main solution concept for simultaneous-move games. 4.3.6 Games with continuous strategy sets We have so far analysed games with discrete strategy sets. However, the above analysis can easily extend to certain classes of games with continuous strategy sets. We will analyse a few such games (the Cournot game, the Bertrand game, and the Bertrand game with product differentiation) in detail later in discussing oligopoly (in Chapter 9 of the subject guide). Also see N&S Section 5.7 for an example of a ‘commons problem’ game with continuous strategy sets. We will also refer to this game when discussing externalities (in Chapter 12 of the subject guide). 4.4 Sequential-move or extensive-form games Let us now consider sequential move games. In this case, we need to draw a game tree to depict the sequence of actions. Games depicted in such a way are also known as extensive-form games. In this subject guide we will consider the phrases ‘sequential-move game’ and ‘extensive-form game’ as interchangeable. To start with, we assume that each player can observe the moves of players who act before them. First, we need to understand the difference between actions and strategies in such games. Once we clarify this, we show how to derive Nash equilibria. Finally, we propose a refinement of Nash equilibrium: subgame perfect Nash equilibrium. In games where all moves of previous players can be observed, subgame perfect Nash equilibria can be derived by backward induction. We then introduce some simple cases where information is imperfect, and show how the notion of strategies differs, and how to derive subgame perfect Nash equilibria. Consider the following extensive-form game. Each player has two actions: player 1’s actions are a1 and a2 and player 2’s actions are b1 and b2 . Player 1 moves before player 2. Player 2 can observe player 1’s action and, therefore, can vary his action depending on the action of player 1. For each profile of actions by player 1 and player 2 there are payoffs at the end. As usual, the first number is the payoff of the first mover (in this case player 1) and the second number is the payoff of the second mover (here player 2). We can define the game as a graph: it has decision nodes and branches from decision nodes to successor nodes. However, such formal definitions are useful only at a later stage. If you simply look at Figure 4.2 below, the depiction of the sequence of players, their action choices at each stage and their final payoffs should be clear to you. 4.4.1 Actions and strategies The notion of a strategy is fairly straightforward in a normal form game. However, for an extensive-form game, it is a little bit more complicated. A strategy in an extensive form game is a complete plan of actions. In other words, a strategy for player i must 59 4. Game theory Figure 4.2: An extensive-form game. specify an action at every node at which i can possibly move. Consider the game above. As already noted, each player has 2 actions. For player 1, the set of actions and the set of strategies is the same. Player 1 can simply decide between a1 and a2 . Therefore, the strategy set of player 1 is simply {a1 , a2 }. Player 2, on the other hand, must plan for two different contingencies. He must decide what to do if player 1 plays a1 , and what to do if player 1 plays a2 . Note that such decisions must be made before the game is actually played. Essentially, game theory tries to capture the process of decision-making of individuals. Faced with a game such as the one above, player 2 must consider both contingencies. This is what we capture by the notion of a strategy. It tells us what player 2 would do in each of the two possible cases. Now player 2 can choose 2 possible actions at the left node (after player 1 plays a1 ) and 2 possible actions at the right node (after a2 ). So there are 2 × 2 = 4 possible strategies for player 2. These are: 1. If player 1 plays a1 , play b1 and if player 1 plays a2 , play b1 . 2. If player 1 plays a1 , play b1 and if player 1 plays a2 , play b2 . 3. If player 1 plays a1 , play b2 and if player 1 plays a2 , play b1 . 4. If player 1 plays a1 , play b2 and if player 1 plays a2 , play b2 . For the sake of brevity of notation, we write these as follows. Just as we read words from left to right, we read strategies from left to right. So we write the strategy ‘if player 1 plays a1 , play b2 and if player 1 plays a2 , play b1 ’ as b2 b1 . Reading from left to right, this implies that the plan is to play b2 at the left node and play b1 at the right node. This is precisely what the longer specification says. So the strategy set of player 2 is {(b1 b1 ), (b1 b2 ), (b2 b1 ), (b2 b2 )}. Suppose instead of 2 actions, player 2 could choose between b1 , b2 and b3 at each node. In that case, player 2 would have 3 × 3 = 9 strategies. 60 4.4. Sequential-move or extensive-form games Suppose player 1 had 3 strategies a1 , a2 and a3 , and after each of these, player 2 could choose between b1 and b2 . Then player 2 would have 2 × 2 × 2 = 8 strategies. 4.4.2 Finding Nash equilibria using the normal form The matrix-form we used to write simultaneous-move games in Section 4.3 is known as the ‘normal form’. To find Nash equilibria in an extensive-form game, the most convenient method is to transform it into its normal form. This is as follows. Note that player 1 has two strategies and player 2 has four strategies. Therefore, we have a 2-by-4 matrix of payoffs as follows: Player 1 a1 a2 b1 b1 3, 1 4, 1 Player 2 b1 b2 b2 b1 3, 1 1, 0 0, 1 4, 1 b2 b2 1, 0 0, 1 Note that if we pair, say, a1 with b2 b1 , only the first component of player 2’s strategy is relevant for the payoff. In other words, since player 1 plays a1 , the payoff is generated by player 2’s response to a1 , which in this case is b2 . Similarly, if we pair a2 with b2 b1 , the second component of player 2’s strategy is relevant for the payoff. Since player 1 plays a2 , we need player 2’s response to a2 , which in this case is b1 , to determine the payoff. Once we write down the normal form, it is easy to find Nash equilibria. Here let us only consider pure strategy Nash equilibria. There are three pure strategy Nash equilibria in this game. Finding these is left as an exercise. Example 4.1 Find the pure strategy Nash equilibria of the extensive-form game above. It is easiest to check each box. If we start at the top left-hand box, player 1 would switch to a2 . So this is not a Nash equilibrium. From (a2 , b1 b1 ) no player can do better by deviating. Therefore, (a2 , b1 b1 ) is a Nash equilibrium. Next try (a1 , b1 b2 ). This is indeed a Nash equilibrium. Note that you must write down the full strategy of player 2. It is not enough to write (a1 , b1 ). Unless we know what player 2 would have played in the node that was not reached (in this case the node after a2 was not reached), we cannot determine whether a strategy is part of a Nash equilibrium. So while (a1 , b1 b2 ) is indeed a Nash equilibrium, (a1 , b1 b1 ) is not. Finally, (a2 , b2 b1 ) is also a Nash equilibrium. These are the three pure strategy Nash equilibria of the game. 4.4.3 Imperfect information: information sets So far we have assumed perfect information: each player is perfectly informed of all previous moves. Let us now see how to represent imperfect information: players may not be perfectly informed about some of the (or all of the) previous moves. Games of imperfect information give rise to certain types of problems that require more sophisticated refinements of Nash equilibria than we study here. In particular, if a 61 4. Game theory player does not observe some past moves, what he believes took place becomes important. We do not study these problems here – the analysis below simply shows you how to represent situations of imperfect information in some simple cases. Consider the extensive-form game introduced at the start of this section. Suppose at the time of making a decision, player 2 does not know what strategy player 1 has chosen. Since player 2 does not know what player 1 has chosen, at the time of taking an action player 2 does not know whether he is at the left node or at the right node. To capture this situation of imperfect information in our game-tree, we say that the two decision nodes of player 2 are in an information set. We represent this information set as in Figure 4.3: by connecting the two nodes by a dotted line (another standard way to do this is to draw an elliptical shape around the two nodes – you will see this in N&S). Figure 4.3: An extensive-form game with an information set. Note that player 2 knows he is at the information set (after player 1 moves), but does not know where he is in the information set (i.e. he does not know what player 1 has chosen). Since player 2 cannot distinguish between the two nodes inside his information set, he cannot take different actions at the two nodes. Therefore, the strategy set of 2 is simply {b1 , b2 }. In other words, now, for both players the strategy set coincides with the action set. This is not surprising: since player 2 takes an action without knowing what player 1 has done, the game is the same as a simultaneous-move game. Indeed, the normal form of the game above coincides with a game in which the two players choose strategies simultaneously. This is shown below. Now the Nash equilibrium is simply (a2 , b1 ). a1 Player 1 a2 62 Player 2 b1 b2 3, 1 1, 0 4, 1 0, 1 4.5. Incredible threats in Nash equilibria and subgame perfection 4.5 Incredible threats in Nash equilibria and subgame perfection Let us now consider whether Nash equilibrium is a satisfactory solution concept for extensive-form games. As you will see, when players move sequentially, Nash equilibrium allows for some strategies by later movers that seem like threats which are incredible. Parents often try to control unruly children by saying things like ‘sit quietly, or we will never let you . . . (insert favourite activity)’, even though they have no intention of carrying out the threat. Children sometimes believe their parents and respond to the threat, but they are often clever enough to see through the ruse and ignore incredible threats. As we will see, Nash equilibria often depend precisely on such incredible threats by later movers. In Nash equilibrium, a player is just supposed to take a best response to the other players’ strategy choices – so the way Nash equilibrium is constructed does not allow the player to ignore certain strategy choices of others as incredible. Once we look at some examples of the problem, we will try to see whether we can refine the set of Nash equilibria to eliminate the possibility of such threats (i.e. come up with extra conditions that an equilibrium must satisfy so that equilibria which depend on incredible threats will not satisfy these extra conditions). Consider the game shown in Figure 4.4. Firm E, where E stands for entrant, is deciding whether to enter a market. The market has an incumbent firm (Firm I). If the entrant enters, the incumbent firm must decide whether to fight (start a price war, say) or accommodate the entrant. The sequence of actions and payoffs is as follows. Figure 4.4: A game with an entrant and an incumbent. The normal form is as follows. Firm E In Out Firm I A F 2, 1 −1, −2 0, 2 0, 2 Note that there are two pure strategy Nash equilibria: (In, A) and (Out, F). The latter equilibrium involves an incredible threat. Clearly, if the entrant does decide to enter the 63 4. Game theory market, the incumbent has no incentive to choose F. Hence the threat of F is incredible. Yet, Nash equilibrium cannot preclude this possibility. Out is the best response to F, and once Firm E decides to stay out, anything (and in particular F) is a best response for Firm I. The game in Figure 4.5 presents another example. Figure 4.5: An extensive-form game. Activity 4.2 Consider the extensive-form game in Figure 4.5. (a) Write down the strategies available to each player. (b) Write down the normal form of the game. (c) Identify the pure strategy Nash equilibria. When you write the normal form and work out the Nash equilibria, you should see that (R, rr) is a Nash equilibrium. Look at the game above and see that this involves an incredible threat. Player 2’s strategy involves playing r after L. Player 1’s strategy is taking a best response given this, and so player 1 is playing R. Given that player 1 is playing R, the threat is indeed a best response for player 2 (indeed, given that player 1 plays R, anything that player 2 can choose after L is trivially a best response since it does not change the payoff, but, of course, not every choice would lead to an equilibrium in which player 1’s best response is R). 4.5.1 Subgame perfection: refinement of Nash equilibrium Let us now describe a solution concept that imposes extra conditions (i.e. further to the requirement that strategies be mutual best responses) for equilibrium and leads to a refinement of the set of Nash equilibria. Several such refinements have been proposed by game theorists. Here, we will look at only one such refinement, namely subgame perfection. To understand the refinement, you first need to understand the idea of a subgame. 64 4.5. Incredible threats in Nash equilibria and subgame perfection Subgame A subgame is a part of a game that starts from a node which is a singleton (i.e. a subgame does not start at an information set), and includes all successors of that node. If one node in an information set belongs to a subgame, so do all other nodes in that information set. In other words, you cannot cut an information set so that only part of it belongs to a subgame. That would clearly alter the information structure of the game, which is not allowed. We can now define a subgame perfect equilibrium. Subgame perfect Nash equilibrium A strategy combination is a subgame perfect Nash equilibrium (SPNE) if: it is a Nash equilibrium of the whole game it induces a Nash equilibrium in every subgame. It should be clear from the definition that the set of subgame perfect equilibria is a refinement of the set of Nash equilibria. 4.5.2 Perfect information: backward induction How do we find subgame perfect equilibria? In perfect information games (recall that, in a game of perfect information, each player knows all past moves of other players), this is easy. Subgame perfect Nash equilibria can be derived simply by solving backwards, i.e. by using backward induction. Solving backwards in the entry game, we see that Firm I would choose A if Firm E chose In. Knowing this, Firm E would compare 0 (from Out) with 2 (from In and A), and choose In. Therefore, the SPNE is (In, A). Let us see that this equilibrium derived using backward induction fits with the definition of SPNE given above. The game has a subgame starting at the node after Firm E plays In (also, the whole game is always trivially a subgame). In the subgame after E plays In, there is only one player (Firm I), and the Nash equilibrium in this subgame is simply the optimal action of Firm I, which in this case is to choose A. Therefore the Nash equilibrium in the subgame is A. It follows that any Nash equilibrium of the whole game that involves playing A in the subgame is a subgame perfect Nash equilibrium. Here, the only Nash equilibrium of the whole game that satisfies this property is (In, A). Therefore, this is the only SPNE. Next, consider the game in which player 1 chooses between L, R and player 2 moves second and chooses between `, r. In this game there are two strict subgames, one starting after each action of player 1. In the left subgame, player 2’s optimal choice is `, and in the right subgame player 2’s optimal choice is r. Given this, player 1 would compare 3 from R and 0 from L, and choose R. The choices obtained by backward induction are shown in Figure 4.6. It follows that the SPNE is (R, `r). Note that it is not sufficient to write (R, `) – the equilibrium specification is meaningless unless you specify the full strategy for player 2. 65 4. Game theory Figure 4.6: Finding subgame perfect Nash equilibria. What player 2 plays at the unreached node is crucial. If player 2 played ` after L, R would not be the optimal choice. Therefore, you must specify player 2’s full strategy and identify (R, `r) as the SPNE. In these games, the SPNE is unique, but it need not be. The next activity presents an example. Activity 4.3 Derive the pure strategy subgame perfect Nash equilibria of the game in Figure 4.2. 4.5.3 Subgame perfection under imperfect information Backward induction need not work under imperfect information: you cannot fold backwards when you come up against an information set. Indeed, this is why the concept of a subgame perfect Nash equilibrium is more general compared to backward induction. If we always had perfect information, we could simply define backward induction equilibria. However, we present below an example to show you that subgame perfection is more general than backward induction, and works in many games in which backward induction does not give us any result. Before we present the example referred to above, consider the imperfect information game introduced in Section 4.4.3. Note that this game does not have any strict subgames (recall that you cannot start a subgame from an information set or cut an information set), so the only subgame is the whole game. Therefore, any Nash equilibrium of the whole game is trivially subgame perfect. As discussed above, the pure strategy Nash equilibrium in this game is (a2 , b1 ). This is also the subgame perfect Nash equilibrium. Next, consider the game in Figure 4.7. Initially player 1 decides whether to come in (and play some game with player 2) or stay out (in which case player 2 plays no role). If the choice is to come in, player 1 66 4.5. Incredible threats in Nash equilibria and subgame perfection Figure 4.7: A game tree. decides between A and B, and player 2 decides between C and D. When player 2 makes his decision, he knows that player 1 has decided to come in (if not then player 2 would not have been asked to play), but without knowing what player 1 has chosen between A and B. In other words, the situation is just as if once player 1 comes in, player 1 and player 2 play a simultaneous-move game (as the game structure shows, they do not move simultaneously – player 1 moves before player 2, but since player 2 has no knowledge of player 1’s move, it is similar to the decision problem faced in a simultaneous move game). 1. Pure strategy Nash equilibria. Let us first identify the pure strategy Nash equilibria. Note that player 1 has 4 strategies: Out A, Out B, In A and In B, while player 2 has 2 strategies: C and D. You might think strategies like Out A do not make sense, but in game theory we try to model the thought process of players, and even if player 1 stays out, she would do so only after thinking about what she would have done had she entered. Strategies reflect such thinking (it is as if player 1 is saying ‘I have decided to finally stay out, but had I come in I would have chosen A’). Let us now write down the normal form of the game. Player 1 Out A Out B In A In B Player 2 C D 1, 3 1, 3 1, 3 1, 3 −2, −2 2, 0 0, 2 −5, −5 You should be able to see from this that the pure strategy Nash equilibria are (Out A, C), (Out B, C) and (In A, D). 67 4. Game theory 2. Pure strategy subgame perfect Nash equilibria. Note that backward induction does not work here: we cannot fold back given the information set of player 2. However, subgame perfection still works. Let us see how applying subgame perfection can refine the set of Nash equilibria. Note that apart from the whole game, there is just one strict subgame, which starts at the node after player 1 chooses In. Below we write down the normal form of the subgame. Player 1 A B Player 2 C D −2, −2 2, 0 0, 2 −5, −5 As you can see, the subgame has two pure strategy Nash equilibria: (B, C) and (A, D). If (B, C) is played in the subgame, player 1 compares 1 (from Out) with 0 (from In followed by (B, C)) and decides to stay out. Therefore, a SPNE of the whole game is (Out B, C). If, on the other hand, (A, D) is played in the subgame, player 1 compares 1 with 2 and decides to come in. Therefore, another SPNE of the whole game is (In A, D). It follows that the pure strategy SPNE of the whole game are (Out B, C) and (In A, D). Another way to derive these is as follows. Since the set of SPNE is a subset of the set of Nash equilibria, and since (B, C) and (A, D) are the Nash equilibria in the subgame, it must be that any Nash equilibria of the whole game that involves playing either (B, C) and (A, D) in the subgame are subgame perfect. Considering the set of Nash equilibria derived above, we can immediately infer that (Out A, C) is Nash but not subgame perfect, while the other two are subgame perfect. 4.6 Repeated Prisoners’ Dilemma Consider the following Prisoners’ Dilemma game: Player 1 C D Player 2 C D 2, 2 0, 3 3, 0 1, 1 In a one-shot game, rational players simply play their dominant strategies. So (D, D) is the only possible equilibrium. Suppose the game is repeated. Can we say something about the behaviour of players in such ‘supergames’ that differs from the behaviour in the one-shot game? First, consider the case of a finite number of repetitions. Say the game is played twice. Would anything change? The answer is no. In the second round, players simply face a one-shot game and they would definitely play their dominant strategies. Given that (D, D) will be played in the next period, playing anything other than D today makes no sense. Therefore, in each period players would play (D, D). But this logic extends to any 68 4.6. Repeated Prisoners’ Dilemma finite number of repetitions. If the game is played a 100 times, in the last period (D, D) will be played. This implies that (D, D) will be played in the 99th period, and so on. While the logic is inescapable, actual behaviour in laboratory settings differs from this. Faced with a large finite number of repetitions, players do cooperate for a while at least. Therefore, it is our modelling that is at fault. To escape from the logic of backward induction, we can assume that when a game is repeated many times, players play them as if the games are infinitely repeated. In that case, we must apply forward-looking logic as there is no last period from which to fold backwards. (A quick note: you should be aware that there are other games with multiple Nash equilibria where some cooperation can be sustained even under a finite number of repetitions. You will encounter these in more advanced courses. Here we only consider the repeated Prisoners’ Dilemma.) Let us now analyse an infinitely repeated Prisoners’ Dilemma game. Payoffs: discounted present value First, we need to have an appropriate notion of payoffs in the infinitely repeated game. Each player plays an action (in this case either C or D) in each period. So in each period, the players end up playing one of the four possible action profiles (C, C), (C, D), (D, C) or (D, D). Let at denote the action profile played in period t. Then in period t, player i receives the payoff ui (at ). The payoff of player i in the repeated game is simply the discounted present value of the stream of payoffs. Let δ denote the common discount factor across players, where 0 < δ < 1. If today’s date is 0, and a player receives x in period t, the present value of that payoff is δ t x. The discount factor can reflect players’ time preference. This can also arise from a simple rate of interest calculation, in which case δ can be interpreted as 1/(1 + r), where r is the rate of interest. Note that higher values of δ indicate that players are more patient (i.e. value future payoffs more). If δ is very low, the situation is almost like a one-shot game, since players only value today’s payoff, and place very little value on any future payoff. Given such discounting, the payoff of player i in the repeated game is: ui (a0 ) + δui (a1 ) + δ 2 ui (a2 ) + · · · . More concisely, the payoff is: ∞ X δ t ui (at ). t=0 If the payoff is the same every period (x, say), this becomes: x(1 + δ + δ 2 + · · · ) = 4.6.1 x . 1−δ Cooperation through trigger strategies Next, we need to consider strategies by players. The history at t is the action profile played in every period from period 0 to t ≥ 1. A strategy of a player consists of an initial action, and after that, an action after every history. Consider the following trigger strategy. 69 4. Game theory Trigger strategy Start by playing C (that is, cooperate at the very first period, when there is no history yet). In period t > 1: • if (C, C) was played last period, play C • if anything else was played last period, play D. Suppose each player follows this strategy. Note that cooperation (playing (C, C)) would work only until someone deviates to D. After the very first deviation, each player switches to D. Since anything other than (C, C) implies playing (D, D) next period, once a switch to (D, D) has been made, there is no way back: the players must play (D, D) forever afterwards. This is why this is a trigger strategy. Another way of stating the trigger strategy is to write in terms of strategy profiles. Start by playing (C, C). In period t: • if (C, C) is played in t − 1, play (C, C) • otherwise play (D, D). Let us see if this will sustain cooperation. Suppose a player deviates in period t. We only need to consider what happens from t onwards. The payoff starting at period t is given by: δ . 3 + δ + δ2 + · · · = 3 + 1−δ If the player did not deviate in period t, the payoff from t onwards would be: 2 + 2δ + 2δ 2 + · · · = 2 . 1−δ For deviation to be suboptimal, we need: δ 2 >3+ 1−δ 1−δ which implies: 1 δ> . 2 Thus if the players are patient enough, cooperation can be sustained in equilibrium. In other words, playing (C, C) always can be the outcome of an equilibrium if the discount factor δ is at least 1/2. 4.6.2 Folk theorem We showed above that the cooperative payoff (2, 2) can be sustained in equilibrium. However, this is not the only possible equilibrium outcome. Indeed, many different payoffs can be sustained in equilibrium. 70 4.6. Repeated Prisoners’ Dilemma For example, note that always playing (D, D) is an equilibrium no matter what the value of δ is. Each player simply adopts the strategy ‘play D initially, and at any period t > 1, play D irrespective of history’. Note that both players adopting this strategy is a mutual best response. Therefore, we can sustain (1, 1) in equilibrium. In fact, by using suitable strategies, we can sustain many more – in fact infinitely more – payoffs as equilibrium outcomes. For the Prisoners’ Dilemma game, we will describe the set of sustainable payoffs below. The result about the large set of payoffs that can be sustained as equilibrium outcomes is known as the ‘folk theorem’. These types of results were known to many game theorists from an early stage of development of non-cooperative game theory. While formal proofs were written down later, we cannot really trace the source of the idea, which explains the name. Here we state a folk theorem adapted to the repeated Prisoners’ Dilemma game. To state this, we will need to compare payoffs in the repeated game to payoffs in the one-shot game that is being repeated. The easiest way to do that is to normalise the repeated game payoff by multiplying by (1 − δ). Then if a player gets 2 every period, the repeated game payoff is (1 − δ) × 2/(1 − δ) = 2. As you can see, this normalisation implies that the set of normalised repeated game payoffs now coincide with the set of payoffs in the underlying one-shot game. So now we can just look at the set of payoffs of the one-shot game and ask which of these are sustainable as the normalised payoff in some equilibrium of the repeated game. For the rest of this section, whenever we mention a repeated game payoff, we always refer to normalised payoffs. Note that in this game, a player can always get at least 1 by simply playing D. It follows that a player must get at least 1 as a (normalised) repeated game payoff. To see the whole set of payoffs that can be sustained, let us plot the payoffs from the four different pure strategy profiles. These are shown in Figure 4.8 below. Now join the payoffs and form a convex set as shown in the figure. We now have a set of payoffs that can arise from pure or mixed strategies. The Folk theorem is the following claim. Consider any pair of payoffs (π1 , π2 ) such that πi > 1 for i = 1, 2. Any such payoff can be supported as an equilibrium payoff for high enough δ. As noted above, for the Prisoners’ Dilemma game we can also sustain the payoff (1, 1) as an equilibrium outcome irrespective of the value of δ. The set of payoffs that can be supported as equilibrium payoffs in our example is shown as the shaded part in Figure 4.8. Example 4.2 Consider the following game. Player 1 C D Player 2 C D 3, 2 0, 1 7, 0 2, 1 (a) Find conditions on the discount factor under which cooperation can be sustained in the repeated game in which the above game is repeated infinitely. 71 4. Game theory Figure 4.8: The set of payoffs that can be supported as equilibrium payoffs under an infinitely-repeated game. (b) Under what conditions is there an equilibrium in the infinitely-repeated game in which players alternate between (C, C) and (D, D), starting with (C, C) in the first period? (c) Draw the set of payoffs sustainable in a repeated game equilibrium according to the folk theorem. (a) First, note that player 2 has no incentive to deviate from (C, C). To ensure that player 1 does not deviate, consider the following strategy profile. Play (C, C) initially. At any period t > 0, if (C, C) has been played in the last period, play (C, C). Otherwise, switch to (D, D). Under this strategy profile, player 1 will not deviate if: 7+2× δ 3 6 1−δ 1−δ which implies δ > 4/5. (b) We want to support alternating between (C, C) and (D, D), starting with (C, C) in period 0, as an equilibrium. Note first that player 2 has no incentive to deviate in odd or even periods. Player 1 cannot gain by a one-shot deviation in odd periods, when (D, D) is supposed to be played. So the only possible deviation is by player 1 in even periods, when (C, C) is supposed to be played. To prevent such a deviation, consider the following strategy profile. • Start by playing (C, C) in period 0. • In any odd period t (where t = 1, 3, 5, . . .) play (D, D) (irrespective of history). 72 4.6. Repeated Prisoners’ Dilemma • In any even period t (where t = 2, 4, 6, . . .): ◦ play (C, C) if (C, C) has been played in the previous even period t − 2 ◦ otherwise play (D, D). Note that this is a version of the trigger strategy. After any deviation from cooperation in even periods, (D, D) is triggered forever. If player 1 does not deviate in any even period t, player 1’s payoff (t onwards) is: Vt = 3 + 2δ + 3δ 2 + 2δ 3 + · · · = 3(1 + δ 2 + δ 4 + · · · ) + 2δ(1 + δ 2 + δ 4 + · · · ) = 3 + 2δ . 1 − δ2 If player 1 does deviate in even period t, player 1’s payoff (t onwards) is: Vtdev = 7 + 2δ + 2δ 2 + · · · = 7 + 2 × δ . 1−δ Therefore, player 1 prefers not to deviate in either period 0 or in any even period if: δ 3 + 2δ > 7 + 2 × 1 − δ2 1−δ which simplifies to 5δ 2 − 4 > 0, implying: r 4 ≈ 0.89. δ> 5 Note also the repeated game payoff generated in this equilibrium. To see this, first normalise the payoff by multiplying by (1 − δ). The normalised payoff of player 1 starting any even period is: (1 − δ) × 3 + 2δ 3 + 2δ = . 2 1−δ 1+δ Similarly, the normalised payoff of player 1 starting any odd period is (2 + 3δ)/(1 + δ). Note that either payoff goes to 5/2 as δ → 1. For player 2, the normalised payoff from the equilibrium is (2 + δ)/(1 + δ) starting any even period, and (1 + 2δ)/(1 + δ) starting any odd period. Note that either payoff goes to 3/2 as δ → 1. In other words, this exercise shows you an example of an equilibrium that sustains a payoff in the interior of the set of payoffs which is sustainable according to the folk theorem (which, in this case, is anything strictly above (2, 1), or the point (2, 1) itself). 73 4. Game theory 4.7 A reminder of your learning outcomes Having completed this chapter, the Essential reading and activities, you should be able to: analyse simultaneous-move games using dominant strategies or by eliminating dominated strategies either once or in an iterative fashion calculate Nash equilibria in pure strategies as well as Nash equilibria in mixed strategies in simultaneous-move games explain why Nash equilibrium is the central solution concept and explain the importance of proving existence specify strategies in extensive-form games analyse Nash equilibria in extensive-form games explain the idea of refining Nash equilibria in extensive-form games using backward induction and subgame perfection analyse the infinitely-repeated Prisoners’ Dilemma game with discounting and analyse collusive equilibria using trigger strategies explain the multiplicity of equilibria in repeated games and state the folk theorem for the Prisoners’ Dilemma game. 4.8 4.8.1 Test your knowledge and understanding Sample examination questions 1. Consider the strategic form game below with two players, 1 and 2. Solve the game by iteratively eliminating dominated strategies. A1 Player 1 B1 C1 Player 2 A2 B2 C2 3, 3 −1, 4 0, 5 2, 1 3, 2 −1, 0 −1, 0 0, 1 1, 0 2. Identify actions and strategies of each player in each of the following games (Figure 4.9 to Figure 4.11). 74 4.8. Test your knowledge and understanding Figure 4.9: Perfect information. Figure 4.10: Imperfect information for player 2. Figure 4.11: Imperfect information for both players. 75 4. Game theory 3. Consider the extensive-form game of imperfect information in Figure 4.12. Figure 4.12: An extensive-form game. (a) Write down the actions and strategies for each player. (b) Identify the pure and mixed strategy Nash equilibria. (c) Identify the pure and mixed strategy subgame perfect Nash equilibria. 4. Consider the extensive-form game in Figure 4.13. Figure 4.13: An extensive-form game. (a) Write down the actions and strategies for each player. (b) Identify the pure strategy Nash equilibria. (c) Identify the pure strategy subgame perfect Nash equilibria. 76 4.8. Test your knowledge and understanding 5. Consider the extensive-form game in Figure 4.14. Figure 4.14: An extensive-form game. (a) Write down the actions and strategies for each player. (b) Identify the pure strategy Nash equilibria. (c) Identify the pure strategy subgame perfect Nash equilibria. 6. Find the pure strategy subgame perfect Nash equilibria for the game in Figure 4.9 in Question 2. 7. Suppose the following game is repeated infinitely. The players have a common discount factor δ, where 0 < δ < 1. Player 1 C D Player 2 C D 3, 2 0, 3 5, 0 2, 1 (a) Find conditions on the discount factor under which cooperation (which implies playing (C, C) in each period) can be sustained as a subgame perfect Nash equilibrium of the infinitely-repeated game. Your answer must specify the strategies of players clearly. (b) In a diagram, show the set of payoffs that can be supported in a repeated game equilibrium according to the folk theorem. 77 4. Game theory 78
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