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MicroeconomicsA. DaripaEC2066

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. If you have any comments on this subject guide, favourable or unfavourable, please use the form at the back of this guide.

University of London International Programmes Publications Office Stewart House 32 Russell Square London WC1B 5DN United Kingdom www.londoninternational.ac.uk

Published by: University of London

© University of London 2016

The University of London asserts copyright over all material in this subject guide except where otherwise indicated. All rights reserved. No part of this work may be reproduced in any form, or by any means, without permission in writing from the publisher. We make every effort to respect copyright. If you think we have inadvertently used your copyright material, please let us know.

Contents

Contents

1 Introduction 1

1.1 Routemap to the subject guide . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to the subject area and prior knowledge . . . . . . . . . . . 2

1.3 Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Aims of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Learning outcomes for the course . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Overview of learning resources . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6.1 The subject guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6.2 Essential reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6.4 Online study resources . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.5 The VLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.6 Making use of the Online Library . . . . . . . . . . . . . . . . . . 7

1.7 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

2 Consumer theory 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Aims of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 11


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

2.4 Demand curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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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 Sample examination questions . . . . . . . . . . . . . . . . . . . . 35

3 Choice under uncertainty 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37





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




4 Game theory 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49




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4.1.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

4.4 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

4.5 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

4.6 Repeated Prisoners’ Dilemma . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6.1 Cooperation through trigger strategies . . . . . . . . . . . . . . . 69

4.6.2 Folk theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70




5 Production, costs and profit maximisation 79

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79





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 Isoquants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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5.6.2 Diminishing MRTS . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6.3 Returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.6.4 Optimal long-run input choice . . . . . . . . . . . . . . . . . . . . 84

5.7 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




6 Perfect competition in a single market 93

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93





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

6.5 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


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7 General equilibrium and welfare 111

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111





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




8 Monopoly 129

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129





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




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9 Oligopoly 137

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137





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

9.4 Bertrand competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.4.1 Collusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.5 Bertrand competition with product differentiation . . . . . . . . . . . . . 143

9.5.1 Sequential pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 144




10 Asymmetric information: adverse selection 149

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149





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




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11 Asymmetric information: moral hazard 161

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161





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




12 Externalities and public goods 173

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173




12.1.4 References cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174


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


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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 andlearning 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. Wespecify properties of preferences, how to go from preferences to utility and optimalchoice by maximising utility under a budget constraint. We show how to obtain demandfunctions from optimal solutions to the consumer choice problem. We discuss whathappens to the consumer’s welfare when prices and/or income change, and appropriatemeasures to evaluate these changes. We also consider the labour supply decision ofindividuals, and the intertemporal choice problem faced by savers and borrowers. Weprovide a brief exposition of some basic algebra of intertemporal choice problems withmore than two periods. In Chapter 3, we model the agent’s behaviour in situationsinvolving risk, and analyse insurance problems. We then introduce strategic interactionin Chapter 4 and provide an exposition of the basic tools of game theory. Next, we turnto 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 thesubject guide is essentially a toolbox for analysis in subsequent chapters. Using thesetools, we analyse the problem of competitive firms as well as the equilibrium in acompetitive market in Chapter 6. In Chapter 7, we then introduce the generalequilibrium across all competitive markets and study the two welfare theorems that arefundamental to our understanding of market economies. We also consider someapplications 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 analysethe problem of monopoly, the associated inefficiencies and policy prescriptions aimed atrestoring efficiency. This is followed by an analysis of the problem of oligopolisticcompetition in Chapter 9. Next, we turn to the problem of asymmetric information andanalyse the problems of adverse selection (in Chapter 10) and moral hazard (in Chapter11). Finally, in Chapter 12, we consider the problem of externalities and public goodsand consider policy prescriptions arising from associated market failures.

Given the emphasis of the EC2066 Microeconomics syllabus on using analyticalmethods to solve economic problems, you are encouraged to spend a considerableamount of time doing the problems or questions given in the textbooks. Learning bydoing 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 recommendedtextbook and the subject guide is a prerequisite for successful problem solving. Thesubject guide aims to indicate as clearly as possible the key elements in microeconomicanalysis that you need to learn. The subject guide also presents detailed algebraicderivations for a variety of topics. For each topic, you should consult both the subjectguide and the suggested parts of the textbook to understand fully the economicprinciples involved.

1.2 Introduction to the subject area and priorknowledge

The syllabus and subject guide assume that you are competent in basic economicanalysis up to the level of the prerequisite courses EC1002 Introduction toeconomics, ST104a Statistics 1, MT105a Mathematics 1 or MT1174 Calculus.They build on the foundations provided in these courses by specifying how yourunderstanding of the microeconomic principles developed so far should be deepened andextended. Like EC1002 Introduction to economics, EC2066 Microeconomics isdesigned to equip you with the economic principles necessary to analyse a whole rangeof economic problems. To maximise your benefit from the subject, you should continueto 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, youare not expected to know the detailed facts about specific economic issues and policiesmentioned in textbook examples. Rather, you should use these examples (and theend-of-chapter Sample examination questions) to aid your understanding of howeconomic 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 ST104aStatistics 1 and MT105a Mathematics 1 or MT1174 Calculus before beginningthis course. Every part of the syllabus can be mastered with the aid of diagrams andrelatively simple algebra. The subject guide indicates the minimum level ofmathematical knowledge that is required. Knowledge (and use in the examination) ofsophisticated mathematical techniques is not required. However, if you aremathematically competent you are encouraged to use mathematical techniques whenthese are appropriate, as long as you recognise that some verbal explanation is alwaysnecessary.

1.3 Syllabus

The course examines how economic decisions are made by households and firms, andhow they interact to determine the quantities and prices of goods and factors ofproduction 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 therole 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 andsubgame 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, resultingmarket failures and the role of contracts and institutions.

Market failures arising from monopoly, externalities and public goods. The role ofpolicy.

1.4 Aims of the course

This subject guide enables you to fully interpret the published syllabus for EC2066Microeconomics. It identifies what you are expected to know within each area of thesyllabus by emphasising the relevant concepts and models and by stating where inspecific textbooks that material can be found. This subject guide aims to help you makethe best use of textbooks to secure a firm understanding of the microeconomic analysiscovered by the syllabus. The subject guide also complements your textbook in certainareas 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 andchoice 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

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1. Introduction

• the nature of economic interaction under asymmetric information

be able to analyse and assess:

• efficiency and welfare optimality of perfectly and imperfectly competitivemarkets

• 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 developingthe ability of linking the concepts introduced in different chapters, in order to approachthe examination well.

1.6 Overview of learning resources

1.6.1 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 throughoutthe chapter.

The sections that follow specify in detail what you are expected to know abouteach topic. The relevant sections of the recommended textbooks are referred to.Wherever necessary, the sections integrate the textbook with additional materialand explanations. Finally, they draw attention to any problems that occur intextbook expositions and explain how these can be overcome.

The boxes that appear in some of the sections give you exercises based on thematerial discussed.

The learning outcomes show you what you should be able to do by the end of thechapter.

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.’

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1.6. Overview of learning resources

This is available as an e-book at a discounted price via the VLE. Please visitthe course page for details.

Students may use the previous edition instead:

Nicholson, W. and C. Snyder Theory and Applications of IntermediateMicroeconomics. (Cengage Learning, 2010) 11th edition, international edition[ISBN 9780324599497].

Note that the title of the twelfth edition differs slightly from that of the eleventh. Ifusing this edition, students should refer to the reading supplement on the VLE forcustomised references.

N&S is more adequate for some parts of the syllabus while less so for others; as aconsequence we integrate some topics more with the extra material provided in thesubject guide. The textbook employs verbal reasoning as the main method ofpresentation, supplemented by diagrammatic analyses. The textbook’s use of algebra isnot uniformly satisfactory. The subject guide supplements the textbook in many casesin 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 textbookslisted above. New editions of these textbooks may have been published by the time youstudy this course. You can use a more recent edition of any of the textbooks; use thedetailed chapter and section headings and the index to identify relevant readings. Also,check the virtual learning environment (VLE) regularly for updated guidance onreadings.

Unless otherwise stated, all websites in this subject guide were accessed in February2016. We cannot guarantee, however, that they will stay current and you may need toperform 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 readaround the subject area in any textbook, paper or online resource. You will need tosupport your learning by reading as widely as possible and by thinking about how theseprinciples apply in the real world. To help you read extensively, you have free access tothe virtual learning environment (VLE) and University of London Online Library (seebelow).

Other useful textbooks for this course include:

Besanko, D. and R. Braeutigam Microeconomics. (John Wiley & Sons, 2014) 5thedition, 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 takeadvantage of the study resources that are available online for this course, including theVLE and the Online Library.

You can access the VLE, the Online Library and your University of London emailaccount via the Student Portal at:http://my.londoninternational.ac.uk

You should have received your login details for the Student Portal with your officialoffer, which was emailed to the address that you gave on your application form. Youhave probably already logged in to the Student Portal in order to register! As soon asyou registered, you will automatically have been granted access to the VLE, OnlineLibrary 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 yourlearning experience, providing additional support and a sense of community. It forms animportant part of your study experience with the University of London and you shouldaccess it regularly.

The VLE provides a range of resources for EMFSS courses:

Electronic study materials: All of the printed materials which you receive fromthe University of London are available to download, to give you flexibility in howand where you study.

Discussion forums: An open space for you to discuss interests and seek supportfrom your peers, working collaboratively to solve problems and discuss subjectmaterial. Some forums are moderated by an LSE academic.

Videos: Recorded academic introductions to many subjects; interviews anddebates with academics who have designed the courses and teach similar ones atLSE.

Recorded lectures: For a few subjects, where appropriate, various teachingsessions 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 latercourses such as Introduction to Economics, Statistics, Mathematics and Principlesof Banking and Accounting, audio-visual tutorials are available to help you workthrough key concepts and to show the standard expected in examinations.

Self-testing activities: Allowing you to test your own understanding of subjectmaterial.

Study skills: Expert advice on getting started with your studies, preparing forexaminations and developing your digital literacy skills.

6

http://my.londoninternational.ac.uk

1.7. Examination advice

Note: Students registered for Laws courses also receive access to the dedicated LawsVLE.

Some of these resources are available for certain courses only, but we are expanding ourprovision 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 tounderstand 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 ofjournal 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 useyour University of London Student Portal login details, or you will be required toregister and use an Athens login.

The easiest way to locate relevant content and journal articles in the Online Library isto use the Summon search engine.

If you are having trouble finding an article listed in a reading list, try removing anypunctuation 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 OnlineLibrary team: [email protected]

1.7 Examination advice

1.7.1 Format of the examination

Important: the information and advice given here are based on the examinationstructure used at the time this subject guide was written. Please note that subjectguides may be used for several years. Because of this we strongly advise you to alwayscheck both the current Regulations for relevant information about the examination, andthe VLE where you should be advised of any forthcoming changes. You should alsocarefully check the rubric/instructions on the paper you actually sit and follow thoseinstructions.

In this examination you should answer eleven of fourteen questions: all eight questionsfrom 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 appearnot 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 whichare not necessarily specific to one particular topic. For example, a question may requireknowledge about markets which are oligopolistic as well as those which are monopolisticor competitive.

7

http://onlinelibrary.london.ac.uk

http://onlinelibrary.london.ac.uk/resources/summon

1. Introduction

Numerical questions will sometimes require the use of a calculator. A calculator may beused when answering questions on the examination paper for this course and it mustcomply 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 institutionalmaterial 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 annualExaminers’ commentaries. For this course, the following advice is also worth noting:

Prepare thoroughly for the examination by attempting the problems/questions inthe 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 someelement of doubt about the interpretation of a question, state at the beginning ofyour answer how you interpret the question. If you are very uncertain of what isrequired, 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 ornumbers 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 forcorrect 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 notknow the answer, at least define any terms or concepts which you think may berelevant (including those in the question!) and, if possible, present the question indiagrammatic or algebraic form. The same applies to a specific part of a multi-partquestion in Section B. The examiners can give no marks for an unattemptedquestion, but they can award marks for relevant points. A single mark may makethe difference between passing and failing the examination.

Although you should attempt all the questions and parts of questions that arerequired, to avoid wasting time you should make sure that you do no more than isrequired. For example, if only two parts of a three-part question need to beanswered, 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 whythe statement is ‘false’. Notice that this is simply a way in which you can startapproaching the problem, but there is no way to know in advance the correctanswer without analysing every specific question.

It is worth noting that in this type of question, simply writing ‘true’ or ‘false’ willnot earn you any marks, even if it happens to be the right answer. The examinersare looking for reasoning, not blind guesses.

8

1.7. Examination advice

Good answers to most questions require relevant assumptions to be stated andterms to be defined. Also, do use the term ceteris paribus (meaning ‘other thingsbeing equal’), where appropriate. If you are asked to examine the effects of achange in a particular exogenous variable, you should not complicate your answersunnecessarily by positing simultaneous changes in other exogenous variables.

For many questions, good answers will require diagrammatic and/or algebraicanalysis to complement verbal reasoning. Good diagrams can often save muchexplanation but free-standing diagrams, however well-drawn and labelled, do notportray sufficient information to the examiners. Diagrams need to be explained inthe text of the answer. Similarly, symbols in algebraic expressions should be definedand 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 appropriatehypothesis, 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 thecourse 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 thisprocess directly, but can we come up with a model to capture the decision-makingprocess so that the predictions from the model match the way people behave? If we canbuild a sensible model, we should be able to use the model to understand how choicevaries when the economic environment changes. This should also help us designappropriate policy. This is the task in this chapter – and as we go through the varioussteps, you should keep this overarching goal in your mind and try to see how each pieceof analysis fits in the overall scheme.

Once we build such a model, we use it to analyse how optimal consumption choiceresponds to price and income variations. We also extend the analysis to coverlabour-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 familiarwith many of the ideas here from EC1002 Introduction to economics, but we aimto investigate certain aspects at relatively greater depth. The chapter also aims toencourage you to ask questions about the meaning of concepts and their usefulness inunderstanding 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? Whyis this concept useful? You should not accept such concepts just because they appear intextbooks 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 aconsumer 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 consumersurplus, equivalent variation and compensating variation, and use these measuresto 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.


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 economicmodels and some basic techniques. You should be familiar with this material fromearlier courses. Nevertheless, you should read this chapter and the appendix and makesure that you understand the content. Throughout this subject guide, we will assumethat 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 constructdemand 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 savingand borrowing with two periods. Finally, we learn to carry out present valuecalculations 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 theconnection between preferences and utility.

2.3.1 Preferences and utility

The theory of choice starts with rational preferences. Generally, preferences areprimitives in economics – you take these as given and proceed from there. The task ofexplaining why certain preferences exist in certain societies falls largely under thedomain 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 onpreferences to rule out irrational behaviour. Just a few relatively sensible restrictionsallow us to build a model of choice that has great analytical power. Indeed, this idea ofcreating an analytical structure that can be manipulated to understand how changes inthe economic environment affect economic outcomes underlies the success of economicsas a tool for investigating the world around us.

Section 2.2 of N&S sets out three restrictions on preferences: completeness, transitivityand non-satiation (‘more is better’). These restrictions allow us to do something veryuseful. Once we add some further technical requirements (a full specification must awaitMasters level courses but the main extra condition we need is that preferences havecertain continuity properties), these restrictions allow us to represent preferences by acontinuous function. This is known as a utility function.

Note that a utility function is an artificial concept – no-one actually has a utilityfunction (you knew that of course, since you surely do not have one). But because wecan 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 absolutemeasure of satisfaction derived from bundles. You might prefer apple juice to orangejuice but would have difficulty saying exactly how much more satisfaction you derivefrom the former compared to the latter. Preferences, therefore, typically give us an‘ordinal’ ranking among bundles of goods. Since utility is simply a representation ofpreferences, it is also an ordinal measure. This means that if your preferences can berepresented by a utility function, then a positive transformation of this function whichpreserves the ordering among bundles is another function that is also a valid utilityfunction. In other words, there are many possible utility functions that can represent agiven set of preferences equally well.

However, there are some instances where we use cardinal utility and make absolutecomparisons among bundles. Money, for example, is a cardinal measure – you know that20 pounds is twice as good as 10 pounds. In general, though, you should understandutility as an ordinal concept.

2.3.2 Indifference curves

Once we can represent preferences using a continuous utility function, we can drawindifference curves. An indifference curve is the locus of different bundles of goods thatyield the same level of utility. In other words, an indifference curve for utility functionu(x, y) is given by u(x, y) = k, where k is some constant. As we vary k, we can trace outthe indifference map. Note that an indifference curve is simply a level curve of a utilityfunction. Just as you draw contours on a map to represent, say, a mountain, soindifference curves drawn for two goods are contour plots of a utility function over thesetwo goods. You can see Figure 3.3 in Perloff for a pictorial representation. You shouldread 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 indifferencecurves.

Activity 2.1 For each of the following utility functions, write down the equation foran 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 implyfor indifference curves? We have the following properties:

1. If an indifference curve is further from the origin compared to another indifferencecurve, any point on the former is preferred to any point on the latter (implied bythe 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 tosubstitute one good for another along an indifference curve. The marginal rate ofsubstitution of a consumer between goods x and y is the units of y the consumer iswilling 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:

dy

dx

∣∣∣∣u constant

= −MUx

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 andvery little x is being consumed. Starting from any such point, a consumer is willing togive up a lot of y in exchange for another unit of x while retaining the same level ofutility as before. As we keep adding units of x and reducing y while keeping utilityconstant (i.e. we are moving down an indifference curve), consumers are willing to giveup less and less of y in return for a further unit of x. One way to interpret this is that

14


people typically have a taste for variety and want to avoid extremes (i.e. avoidsituations 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 asindifference curves being ‘convex to the origin’ in some textbooks. This works as avisual description, but you should be aware that in terms of mathematics, this is not ameaningful description – there is no mathematical concept where something is convexrelative 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 s1and s2 are in S, then a convex combination λs1 + (1− λ)s2 is also in the set S for any0 < λ < 1.

Now consider any indifference curve yielding utility level u. Consider the set of allpoints that yield utility u or more. This is the set of all points on an indifference curveplus 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 whichis shaded) is a convex set. Try taking any two points in B and then making a convexcombination. 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 setB of points on or above the indifference curve is not convex. If you combine points suchas a1 and a2 in the diagram, for some values of λ, the convex combinations fall outsidethe set B.

Note that when two goods are perfect substitutes, you get a straight line indifferencecurve. At the other extreme, the two goods are perfect complements (nosubstitutability) and the indifference curve is L-shaped. Indifference curves withdiminishing 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 thata consumer can afford to buy. This is captured by the budget constraint. Sinceconsumers are generally taken to be price-takers (i.e. what an individual consumerpurchases does not affect the market price for any good), the budget line is a straightline. See Section 2.7 of N&S for the construction of budget sets. You should be awarethat budget lines would no longer be a straight line if a consumer buys different units atdifferent prices. This could happen if a consumer is a large buyer in a market or if theconsumer 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 aresuch that indifference curves have the usual convex shape, the best point is where anindifference 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


Figure 2.3: Consumer optimisation.

constraint. So we have:

−MUx

MUy

= −PxPy.

Multiplying both sides by −1 we can write this as the familiar condition:

MRSxy =PxPy.

Let us derive this condition formally using a Lagrange multiplier approach. This is theapproach 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 (ifnot, 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 maximisesu(x, y) subject to the budget constraint Pxx+ Pyy = M . Set up the Lagrangian:

L = u(x, y) + λ(M − Pxx+ Pyy).

The first-order conditions for a constrained maximum are:

∂L∂x

=∂u

∂x− λPx = 0

∂L∂y

=∂u

∂y− λPy = 0

∂L∂λ

= M − Pxx+ Pyy = 0.

From the first two conditions, we get:

PxPy

=∂u/∂x

∂u/∂y= MRSxy.

17

2. Consumer theory

Second-order condition

The first-order conditions above are, by themselves, not sufficient to guarantee amaximum. We also need the second-order condition to hold. It is better to derive thisformally once you have learned matrix algebra, which allows a relatively simpleexposition of the second-order condition. For our purposes here, note that thediminishing MRS condition is sufficient to guarantee that a maximum occurs at thepoint satisfying the first-order conditions. This should also be clear to you from thegraph. If indifference curves satisfy the usual convexity property, there is an interiortangency point with the budget constraint line at which the maximum utility isattained.

Figure 2.4 below demonstrates that if preferences are not convex, the first-orderconditions are not sufficient to guarantee optimality.

Figure 2.4: Violation of the second-order condition under non-convex preferences. Notethat MRS is not always diminishing. Point A satisfies the first-order condition MRS equalto price ratio, but is not optimal. (Source: Schmalensee, R. and R.N. Stavins (2013) ‘TheSO2 allowance trading system: the ironic history of a grand policy experiment,’ Journalof Economic Perspectives, Vol. 27, pp.103–21. Reproduced by kind permission of theAmerican 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 conditiondoes not apply. For perfect substitutes, there is either a corner solution or the budgetline coincides with the indifference curve. In the latter case, any point on the budgetline is optimal. For the case of perfect complements, the optimum occurs where the kinkin the indifference curve just touches the budget line. Note that this is not a tangencypoint – the slope of the indifference curve is undefined at the kink. N&S clarifies thesecases with appropriate diagrams.

18


Demand functions

The maximisation exercise above gives us the demand for goods x and y at given pricesand income. As we vary the price of good x, we can trace out the demand curve forgood x. See Section 3.6 of N&S for a discussion. The activities below compute demandfunctions in specific examples.

Example 2.1 A consumer has the following Cobb–Douglas utility function:

u(x, y) = xαyβ

where α, β > 0. The price of x is normalised to 1 and the price of y is p. Theconsumer’s income is M . Derive the demand functions for x and y.

The consumer’s problem is as follows:

maxx, y

xαyβ subject to x+ py = M.

Using the Lagrange multipliers method, we get:

MUx

MUy

=PxPy.

This implies:αxα−1yβ

βxαyβ−1=

1

p.

Simplifying:αy

βx=

1

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. Theconsumer’s income is M . Derive the demand functions for x and y.

The consumer would choose the bundle at which the highest indifference curve isreached while not exceeding the budget. This is point E in Figure 2.5 where theindifference curve just touches the budget constraint (Figure 2.5 is drawn usingα/β = 1/2). Note that this is not a tangency point as the slope of the indifferencecurve 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 budgetconstraint, 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. Theslope of the indifference curve at the kink is undefined.

2.4 Demand curves

2.4.1 The impact of income and price changes

See N&S Chapter 3. Now that we have derived demand curves, we can try tounderstand 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 ofdemand 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 isbetter’ assumption. The full argument is left as an exercise.

Activity 2.3 ‘It is not possible for all goods to be inferior.’ Provide a carefulexplanation of this statement.

The income-consumption curve

The income-consumption curve of a consumer traces out the path of optimal bundles asincome varies (keeping all prices constant). Using this exercise, we also plot therelationship between quantity demanded and income directly. The curve that shows thisrelationship is called the Engel curve. See Perloff Section 4.2 for an exposition of theincome-consumption curve and the Engel curve.

The slope of the income-consumption curve indicates the sign of the income elasticity ofdemand (explained below).

Price changes

See Sections 3.3 to 3.8 of N&S. It is very important to understand fully thedecomposition of the total price effect into income and substitution effects. Thisdecomposition is, of course, a purely artificial thought experiment. But this thoughtexperiment is extremely useful in understanding how the demand for different goodsresponds to a change in price at different levels of income and given differentopportunities to substitute out of a good. You should understand how these effects(and, therefore, the total price effect) differ across normal and inferior goods, andunderstand how the effect known as Giffen’s paradox can arise.

The idea of income and substitution effects can help us understand the design of anoptimal 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 theprice of some other good, and how this effect differs depending on whether the othergood is a substitute or a complement.

Example 2.3 Suppose u(x, y) = x1/2y1/2. Income is M = 72. The price of y is 1and the price of x changes from 9 to 4. Calculate the income effect (IE) and thesubstitution effect (SE).

Let (px, py) denote the original prices and let p′x denote the lower price of x.

Under the original prices, the Marshallian demand functions (you should be able tocalculate these) are:

x∗ =M

2px

and:

y∗ =M

2py.

21

2. Consumer theory

The optimised utility is, therefore:

u∗0 =M

2√pxpy

where the subscript of u is a reminder that this is the original utility level (beforethe price change).

The total price effect (PE) from a price fall is:

PE =M

2p′x− M

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 thatthe consumer is on the original indifference curve. In other words, we must keep theutility at u∗0. In Figure 2.6, the dashed budget line is the one after the compensatingreduction in income. The point B is the optimal point on this compensated budgetline. 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? Thiscan be calculated as follows.

Suppose ∆M is the amount of income we take away. We need ∆M to be such that:

M −∆M

2√p′xpy

= u∗0

which implies:M −∆M

2√p′xpy

=M

2√pxpy

.

Using the values supplied, ∆M = 24 so that M −∆M = 48.

Under a reduced income of 48, and given the new price p′x = 4, the demand for x is6. The original demand for x was 4. Under the compensated price change, thedemand is 6. Therefore, the SE is 2. It follows that the rest of the change must bethe IE. Since the total price effect is 5, the IE is 3.

In terms of algebra:

SE =M −∆M

2p′x− M

2px.

The IE is the remainder of the price effect, so that:

IE =

(M

2p′x− M

2px

)−(M −∆M

2p′x− M

2px

).

Simplifying:

IE =∆M

2p′x.

Looking ahead, the ∆M we calculated here is known as the ‘compensatingvariation’. 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 substitutioneffect, 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 byaggregating 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 variablessuch as prices and income. The elasticity of market demand for a good can be estimatedfrom data, and these elasticity estimates are important for firms in setting prices and forformulation 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 mustread 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 givenpercentage change in the price of the good, given by:

ε =dQ/Q

dP/P=P

Q

dQ

dP.

Note that ε < 0 since demand is typically downward-sloping. Demand is said to beelastic 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. Youshould 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 referredto 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

M

dM

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 lowerthan 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 =PjQi

dQi

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 ofN&S. Practical knowledge of elasticities forms an important part of designing andunderstanding a variety of tax and subsidy policies in different markets.

Activity 2.4 Suppose u(x, y) = xαyβ, where α + β = 1. Income is M . Calculate theprice 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 forgood x, we vary the price of good x but hold constant the prices of other goods andincome. Of course, as the price changes so that the optimal choice changes, the utility ofthe consumer at the optimal point also changes. This is the usual demand curve, and isalso 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, itrefers to the Marshallian, or uncompensated, demand curve.

A compensated, or Hicksian, demand curve can be derived as follows. Suppose as theprice of a good changes, we keep utility constant while allowing income to vary. In otherwords, if the price of x, say, falls (so that the new optimal bundle of the consumerwould be associated with a higher level of utility if income is left unchanged), we takeaway enough income to leave the consumer at the original level of utility. It is clear thatthis 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 theuncompensated demand curve.

For an inferior good, the compensated demand curve is more elastic compared tothe uncompensated demand curve.

You should understand that all three properties result from the fact that only thesubstitution effect matters for the change in compensated demand when the pricechanges.

The next example asks you to calculate the compensated demand curve for aCobb–Douglas utility function.

Example 2.4 Suppose u(x, y) = x1/2y1/2. Income is M . Calculate the compensateddemand curves for x and y.

To do this, we must first calculate the Marshallian demand curves. These are givenby (you should do the detailed calculations to show this):

x =M

2pxand y =

M

2py.

The optimised value of utility is:

V =M

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 levelgiven by the original choice of x and y. It follows that the compensated demandfunctions are:

xc =M∗

2px= V

√pypx

and:

yc =M∗

2py= V

√pxpy.

Note that the Marshallian demand for x does not depend on py, but the Hicksian, orcompensated, demand does. This is because changes in py require incomeadjustments, 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 theother two measures: compensating variation (CV) and equivalent variation (EV). Weprovide definitions and applications below.

25

2. Consumer theory

When drawing demand curves, we typically draw the inverse demand curve (price onthe vertical axis, quantity on the horizontal axis). In such a diagram, the consumersurplus (CS) is the area under the (inverse) demand curve and above the market priceup to the quantity purchased at the market price. This is the most widely-used measureof welfare. We can measure the welfare effect of a price rise by calculating the change inCS (denoted by ∆CS).

Much of our discussion of policy will be based on this measure. Any part of ∆CS thatdoes not get translated into revenue or profits is a deadweight loss. The extent ofdeadweight loss generated by any policy is a measure of inefficiency associated with thatpolicy.

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 understandthe design of ideal policies, but when measuring welfare change in practice, use ∆CS.

CV is the amount of money that must be given to a consumer to offset the harmfrom a price increase, i.e. to keep the consumer on the original indifference curvebefore the price increase.

Compensating variation (CV)

EV is the amount of money that must be taken away from a consumer to cause asmuch 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 curvereached after the price rise.

Equivalent variation (EV)

Comparing the three measures

Consider welfare changes from a price rise. For a normal good, we haveCV > ∆CS > EV, and for an inferior good we have CV < ∆CS < EV. The measureswould 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 ofs per unit. This results in an increase in utility from u0 before the subsidy to u1after the subsidy. Could the government follow an alternative policy that wouldresult 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, theEV of a price fall is lower than the expenditure on heating after the price fall. Theintuition is that a per-unit subsidy distorts choice in favour of consuming moreheating, raising the total cost of the subsidy. To put the same idea differently, anequivalent 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 anadditional substitution effect boosting the demand for heating.

To see this, consider Figure 2.7. The initial choice is point A and after the subsidythe pensioner moves to point B. How much money is the government spending onthe subsidy? Note that after the subsidy, H1 units of heating fuel are beingconsumed. At pre-subsidy prices, buying H1 would mean the pensioner would haveE ′ of other goods. Since the price of the composite good is 1, M is the same as totalincome. It follows that the amount of income that would be spent on heating to buyH1 units of heating at pre-subsidy prices is given by ME ′. Similarly, at thesubsidised price, the amount of income being spent on heating fuel is MB′. Thedifference B′E ′ is then the amount of the subsidy. This is the same length assegment BE.

Once we understand how to show the amount of spending on the subsidy in thediagram, we are ready to compare this spending with an equivalent variation ofincome. 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 budgetlines (segment BE) shows the extent of the expenditure on the subsidy (as explainedabove). An equivalent variation in income, on the other hand, would moveconsumption to C. It follows that DE is the equivalent variation in income, which issmaller than the expenditure on the subsidy. Therefore, a direct income transferpolicy 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 functionu(x1, x2) = x1/2y1/2. He originally faces prices (1, 1) and has income 100. Then theprice 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 thedemand functions are:

x1 =M

2p1and x2 =

M

2p2.

Therefore, utility is:

u∗(p1, p2,M) =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 isgiven by:

M + CV

2√

2=M

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

2√

2=M − EV

2.

Solving:

EV =(√

2− 1)M√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 somewhatbasic coverage in the textbook.

The tools developed above can also be used to analyse the labour supply decision of anagent. Every economic agent has, in a day, 24 hours. An agent must choose how manyof these hours to spend working, and how many hours of leisure to enjoy. Working earnsthe agent income, which represents all goods the agent can consume. But the agent alsoenjoys leisure. If the hourly wage is w, this can be seen as the price that the agent mustpay to enjoy an hour of leisure.

The agent, therefore, faces the following problem. Let Z denote the number of hoursworked, N denote the number of leisure hours, M denote income and M denoteunearned income (inheritance, gifts etc). The utility maximisation problem is:

maxN,M

u(N,M)

subject to:

Z = 24−N

M = wZ + M.

We can simplify the constraints to M = w(24−N) + M , or M + wN = 24w + M .Therefore, we have a familiar utility maximisation problem:

maxN,M

u(N,M)

subject to the budget constraint:

M + wN = 24w + M.

At the optimum we have the slope of the indifference curve (−MRS) equal to the slopeof the budget line (−w). Therefore:

MUN

MUM

= w.

How does the optimal choice of labour respond to a change in w? We can analyse thisusing income and substitution effects. In this case, a change in w also changes incomedirectly (as you can see from the budget constraint), so the exercise is a little differentcompared to that under standard goods.

Suppose w rises.

The rise in w raises income at current levels of labour and leisure. Assuming leisureis a normal good (this should be your default assumption), this raises the demandfor leisure.

Income effect

29

2. Consumer theory

The rise in w makes leisure relatively more expensive, causing the agent to substituteaway from leisure. This reduces demand for leisure.

Substitution effect

The two effects go in opposite directions, therefore the direction of the total effect isunclear. In most cases, the substitution effect dominates, giving us an upward-slopinglabour 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 wewould get a backward-bending labour supply curve which initially slopes upward butthen turns back and has a negative slope.

If leisure is, on the other hand, an inferior good, the two effects would go in the samedirection 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 basicdiscussion in the textbook.

We focus on a two-period problem. Suppose the agent’s endowment is Y0 in period 0and Y1 in period 1. Given a rate of interest r, the present value of income in period 0 isY0 + Y1/(1 + r). If the individual consumes all income in period 0, C0 is equal to thispresent 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. Theproblem is then as follows:

maxC0, C1

u(C0, C1)

subject to the intertemporal budget constraint:

C0 +C1

1 + r= Y0 +

Y11 + r

.

This is similar to a standard optimisation problem with two goods, C0 and C1, wherethe price of the former is 1 and the price of the latter is 1/(1 + r). Unsurprisingly, theoptimum satisfies the property that:

MUC0

MUC1

= 1 + r.

If the optimal consumption bundle is C0 = Y0 and C1 = Y1, the agent is neither a savernor 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 cansee this by decomposing the effect into income and substitution effects. Let us look atthe 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

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 inperiod 0 (borrow less).

Income 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.

Substitution effect

Since the two effects go in the same direction, the direction of change is unambiguous: arise in the rate of interest lowers borrowing.

31

2. Consumer theory

The problem of a saver (lender)

For a saver, Y0 > C0, so that a rise in the interest rate raises income tomorrow. Givenconsumption is normal in both periods, the agent should consume more in period 0(save less).

Income 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 (savemore).

Substitution effect

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 raterises, 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 theendowment point as shown.

Figure 2.10: Intertemporal budget constraint.

Note that consumers reaching an optimum in the part of the budget constraint abovethe endowment point are savers, and those reaching an optimum somewhere in thelower 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 aborrower 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 streamof payoffs. We can extend this easily to multiple (or infinite) periods. This calculation isuseful in many cases – for example, in calculating the repeated game payoff in gametheory. 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 streamof payoffs is given by:

PV = y0 +y1

1 + r+

y2(1 + r)2

+ · · ·+ yn(1 + r)n

.

If we write δ = 1/(1 + r), we can write this as:

PV = y0 + δy1 + δ2y2 + · · ·+ δnyn.

Suppose y0 = y1 = · · · = yn = y. In this case:

PV = y(1 + δ + δ2 + · · ·+ δn).

We can sum this as follows. Let:

S = 1 + δ + δ2 + · · ·+ δn.

Then:

δS = δ + δ2 + · · ·+ δn+1.

We have:

S − δS = 1− δn+1.

Therefore:

S =1− δn+1

1− δ.

So:

PV = y1− δn+1

1− δ.

If the payoff stream is infinite, the present value is very simple:

PV = y(1 + δ + δ2 + · · · ) = 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) untila maturity date T , at which point the face value F is paid. The price of the bond, P , issimply the present value given by:

P = δx+ δ2x+ · · ·+ δTF.

Note that the price declines if δ falls, which happens if r rises. Therefore, the price of abond 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 aconsol has a particularly simple expression:

P = δx+ δ2x+ · · · = xδ

1− δ.

Now δ = 1/(1 + r). Therefore:

δ

1− δ=

1/(1 + r)

r/(1 + r)=

1

r.

It follows that:P =

x

r.

This makes the inverse relationship between P and r clear.

2.8 A reminder of your learning outcomes

Having completed this chapter, the Essential reading and activities, you should be ableto:

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 aconsumer 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 consumersurplus, equivalent variation and compensating variation, and use these measuresto 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 Test your knowledge and understanding

2.9.1 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 Marshalliandemand 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 + y2.

(a) Does this satisfy the property of diminishing MRS? Show algebraically, andalso show by drawing indifference curves.

(b) Show that using the tangency condition (MRS equals price ratio) would notlead to an optimum in this case.

(c) Show (in a diagram) the possible optimal bundles.

5. Consider the quasilinear utility function u(x1, x2) = lnx1 + x2 (this is linear in x2,but not in x1, hence the name ‘quasilinear’). Let p1 and p2 denote the prices of x1and 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 noelement of uncertainty. However, many important economic decisions are made insituations involving some degree of risk. In this chapter, we cover a model ofdecision-making under uncertainty called the expected utility model. The modelintroduces the von Neumann–Morgenstern (vN–M) utility function. This is unlike theordinal utility functions we saw in the previous chapter and has special properties. Inparticular, the curvature of the vN–M utility function can indicate a consumer’sattitude towards risk. Once we introduce the model, we use it to derive the demand forinsurance and also introduce a measure of the degree of risk aversion.


This chapter aims to introduce the expected utility model which tells us how aconsumer evaluates a risky prospect. We aim to show how this helps us understandattitudes towards risk and analyse the demand for insurance. We also aim to set up ameasure of the degree of risk aversion.



calculate the expected value of a gamble

explain the nature of the vN–M utility function and calculate the expected utilityfrom a gamble

explain the different risk attitudes and what they imply for the vN–M utilityfunction

analyse the demand for insurance and show the relationship between insurance andpremium

explain the concept of diversification

calculate the Arrow–Pratt measure of risk aversion for different specifications of thevN–M utility function.

37

3. Choice under uncertainty


N&S Sections 4.1, 4.2 and 4.3 up to and including the discussion on diversification (upto page 135). N&S does not cover the expected utility model or the Arrow–Prattmeasure of risk aversion. We provide details below. These topics are also covered inPerloff Section 16.2 (exclude the last part on willingness to gamble).


This chapter covers expected utility theory and uses the theory to derive the demandfor 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&Sdiscusses these.

A variable that represents the outcomes from a random event. A random variablehas many possible values, and each value occurs with a specified probability.

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:

E(X) = p1x1 + p2x2 + · · ·+ pnxn =∑i

pixi.

Expected value of a random variable

3.4 Expected utility theory

Expected utility theory was developed by John von Neumann and Oscar Morgensternin their book The Theory of Games and Economic Behavior. (Princeton UniversityPress, 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 togive a rough idea of what is involved.

Suppose an agent faces a gamble G that yields an amount x1 with probability p1, x2with probability p2, . . . , and xn with probability pn. How should the agent evaluate thisgamble? 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 undertheir axioms, there exists a function u such that the gamble can be evaluated using thefollowing ‘expected utility’ formulation:

E(U(G)) = p1u(x1) + p2u(x2) + · · ·+ pnu(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 functionlike 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 positiveaffine transformations), without changing the expected utility property but not anyother kinds of transformations (for example, u2 is not allowed). The reason is that, aswe discuss below, the curvature of the vN–M utility function captures attitude towardsrisk. Transformations other than positive affine ones change the curvature of thisfunction, and therefore the transformed u function would not represent the samerisk-preferences as the original. Thus vN–M utility functions are partly cardinal.

Note that the expected utility representation is very convenient. Once we know thevN–M utility function u, we can evaluate any gamble easily by simply taking theexpectation 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 isrisk-averse. Let us show this by establishing that an agent with a concave u functionwould reject a fair gamble.

Recall that a function f(W ) is concave if f ′′(W ) < 0, i.e. the second derivative of thefunction with respect to W is negative.

Suppose G is a gamble which yields 20 with probability 1/2, and 10 with probability1/2. Suppose an agent has wealth 15 and is given the following choice: invest 15 ingamble G, or do nothing. Note that the expected value of the gamble, E(G), is exactly15, so that this is a fair gamble (the expected wealth is the same whether G is acceptedor rejected).

An agent who simply cared about expected value, and not about risk, would beindifferent between accepting and rejecting G. However, a risk-averse individual wouldreject a fair gamble.

The expected utility of an agent from G is:

E(U(G)) =1

2× u(20) +

1

2× u(10).

As Figure 3.1 shows, given a concave u-function:

E(U(G)) < u(15) = u(E(G)).

Therefore, the agent would not accept a fair gamble. This shows that a concaveu-function implies risk aversion.

Note that one of the implications of a concave vN–M utility function is that themarginal utility of wealth is declining. The point is noted in N&S Section 4.2.

39


Figure 3.1: The vN–M utility function for a risk-averse individual. Note that the functionis 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 definethe certainty equivalent (CE) of a gamble. The CE of a gamble is the certain wealththat would make an agent indifferent between accepting the gamble and accepting thecertain 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 and10, 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 muchwould the agent be willing to pay? The amount an agent pays for insurance is called therisk 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 0.5a loss occurs so that wealth becomes 64. In what follows, we will assume that Kim canonly buy full insurance. In other words, the insurance company offers to pay Kim 36whenever 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 maximumand 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 adiagram as above to show that Kim would be prepared to pay a positive premium ifwealth is fully insured. Alternatively, you could point out that the expected utility ofthe 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 asabove and point out that since expected wealth exceeds the certainty equivalent, thepremium is positive.

Let us now calculate the maximum premium that Kim would be willing to pay tobuy 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 is100− 81 = 19.

There is another way of finding this, which considers the maximum premium in terms ofexpected wealth (i.e. how much expected wealth would Kim give up in order to fullyinsure?). You need to understand this, since some textbooks use this way of identifyingthe premium. For example, this is the approach adopted by Perloff. Unfortunately,textbooks (including Perloff) never make clear exactly what they are doing, which canbe very confusing for students. Reading the exposition here should clarify the matteronce and for all.

The maximum premium in terms of expected wealth is calculated as follows. Note thatunder full insurance the gross expected wealth Kim would receive is E(W ) = 82. Wealso know that CE = 81. Therefore, the maximum amount of expected wealth Kimwould give up is 82− 81 = 1. (Note that this should explain why in the diagram on riskaversion in Perloff Section 16.2, and subsequent solved problems, the risk premium isidentified 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 ispaid 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


Once again, the purpose of writing this out in detail is to make you aware thattextbooks vary in their treatment of this. Some talk about premium in terms ofexpected wealth, while others calculate the actual premium, but they do not make itclear what it is that they are doing. In answering questions of this sort in anexamination, it is easiest (and clearest) to calculate the actual premium. You can followthe other route and define the premium in terms of expected wealth, but in that caseyou 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 minimumpremium (or fair premium) is Rmin such that it equals the expected payout, which is0.5× (100− 64) = 18. (Note that this is simply 100− E(W ), where E(W ) is expectedwealth, which is 82 in this case.)

As above, the other way of answering the question is to say that in terms of the expectedwealth that Kim needs to give up, the minimum is zero. Think of this as follows. Kimsimply hands over her actual wealth to the insurance company, and in return receivesthe expected wealth in all states. The insurance company is risk-neutral, and inexpected 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-averseagent a continuous choice of levels of insurance. Can we say something general abouthow much insurance an agent would choose? As it turns out, we can. If insurance isactuarially fair (which is another way of saying that the insurance company just breakseven, so that the premium is equal to the expected payment to the agent), we can showthat 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 thatthere the agent was given a simple choice between full insurance and no insurance, andin that case the maximum willingness to pay for insurance is 19, even though the fairpremium is 18. However, if a more continuous choice of insurance levels was provided tothat agent, the agent would optimally buy full insurance only at a premium of 18, andoptimally 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. Everyrisk-averse agent would buy full insurance. If, on the other hand, the insurancepremium is greater than the fair level, how much insurance an agent buys depends ontheir degree of risk-aversion. No agent with a finite degree of risk-aversion would buyfull insurance anymore, but the extent of insurance purchased increases as the agent’sdegree 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 withprobability p, where 0 < p < 1. The agent can buy a coverage of X by paying thepremium rX.

The wealth if loss occurs is given by WL = W − L+X − rX, and the wealth when noloss 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:

pu′(WL)(1− r)− (1− p)u′(WN)r = 0.

Note that the second-order condition for a maximum is satisfied since the agent isrisk-averse implying that u′′ < 0 (u is concave). Therefore, we have:

u′(WL)

u′(WN)=

(1− p)r(1− r)p

.

If insurance is fair, that implies the insurance company breaks even, i.e. the expectedpayout pX equals the expected receipt rX. Since pX = rX, we have:

p = r.

It follows that:u′(WL) = u′(WN).

Since u′ is a decreasing function (because u′′ < 0), it is not possible to have this equalityif WL 6= WN . (Note that if u′ was a non-monotonic function and was going up anddown, it would be possible to have WL different from WN but have the same value of u′

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 theentire loss) at the fair premium.

Note also what would happen if r > p. Then (1− p)r > (1− r)p. Therefore:

u′(WL)

u′(WN)=

(1− p)r(1− r)p

> 1.

This implies that:u′(WL) > u′(WN).

Again, because u′ is a decreasing function (u′′ < 0), this implies that WL < WN , whichin turn implies that X < L. Thus if the premium exceeds the fair premium, less thanfull 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 convexvN–M utility function (so that we have u′′ > 0) – represents risk-loving behaviour, andthe vN–M utility function is a straight line (u′′ = 0) for a risk-neutral agent.

A risk-neutral agent does not care about risk and only cares about the expected valueof a gamble. In other words, a risk-neutral agent is indifferent between accepting andrejecting a fair gamble. For a risk-neutral agent, we can write the vN–M utility functionof wealth simply as:

u(W ) = W.

43


Figure 3.2: The vN–M utility function for a risk-neutral individual. Note that the functionis linear and u(E(G)) = E(U(G)) so that the agent is indifferent between a fair gambleand the safe alternative of 15.

Figure 3.2 shows the vN–M utility function for a risk-neutral agent. The figure refers tothe gamble introduced in the section on risk aversion (either keep 15 or invest in a fairgamble 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 whenthey have the same expected outcome. In other words, a risk-loving agent would preferto accept a fair gamble. As Figure 3.3 below shows, for a risk-loving agent, the CE of agamble is higher than the expected value of the gamble (you would have to pay arisk-loving agent to give up a risky gamble in favour of the safe alternative of gettingthe 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 measurethe degree of risk aversion? As you might guess, the degree of risk aversion has to dowith the curvature of the vN–M utility function u. The more concave it is, the greaterthe degree of risk aversion. The closer it is to a straight line, the lower the degree of riskaversion. Since the second derivative captures the curvature, a measure of risk aversionmight be u′′. However, this would not be ideal for the following reason. We know that apositive affine transformation of u, say u = a+ bu, where a and b are positive constants,does not change attitude towards risk. But such a transformation would change thesecond derivative and, therefore, change the risk measure. This problem could beavoided if u′′ is divided by u′. Furthermore, since the most common risk attitude is riskaversion, and for this case u′′ < 0, putting a negative sign in front of u′′ 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 functionis convex and u(E(G)) < E(U(G)) so that the agent prefers a fair gamble to the safewealth of 15.

positive risk measure under risk aversion. These help to interpret the Arrow–Prattmeasure of risk aversion, which is given by:

ρ = −u′′

u′.

That is, the Arrow–Pratt measure of risk aversion is −1 times the ratio of the secondderivative and the first derivative of the vN–M utility function. This is the mostcommon measure of risk aversion. There are other measures, which you will encounterin Masters level courses.

It can be shown that the larger the Arrow–Pratt measure of risk aversion, the moresmall gambles an individual will take. A derivation of this result must await a Masterslevel course as well.

As noted above, for a risk-averse individual, u′′ < 0, so the minus sign in front makesthe measure a positive number. For a risk-neutral agent, u′′ = 0 so that ρ = 0, and for arisk-loving agent u′′ > 0 so that the measure is negative.

Example 3.1 Let us calculate the Arrow–Pratt measure for different specificationsof the vN–M utility function.

(a) Suppose u(W ) = lnW . Then u′(W ) = 1/W and u′′(W ) = −1/W 2. It followsthat:

ρ =1

W.

This agent has risk aversion that is decreasing in wealth.

45


(b) Next, suppose u(W ) = Wα, where 0 < α < 1. Then:

ρ = −(α− 1)αWα−2

αWα−1 =1− αW

.

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 isrisk-neutral.

(c) Next, suppose u(W ) = −e−α. Then u′(W ) = αe−α and u′′(W ) = −α2e−α.Therefore, ρ = α. In this case the degree of risk aversion does not depend on thewealth level.

3.9 Reducing risk

Insurance provides a way to reduce risk. Diversification is also another way to reducerisk. You should read carefully the discussion on this in N&S Section 4.3 (you do notneed to study this section beyond diversification).



calculate the expected value of a gamble

explain the nature of the vN–M utility function and calculate the expected utilityfrom a gamble

explain the different risk attitudes and what they imply for the vN–M utilityfunction

analyse the demand for insurance and show the relationship between insurance andpremium

explain the concept of diversification

calculate the Arrow–Pratt measure of risk aversion for different specifications of thevN–M utility function.



1. A risk-averse individual is offered a choice between a gamble that pays 1000 with aprobability of 1/4, and 100 with a probability of 3/4, or a payment of 325. Whichwould they choose? What if the payment was 320?

46


2. Suppose u(W ) = −1/W . What is the risk attitude of this person? Calculate theArrow–Pratt measure of risk aversion for this preference.

3. Suppose an agent has vN–M utility function u(W ). Under what condition woulda+ 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 ) = lnW for an agent. The agent faces the following gamble: withprobability 0.5 wealth is 100, and with probability 0.5 a loss occurs so that wealthbecomes 64. The agent can buy any amount of insurance: a coverage of X can bepurchased by paying premium rX.

(a) Work out the insurance coverage X that the agent would optimally purchaseas 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


48

Chapter 4

Game theory

4.1 Introduction

In many economic situations agents must act strategically by taking into account thebehaviour of others. Game theory provides a set of tools that enables you to analysesuch situations in a logically coherent manner. For each concept introduced, you shouldtry to understand why it makes sense as a tool of analysis and (this is much harder) tryto see what its shortcomings might be. This is not the right place to ask questions suchas ‘what is the policy-relevance of this concept?’. The concepts you come across here arejust tools, and that is the spirit in which you should learn them. As you will see, someof these concepts are used later in this course (as well as in a variety of other economicscourses that you might encounter later) to analyse certain types of economicinteractions.


The chapter aims to familiarise you with a subset of basic game theory tools that areused extensively in modern microeconomic theory. The chapter aims to coversimultaneous-move games as well as sequential-move games, followed by an analysis ofthe repeated Prisoners’ Dilemma game.



analyse simultaneous-move games using dominant strategies or by eliminatingdominated strategies either once or in an iterative fashion

calculate Nash equilibria in pure strategies as well as Nash equilibria in mixedstrategies in simultaneous-move games

explain why Nash equilibrium is the central solution concept and explain theimportance 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 backwardinduction and subgame perfection

49

4. Game theory

analyse the infinitely-repeated Prisoners’ Dilemma game with discounting andanalyse collusive equilibria using trigger strategies

explain the multiplicity of equilibria in repeated games and state the folk theoremfor the Prisoners’ Dilemma game.


N&S Chapter 5. However, the content of this chapter is not sufficient by itself. Thischapter of the subject guide fleshes out the basic tools that you are required to know insome detail, and you should also read this and follow the exercises carefully. Forrepeated games, the interpretation of payoffs presented here is slightly different from thetextbook. While the analyses are formally equivalent, the approach in this subject guidefollows the standard one in the literature. You are likely to find this fits better withanalyses 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 conceptscovered in this subject guide.

Osborne, M.J. An Introduction to Game Theory (Oxford University Press 2009)International edition [ISBN 9780195322484].


The chapter introduces simultaneous-move games and analyses dominance criteria aswell as Nash equilibrium. Next, the chapter introduces sequential-move games andanalyses Nash equilibrium as well as subgame-perfect Nash equilibrium. Finally, thechapter 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 threeelements. First, a set of players. Second, each player must have a set of strategies. Onceeach player chooses a strategy from their strategy set, we have a strategy profile. Forexample, 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 strategyset 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 agame: a payoff function for each player. A payoff function for any player is defined overthe 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 nplayers, a strategy profile is (s1, s2, . . . , sn). For any such strategy profile, there is apayoff 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 andu1({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. Thefirst number in each cell is the payoff of player 1 and the second number is the payoff ofplayer 2. Note that player 1 chooses rows and player 2 chooses columns.

Player 2L R

Player 1 U 1, 2 2, 1D 2, 3 1, 2

It is sometimes useful to write the strategy profile (s1, s2, . . . , sn) as (si, s−i), where s−iis 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 asui(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 payoffcompared to other strategies irrespective of the strategy choices of others. Such astrategy is called a dominant strategy. If a player has a dominant strategy, hisproblem is simple: he should clearly play that strategy. If each player has a dominantstrategy, the equilibrium of the game is obvious: each player plays his own dominantstrategy.

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 eachprofile generates a payoff for each player. The first number in each cell is the payoff ofplayer 1 and the second number is the payoff of player 2. Again, note that player 1chooses the row that is being played and player 2 chooses the column that is beingplayed.

Player 2C D

Player 1 C 2, 2 0, 3D 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 situationwhere 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 isinfinitely-repeated, then under certain conditions cooperation can emerge as anequilibrium. But in a one-shot game (i.e. a game that is played once) the only possibleequilibrium is that each player plays their dominant strategy. In the game above, thedominant strategy equilibrium is (D, D).

In terms of the notation introduced before, we can define a dominant strategy as follows.

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 othersare playing.

Dominant strategy

4.3.2 Dominated strategies and iterated elimination

Even if a player does not have a dominant strategy, he might have one or moredominated strategies. A dominated strategy for i is a strategy of i (say si) that yields alower payoff compared to another strategy (say s′i) irrespective of what others areplaying. In other words, the payoff of i from playing si is always (i.e. under all possiblechoices of other players) lower than the payoff from playing s′i. Since si is a dominatedstrategy, 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 byiteratively 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, butfor 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 than1). So we can eliminate U . We are then left with (D, L), which is the equilibrium byiteratively eliminating dominated strategies. The game is dominance solvable.

Here is another example of a dominance solvable game. We find the equilibrium of thisgame by iteratively eliminating dominated strategies.

Player 2Left Middle Right

Top 4, 3 2, 7 0, 4Player 1 Middle 5, 5 5, −1 −4, −2

Bottom 3, 5 1, 5 −1, 6

We can eliminate dominated strategies iteratively as follows.

52


1. For player 1, Bottom is dominated by Top. Eliminate Bottom.

2. In the remaining game, for player 2, Right is dominated by Middle. EliminateRight.

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. EliminateMiddle.

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 anequilibrium. Players might not have dominant strategies; moreover none of thestrategies of any player might be dominated. The following game provides an example.

Player 2A2 B2 C2

A1 3, 1 1, 3 4, 2Player 1 B1 1, 0 3, 1 3, 0

C1 2, 3 2, 0 3, 2

As noted above, the problem with dominance criteria is that they apply only to somegames. For games that do not have dominant or dominated strategies, the idea ofderiving an equilibrium using dominance arguments does not work. If we cannot derivean equilibrium by using dominant strategies or by (iteratively) eliminating dominatedstrategies, how do we proceed?

If we want to derive an equilibrium that does not rely on specific features such asdominance, we need a concept of equilibrium that applies generally to all games. As weshow below, a Nash equilibrium (named after the mathematician John Nash) isindeed 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 thePrisoners’ 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 Cwith probability p and play D with probability (1− p), where p is some numberbetween 0 and 1. Such a strategy is called a ‘mixed’ strategy; while a strategy that justchooses one action (C or D) is called a ‘pure’ strategy.

We start by analysing Nash equilibrium under pure strategies. Later we introducemixed strategies. We then note that one can prove an existence result: all games have atleast one Nash equilibrium (in either pure or mixed strategies). This is why Nashequilibrium 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 bestresponse 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 byreplacing s∗i with any other strategy.

A more formal definition is as follows.

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.

Nash equilibrium in pure strategies

To find out the Nash equilibrium of the game above, we must look for the mutual bestresponses. Let us check the best response of each player. Player 1’s best response is asfollows:

Player 2’s strategy Player 1’s best responseA2 A1

B2 B1

C2 A1

Player 2’s best response:

Player 1’s strategy Player 2’s best responseA1 B2

B1 B2

C1 A2

Note from these that the only mutual best response is (B1, B2). This is the only Nashequilibrium 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 noNash equilibrium involving A1.

If player 1 plays B1, player 2’s best response is B2. If player 2 plays B2, player 1’sbest 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 Nashequilibrium involving C1.

From the three steps above, we can conclude that (B1, B2) is the only Nash equilibrium.

54


Let us also do a slightly different exercise. Suppose you want to check if a particularstrategy profile is a Nash equilibrium. Suppose you want to check if (A1, C2) is a Nashequilibrium. You should check as follows.

If player 2 plays C2, player 1 cannot do any better by changing strategy from A1 (4 isbetter 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 2from 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 someplayer 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 byswitching given what the other player is playing. Since player 2 is playing B2, player 1gets 3 from B1 which is better than 1 from A1 or 2 from C1. Since player 1 is playingB1, player 2 gets 1 from B2 which is better than 0 from A2 or C2. Therefore (B1, B2) isa Nash equilibrium.

Nash equilibrium is not necessarily unique. Consider the following game.

Player 2A B

A 2, 1 0, 0Player 1 B 0, 0 1, 2

Note there are multiple pure strategy Nash equilibria. Both (A,A) and (B,B) are Nashequilibria.

Dominance criteria and Nash equilibrium

Note that while a dominant strategy equilibrium is also a Nash equilibrium, Nashequilibrium does not require dominance. However, the greater scope of Nash equilibriumcomes 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 requirementis even more problematic when there are multiple Nash equilibria. On the other hand, ifplayers have dominant strategies, they do not need to think at all about what others aredoing. A player would simply play the dominant strategy since it is a best response nomatter what others do. This is why a dominant strategy equilibrium (or one achievedthrough iterative elimination of dominated strategies) is more convincing than a Nashequilibrium. However, as noted before, many games do not have dominant or dominatedstrategies, and are therefore not dominance solvable. We need a solution concept thatapplies generally to all games, and Nash equilibrium is such a concept.

4.3.4 Mixed strategies

Consider the following game.

Player 2A2 B2

A1 3, 1 2, 3Player 1 B1 2, 1 3, 0

55

4. Game theory

Notice that this game has no pure strategy Nash equilibrium. However, as you will seebelow, the game does have a Nash equilibrium in mixed strategies.

Let us first define a mixed strategy.

A mixed strategy si is a probability distribution over the set of (pure) strategies.

Mixed strategy

In the game above, A1 and B1 are the pure strategies of player 1. A mixed strategy ofplayer 1 could be A1 with probability 1/3, and B1 with probability 2/3. Notice that apure 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 strategiesthat constitute a mutual best response.

A mutual best response in mixed strategies has an essential property that makes it easyto find mixed strategy Nash equilibria. Let us consider games with two players tounderstand 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) where0 < 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 andB1? Suppose π1(A1) > π1(B1). Then clearly player 1 should simply play A1, rather thanany strictly mixed strategy, to maximise her payoff. In other words, player 1’s bestresponse in this case would be to choose p = 1, rather than a strictly mixed strategyp < 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. justto 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 isindifferent between the two strategies. How does such indifference come about? This isdown to player 2’s strategy choice. Player 2’s choice of q must be such that player 1 isindifferent between playing A1 or B1. In other words, in equilibrium q must be suchthat 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


to make player 2 indifferent between A2 and B2. In other words, player 1’s choice of p issuch 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 withprobability 1/3 and B1 with probability 2/3, while player 2 plays A2 with probability1/2 and B2 with probability 1/2.

We can also show this in a diagram. Let us first write down the best response functionof 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 strategyNash 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-playergame is that each player’s chosen probability distribution must make the other playerindifferent between the strategies he is randomising over. In a k-player game, the jointdistribution 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 ofthe strategies he plays with positive probability.

The game above has no Nash equilibrium in pure strategies, but has a mixed strategyNash 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 amixed strategy Nash equilibrium in the Battle of the sexes.

Activity 4.1 Find the mixed strategy Nash equilibrium in the following game. Alsoshow all Nash equilibria of the game in a diagram by drawing the best responsefunctions.

Player 2A B

A 2, 1 0, 0Player 1 B 0, 0 1, 2

4.3.5 Existence of Nash equilibrium

Once we include mixed strategies in the set of strategies, we have the followingexistence theorem, proved by John Nash in 1951.

Every game with a finite number of players and finite strategy sets has at least oneNash equilibrium.

Existence theorem

Nash proved the existence theorem for his equilibrium concept using a mathematicalresult called a ‘fixed-point theorem’. Take any strategy profile and compute the bestresponse to it for every player. So the best response is another strategy profile. Supposewe do this for every strategy profile. So long as certain conditions are satisfied, afixed-point theorem says that there is going to be at least one strategy profile which is abest response to itself. This is, of course, a Nash equilibrium. Thus upon setting up thestrategy sets and the best response functions properly, a fixed-point theorem can beused to prove existence. You will see a formal proof along these lines if you study gametheory 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 gamewe look at, we will always be able to derive at least one Nash equilibrium. If existencedid not hold for some equilibrium concept (for example, games do not necessarily have adominant 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

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4.4. Sequential-move or extensive-form games

that we confront might not have an equilibrium at all. But being able to prove existencefor Nash equilibrium removes such problems. Indeed, as noted before as well, this isprecisely what makes Nash equilibrium the main solution concept forsimultaneous-move games.

4.3.6 Games with continuous strategy sets

We have so far analysed games with discrete strategy sets. However, the above analysiscan easily extend to certain classes of games with continuous strategy sets. We willanalyse a few such games (the Cournot game, the Bertrand game, and the Bertrandgame with product differentiation) in detail later in discussing oligopoly (in Chapter 9of 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 discussingexternalities (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 treeto depict the sequence of actions. Games depicted in such a way are also known asextensive-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 actbefore them. First, we need to understand the difference between actions and strategiesin such games. Once we clarify this, we show how to derive Nash equilibria. Finally, wepropose a refinement of Nash equilibrium: subgame perfect Nash equilibrium. Ingames where all moves of previous players can be observed, subgame perfect Nashequilibria can be derived by backward induction. We then introduce some simple caseswhere information is imperfect, and show how the notion of strategies differs, and howto derive subgame perfect Nash equilibria.

Consider the following extensive-form game. Each player has two actions: player 1’sactions are a1 and a2 and player 2’s actions are b1 and b2. Player 1 moves before player2. Player 2 can observe player 1’s action and, therefore, can vary his action dependingon the action of player 1. For each profile of actions by player 1 and player 2 there arepayoffs at the end. As usual, the first number is the payoff of the first mover (in thiscase 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 decisionnodes to successor nodes. However, such formal definitions are useful only at a laterstage. 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, foran extensive-form game, it is a little bit more complicated. A strategy in an extensiveform game is a complete plan of actions. In other words, a strategy for player i must

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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, theset of actions and the set of strategies is the same. Player 1 can simply decide betweena1 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 decidewhat to do if player 1 plays a1, and what to do if player 1 plays a2. Note that suchdecisions must be made before the game is actually played. Essentially, game theorytries to capture the process of decision-making of individuals. Faced with a game suchas the one above, player 2 must consider both contingencies. This is what we capture bythe notion of a strategy. It tells us what player 2 would do in each of the two possiblecases.

Now player 2 can choose 2 possible actions at the left node (after player 1 plays a1) and2 possible actions at the right node (after a2). So there are 2× 2 = 4 possible strategiesfor player 2. These are:

1. If player 1 plays a1, play b1 and if player 1 plays a2, play b1.




For the sake of brevity of notation, we write these as follows. Just as we read wordsfrom left to right, we read strategies from left to right. So we write the strategy ‘ifplayer 1 plays a1, play b2 and if player 1 plays a2, play b1’ as b2b1. Reading from left toright, this implies that the plan is to play b2 at the left node and play b1 at the rightnode. This is precisely what the longer specification says.

So the strategy set of player 2 is {(b1b1), (b1b2), (b2b1), (b2b2)}.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.

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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 couldchoose 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 asthe ‘normal form’. To find Nash equilibria in an extensive-form game, the mostconvenient method is to transform it into its normal form. This is as follows. Note thatplayer 1 has two strategies and player 2 has four strategies. Therefore, we have a 2-by-4matrix of payoffs as follows:

Player 2b1b1 b1b2 b2b1 b2b2

Player 1 a1 3, 1 3, 1 1, 0 1, 0a2 4, 1 0, 1 4, 1 0, 1

Note that if we pair, say, a1 with b2b1, only the first component of player 2’s strategy isrelevant for the payoff. In other words, since player 1 plays a1, the payoff is generated byplayer 2’s response to a1, which in this case is b2. Similarly, if we pair a2 with b2b1, thesecond component of player 2’s strategy is relevant for the payoff. Since player 1 playsa2, 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 onlyconsider pure strategy Nash equilibria. There are three pure strategy Nash equilibria inthis game. Finding these is left as an exercise.

Example 4.1 Find the pure strategy Nash equilibria of the extensive-form gameabove.

It is easiest to check each box. If we start at the top left-hand box, player 1 wouldswitch to a2. So this is not a Nash equilibrium. From (a2, b1b1) no player can dobetter by deviating. Therefore, (a2, b1b1) is a Nash equilibrium.

Next try (a1, b1b2). This is indeed a Nash equilibrium. Note that you must writedown the full strategy of player 2. It is not enough to write (a1, b1). Unless we knowwhat player 2 would have played in the node that was not reached (in this case thenode after a2 was not reached), we cannot determine whether a strategy is part of aNash equilibrium. So while (a1, b1b2) is indeed a Nash equilibrium, (a1, b1b1) is not.

Finally, (a2, b2b1) is also a Nash equilibrium. These are the three pure strategy Nashequilibria of the game.

4.4.3 Imperfect information: information sets

So far we have assumed perfect information: each player is perfectly informed of allprevious moves. Let us now see how to represent imperfect information: players may notbe 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 moresophisticated refinements of Nash equilibria than we study here. In particular, if a

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4. Game theory

player does not observe some past moves, what he believes took place becomesimportant. We do not study these problems here – the analysis below simply shows youhow to represent situations of imperfect information in some simple cases.

Consider the extensive-form game introduced at the start of this section. Suppose at thetime 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 actionplayer 2 does not know whether he is at the left node or at the right node. To capturethis situation of imperfect information in our game-tree, we say that the two decisionnodes of player 2 are in an information set. We represent this information set as inFigure 4.3: by connecting the two nodes by a dotted line (another standard way to dothis 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 doesnot know where he is in the information set (i.e. he does not know what player 1 haschosen). Since player 2 cannot distinguish between the two nodes inside his informationset, he cannot take different actions at the two nodes. Therefore, the strategy set of 2 issimply {b1, b2}. In other words, now, for both players the strategy set coincides with theaction set. This is not surprising: since player 2 takes an action without knowing whatplayer 1 has done, the game is the same as a simultaneous-move game. Indeed, thenormal form of the game above coincides with a game in which the two players choosestrategies simultaneously. This is shown below. Now the Nash equilibrium is simply(a2, b1).

Player 2b1 b2

a1 3, 1 1, 0Player 1 a2 4, 1 0, 1

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4.5. Incredible threats in Nash equilibria and subgame perfection

4.5 Incredible threats in Nash equilibria and subgameperfection

Let us now consider whether Nash equilibrium is a satisfactory solution concept forextensive-form games. As you will see, when players move sequentially, Nashequilibrium allows for some strategies by later movers that seem like threats which areincredible. 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 nointention of carrying out the threat. Children sometimes believe their parents andrespond to the threat, but they are often clever enough to see through the ruse andignore incredible threats. As we will see, Nash equilibria often depend precisely on suchincredible threats by later movers. In Nash equilibrium, a player is just supposed totake a best response to the other players’ strategy choices – so the way Nashequilibrium is constructed does not allow the player to ignore certain strategy choices ofothers as incredible. Once we look at some examples of the problem, we will try to seewhether we can refine the set of Nash equilibria to eliminate the possibility of suchthreats (i.e. come up with extra conditions that an equilibrium must satisfy so thatequilibria 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 decidingwhether to enter a market. The market has an incumbent firm (Firm I). If the entrantenters, the incumbent firm must decide whether to fight (start a price war, say) oraccommodate 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 IA F

In 2, 1 −1, −2Firm E Out 0, 2 0, 2

Note that there are two pure strategy Nash equilibria: (In, A) and (Out, F). The latterequilibrium involves an incredible threat. Clearly, if the entrant does decide to enter the

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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 responsefor Firm I.

The game in Figure 4.5 presents another example.


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 seethat (R, rr) is a Nash equilibrium. Look at the game above and see that thisinvolves an incredible threat. Player 2’s strategy involves playing r after L. Player1’s strategy is taking a best response given this, and so player 1 is playing R. Giventhat 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 abest response since it does not change the payoff, but, of course, not every choicewould 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 therequirement that strategies be mutual best responses) for equilibrium and leads to arefinement of the set of Nash equilibria. Several such refinements have been proposed bygame theorists. Here, we will look at only one such refinement, namely subgameperfection. To understand the refinement, you first need to understand the idea of asubgame.

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Subgame

A subgame is a part of a game that starts from a node which is a singleton (i.e. asubgame 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 thatinformation set. In other words, you cannot cut an information set so that only part ofit belongs to a subgame. That would clearly alter the information structure of thegame, 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 arefinement 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 solvingbackwards, i.e. by using backward induction.

Solving backwards in the entry game, we see that Firm I would choose A if Firm Echose 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 thedefinition of SPNE given above. The game has a subgame starting at the node afterFirm E plays In (also, the whole game is always trivially a subgame). In the subgameafter E plays In, there is only one player (Firm I), and the Nash equilibrium in thissubgame 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 Nashequilibrium of the whole game that involves playing A in the subgame is a subgameperfect Nash equilibrium. Here, the only Nash equilibrium of the whole game thatsatisfies 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 movessecond and chooses between `, r. In this game there are two strict subgames, onestarting 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 wouldcompare 3 from R and 0 from L, and choose R. The choices obtained by backwardinduction are shown in Figure 4.6.

It follows that the SPNE is (R, `r). Note that it is not sufficient to write (R, `) – theequilibrium specification is meaningless unless you specify the full strategy for player 2.

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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, Rwould not be the optimal choice. Therefore, you must specify player 2’s full strategyand identify (R, `r) as the SPNE.

In these games, the SPNE is unique, but it need not be. The next activity presents anexample.

Activity 4.3 Derive the pure strategy subgame perfect Nash equilibria of the gamein Figure 4.2.

4.5.3 Subgame perfection under imperfect information

Backward induction need not work under imperfect information: you cannot foldbackwards when you come up against an information set. Indeed, this is why theconcept of a subgame perfect Nash equilibrium is more general compared to backwardinduction. If we always had perfect information, we could simply define backwardinduction equilibria. However, we present below an example to show you that subgameperfection is more general than backward induction, and works in many games in whichbackward induction does not give us any result.

Before we present the example referred to above, consider the imperfect informationgame introduced in Section 4.4.3. Note that this game does not have any strictsubgames (recall that you cannot start a subgame from an information set or cut aninformation set), so the only subgame is the whole game. Therefore, any Nashequilibrium of the whole game is trivially subgame perfect. As discussed above, the purestrategy Nash equilibrium in this game is (a2, b1). This is also the subgame perfect Nashequilibrium.

Next, consider the game in Figure 4.7.

Initially player 1 decides whether to come in (and play some game with player 2) orstay out (in which case player 2 plays no role). If the choice is to come in, player 1

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Figure 4.7: A game tree.

decides between A and B, and player 2 decides between C and D. When player 2 makeshis decision, he knows that player 1 has decided to come in (if not then player 2 wouldnot have been asked to play), but without knowing what player 1 has chosen between Aand B. In other words, the situation is just as if once player 1 comes in, player 1 andplayer 2 play a simultaneous-move game (as the game structure shows, they do notmove simultaneously – player 1 moves before player 2, but since player 2 has noknowledge of player 1’s move, it is similar to the decision problem faced in asimultaneous move game).

1. Pure strategy Nash equilibria. Let us first identify the pure strategy Nashequilibria. Note that player 1 has 4 strategies: Out A, Out B, In A and In B, whileplayer 2 has 2 strategies: C and D. You might think strategies like Out A do not makesense, but in game theory we try to model the thought process of players, and even ifplayer 1 stays out, she would do so only after thinking about what she would have donehad she entered. Strategies reflect such thinking (it is as if player 1 is saying ‘I havedecided 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 2C D

Out A 1, 3 1, 3Player 1 Out B 1, 3 1, 3

In A −2, −2 2, 0In B 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).

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4. Game theory

2. Pure strategy subgame perfect Nash equilibria. Note that backward inductiondoes 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 refinethe set of Nash equilibria.

Note that apart from the whole game, there is just one strict subgame, which starts atthe node after player 1 chooses In. Below we write down the normal form of thesubgame.

Player 2C D

Player 1 A −2, −2 2, 0B 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 Infollowed by (B,C)) and decides to stay out. Therefore, a SPNE of the whole game is(OutB,C).

If, on the other hand, (A,D) is played in the subgame, player 1 compares 1 with 2 anddecides to come in. Therefore, another SPNE of the whole game is (InA,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 ofNash equilibria, and since (B,C) and (A,D) are the Nash equilibria in the subgame, itmust 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 equilibriaderived above, we can immediately infer that (Out A,C) is Nash but not subgameperfect, while the other two are subgame perfect.

4.6 Repeated Prisoners’ Dilemma

Consider the following Prisoners’ Dilemma game:

Player 2C D

Player 1 C 2, 2 0, 3D 3, 0 1, 1

In a one-shot game, rational players simply play their dominant strategies. So (D,D) isthe only possible equilibrium. Suppose the game is repeated. Can we say somethingabout the behaviour of players in such ‘supergames’ that differs from the behaviour inthe 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 aone-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 nosense. Therefore, in each period players would play (D,D). But this logic extends to any

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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 backwardinduction, we can assume that when a game is repeated many times, players play themas if the games are infinitely repeated. In that case, we must apply forward-looking logicas there is no last period from which to fold backwards.

(A quick note: you should be aware that there are other games with multiple Nashequilibria where some cooperation can be sustained even under a finite number ofrepetitions. You will encounter these in more advanced courses. Here we only considerthe 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 eachperiod, 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 simplythe discounted present value of the stream of payoffs.

Let δ denote the common discount factor across players, where 0 < δ < 1. If today’sdate is 0, and a player receives x in period t, the present value of that payoff is δtx. Thediscount factor can reflect players’ time preference. This can also arise from a simplerate of interest calculation, in which case δ can be interpreted as 1/(1 + r), where r is therate 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) + δ2ui(a2) + · · · .

More concisely, the payoff is:∞∑t=0

δtui(at).

If the payoff is the same every period (x, say), this becomes:

x(1 + δ + δ2 + · · · ) =x

1− δ.

4.6.1 Cooperation through trigger strategies

Next, we need to consider strategies by players. The history at t is the action profileplayed in every period from period 0 to t ≥ 1. A strategy of a player consists of aninitial action, and after that, an action after every history. Consider the followingtrigger strategy.

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4. Game theory

Trigger strategy

Start by playing C (that is, cooperate at the very first period, when there is nohistory 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)) wouldwork only until someone deviates to D. After the very first deviation, each playerswitches 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. Weonly need to consider what happens from t onwards. The payoff starting at period t isgiven by:

3 + δ + δ2 + · · · = 3 +δ

1− δ.

If the player did not deviate in period t, the payoff from t onwards would be:

2 + 2δ + 2δ2 + · · · = 2

1− δ.

For deviation to be suboptimal, we need:

2

1− δ> 3 +

δ

1− δ

which implies:

δ >1

2.

Thus if the players are patient enough, cooperation can be sustained in equilibrium. Inother words, playing (C,C) always can be the outcome of an equilibrium if the discountfactor 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 differentpayoffs can be sustained in equilibrium.

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For example, note that always playing (D,D) is an equilibrium no matter what thevalue of δ is. Each player simply adopts the strategy ‘play D initially, and at any periodt > 1, play D irrespective of history’. Note that both players adopting this strategy is amutual best response. Therefore, we can sustain (1, 1) in equilibrium. In fact, by usingsuitable strategies, we can sustain many more – in fact infinitely more – payoffs asequilibrium outcomes. For the Prisoners’ Dilemma game, we will describe the set ofsustainable payoffs below.

The result about the large set of payoffs that can be sustained as equilibrium outcomesis known as the ‘folk theorem’. These types of results were known to many gametheorists from an early stage of development of non-cooperative game theory. Whileformal 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. Tostate this, we will need to compare payoffs in the repeated game to payoffs in theone-shot game that is being repeated. The easiest way to do that is to normalise therepeated 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 normalisationimplies that the set of normalised repeated game payoffs now coincide with the set ofpayoffs in the underlying one-shot game. So now we can just look at the set of payoffs ofthe one-shot game and ask which of these are sustainable as the normalised payoff insome equilibrium of the repeated game.

For the rest of this section, whenever we mention a repeated game payoff, we alwaysrefer to normalised payoffs. Note that in this game, a player can always get at least 1 bysimply playing D. It follows that a player must get at least 1 as a (normalised) repeatedgame payoff.

To see the whole set of payoffs that can be sustained, let us plot the payoffs from thefour different pure strategy profiles. These are shown in Figure 4.8 below. Now join thepayoffs and form a convex set as shown in the figure. We now have a set of payoffs thatcan arise from pure or mixed strategies.

The Folk theorem is the following claim. Consider any pair of payoffs (π1, π2) suchthat πi > 1 for i = 1, 2. Any such payoff can be supported as an equilibrium payoff forhigh 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 shownas the shaded part in Figure 4.8.

Example 4.2 Consider the following game.

Player 2C D

Player 1 C 3, 2 0, 1D 7, 0 2, 1

(a) Find conditions on the discount factor under which cooperation can besustained in the repeated game in which the above game is repeated infinitely.

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4. Game theory

Figure 4.8: The set of payoffs that can be supported as equilibrium payoffs under aninfinitely-repeated game.

(b) Under what conditions is there an equilibrium in the infinitely-repeated game inwhich players alternate between (C,C) and (D,D), starting with (C,C) in thefirst period?

(c) Draw the set of payoffs sustainable in a repeated game equilibrium according tothe folk theorem.

(a) First, note that player 2 has no incentive to deviate from (C,C). To ensure thatplayer 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

1− δ6

3

1− δwhich 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) issupposed to be played. So the only possible deviation is by player 1 in evenperiods, 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 ofhistory).

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• 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 fromcooperation 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 + 3δ2 + 2δ3 + · · ·

= 3(1 + δ2 + δ4 + · · · ) + 2δ(1 + δ2 + δ4 + · · · )

=3 + 2δ

1− δ2.

If player 1 does deviate in even period t, player 1’s payoff (t onwards) is:

V devt = 7 + 2δ + 2δ2 + · · · = 7 + 2× δ

1− δ.

Therefore, player 1 prefers not to deviate in either period 0 or in any evenperiod if:

3 + 2δ

1− δ2> 7 + 2× δ

1− δwhich simplifies to 5δ2 − 4 > 0, implying:

δ >

√4

5≈ 0.89.

Note also the repeated game payoff generated in this equilibrium. To see this, firstnormalise the payoff by multiplying by (1− δ). The normalised payoff of player 1starting any even period is:

(1− δ)× 3 + 2δ

1− δ2=

3 + 2δ

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) startingany even period, and (1 + 2δ)/(1 + δ) starting any odd period. Note that eitherpayoff goes to 3/2 as δ → 1.

In other words, this exercise shows you an example of an equilibrium that sustains apayoff in the interior of the set of payoffs which is sustainable according to the folktheorem (which, in this case, is anything strictly above (2, 1), or the point (2, 1)itself).

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4. Game theory



analyse simultaneous-move games using dominant strategies or by eliminatingdominated strategies either once or in an iterative fashion

calculate Nash equilibria in pure strategies as well as Nash equilibria in mixedstrategies in simultaneous-move games

explain why Nash equilibrium is the central solution concept and explain theimportance 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 backwardinduction and subgame perfection

analyse the infinitely-repeated Prisoners’ Dilemma game with discounting andanalyse collusive equilibria using trigger strategies

explain the multiplicity of equilibria in repeated games and state the folk theoremfor the Prisoners’ Dilemma game.



1. Consider the strategic form game below with two players, 1 and 2. Solve the gameby iteratively eliminating dominated strategies.

Player 2A2 B2 C2

A1 3, 3 −1, 4 0, 5Player 1 B1 2, 1 3, 2 −1, 0

C1 −1, 0 0, 1 1, 0

2. Identify actions and strategies of each player in each of the following games (Figure4.9 to Figure 4.11).

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Figure 4.9: Perfect information.

Figure 4.10: Imperfect information for player 2.

Figure 4.11: Imperfect information for both players.

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4. Game theory

3. Consider the extensive-form game of imperfect information in Figure 4.12.


(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.



(b) Identify the pure strategy Nash equilibria.

(c) Identify the pure strategy subgame perfect Nash equilibria.

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5. Consider the extensive-form game in Figure 4.14.



(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 4.9in Question 2.

7. Suppose the following game is repeated infinitely. The players have a commondiscount factor δ, where 0 < δ < 1.

Player 2C D

Player 1 C 3, 2 0, 3D 5, 0 2, 1

(a) Find conditions on the discount factor under which cooperation (which impliesplaying (C,C) in each period) can be sustained as a subgame perfect Nashequilibrium of the infinitely-repeated game. Your answer must specify thestrategies of players clearly.

(b) In a diagram, show the set of payoffs that can be supported in a repeatedgame equilibrium according to the folk theorem.

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4. Game theory

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