This textbook incorporates the author’s previous book "The Economics of Uncertainty and Insurance" and extends it with the addition of several new chapters on risk sharing, asymmetric information, adverse selection, signaling and moral hazard. It provides a comprehensive introduction to the analysis of economic decisions under uncertainty and to the role of asymmetric information in contractual relationships. It is richly illustrated with 150 figures. It is suitable for both self-study and as the basis for an upper-division undergraduate course. The book is written to be accessible to anyone with minimum knowledge of calculus, in particular the ability to calculate the (partial) derivative of a function of one or two variables. The book contains a total of 150 fully solved exercises.

1 Introduction.................................................................................................... 9

Part I: Insurance

2 Insurance: basic notions.................................................................... 15

2.1 Uncertainty and lotteries 15

2.2 Money lotteries and attitudes to risk 16

2.3 Certainty equivalent and the risk premium 19

2.4 Insurance: basic concepts 21

2.5 Isoprofit lines 25

2.6 Profitable insurance requires risk aversion 30

2.6.1 Insuring a risk-neutral individual....................................................................... 30

2.6.2 Insuring a risk-averse individual....................................................................... 31

2.6.3 The profit-maximizing contract for a monopolist.......................................... 32

2.6.4 Perfectly competitive industry with free entry............................................... 34

2.7 Exercises 36

2.7.1 Exercises for Section 2.2: Money lotteries and attitudes to risk................... 36

2.7.2 Exercises for Section 2.3: Certainty equivalent and risk premium.............. 38

2.7.3 Exercises for Section 2.4: Insurance: basic concepts................................... 39

2.7.4 Exercises for Section 2.5: Isoprofit lines........................................................... 40

2.7.5 Exercises for Section 2.6: Profitable insurance requires risk aversion........... 42

2.8 Solutions to Exercises 42

3 Expected Utility Theory......................................................................... 55

3.1 Expected utility: theorems 55

3.2 Expected utility: the axioms 63

3.3 Exercises 71

3.3.1 Exercises for Section 3.1: Expected utility: theorems.................................... 71

3.3.2 Exercises for Section 3.2: Expected utility: the axioms................................. 73

3.4 Solutions to Exercises 74

4 Money lotteries revisited...................................................................... 79

4.1 von Neumann Morgenstern preferences over money lotteries 79

4.1.1 The vNM utility-of-money function of a risk-neutral agent........................... 79

4.1.2 Concavity and risk aversion............................................................................ 80

4.1.3 Convexity and risk loving.................................................................................. 83

4.1.4 Mixtures of risk attitudes.................................................................................... 84

4.1.5 Attitude to risk and the second derivative of the utility function............... 85

4.2 Measures of risk aversion 86

4.3 Some noteworthy utility functions 92

4.4 Higher risk 93

4.4.1 First-order stochastic dominance................................................................... 94

4.4.2 Mean preserving spread and second-order stochastic dominance . . . 95

4.5 Exercises 99

4.5.1 Exercises for Section 4.1: vNM preferences over money lotteries............... 99

4.5.2 Exercises for Section 4.2: Measures of risk aversion..................................... 100

4.5.3 Exercises for Section 4.3: Some noteworthy utility functions...................... 102

4.5.4 Exercises for Section 4.4: Higher risk.............................................................. 103

4.6 Solutions to Exercises 104

5 Insurance: Part 2.................................................................................... 113

5.1 Binary lotteries and indifference curves 113

5.1.1 Case 1: risk neutrality...................................................................................... 114

5.1.2 Case 2: risk aversion....................................................................................... 115

5.1.3 Case 3: risk love............................................................................................... 117

5.1.4 The slope of an indifference curve............................................................... 118

5.2 Back to insurance 121

5.2.1 The profit-maximizing contract for a monopolist........................................ 124

5.2.2 Perfectly competitive industry with free entry............................................. 125

5.3 Choosing from a menu of contracts 127

5.3.1 Choosing from a finite menu......................................................................... 127

5.3.2 Choosing from a continuum of options....................................................... 128

5.4 Mutual insurance 138

5.5 Exercises 140

5.5.1 Exercises for Section 5.1: Binary lotteries and indifference curves............ 140

5.5.2 Exercises for Section 5.2: Back to insurance................................................ 141

5.5.3 Exercises for Section 5.3: Choosing from a menu of contracts................. 143

5.5.4 Exercises for Section 5.4: Mutual insurance................................................. 146

5.6 Solutions to Exercises 147

Part II: Risk Sharing

6 Risk Sharing and Efficiency............................................................ 165

6.1 Sharing an uncertain surplus 165

6.2 The Edgeworth box 167

6.3 Points of tangency 175

6.3.1 Risk averse Principal and risk neutral Agent................................................ 175

6.3.2 Risk neutral Principal and risk averse Agent................................................ 176

6.3.3 A general principle......................................................................................... 178

6.3.4 Both parties risk averse................................................................................... 178

6.3.5 Both parties risk neutral................................................................................... 181

6.3.6 Pareto efficiency for contracts in the interior of the Edgeworth box . . . 182

6.4 Pareto efficient contracts on the sides of the Edgeworth box 183

6.4.1 Risk averse Principal and risk neutral Agent................................................ 183

6.4.2 Risk neutral Principal and risk averse Agent................................................ 184

6.4.3 Both parties risk averse................................................................................... 186

6.5 The Edgeworth box when the parties have positive initial wealth 187

6.6 More than two outcomes 193

6.6.1 Risk-neutral Principal and risk-averse Agent................................................ 194

6.6.2 Risk-averse Principal and risk-neutral Agent................................................ 197

6.6.3 Both parties risk neutral................................................................................... 197

6.6.4 Both parties risk averse................................................................................... 198

6.7 Exercises 199

6.7.1 Exercises for Section 6.1: Sharing an uncertain surplus.............................. 199

6.7.2 Exercises for Section 6.2: The Edgeworth box.............................................. 199

6.7.3 Exercises for Section 6.3: Points of tangency............................................... 201

6.7.4 Exercises for Section 6.4: Pareto efficient contracts on the sides of the Edgeworth box 204

6.7.5 Exercises for Section 6.5: The Edgeworth box when the parties have positive initial wealth 206

6.7.6 Exercises for Section 6.6: More than two outcomes................................... 207

6.8 Solutions to Exercises 209

Part III: Asymmetric Information: Adverse Selection

7 Adverse Selection................................................................................. 227

7.1 Adverse selection or hidden type 227

7.2 Conditional probability and belief updating 229

7.2.1 Conditional probability.................................................................................. 230

7.2.2 Belief updating................................................................................................ 231

7.3 The market for used cars 234

7.3.1 Possible remedies............................................................................................ 241

7.3.2 Further remarks................................................................................................ 241

7.4 Exercises 243

7.4.1 Exercises for Section 7.2.2: Conditional probability and belief updating 243

7.4.2 Exercises for Section 7.3: The market for used cars..................................... 245

7.5 Solutions to Exercises 247

8 Adverse Selection in Insurance.................................................... 253

8.1 Adverse selection in insurance markets 253

8.2 Two types of customers 254

8.2.1 The contracts offered by a monopolist who can tell individuals apart 256

8.3 The monopolist under asymmetric information 257

8.3.1 The monopolist’s profit under Option 1........................................................ 258

8.3.2 The monopolist’s profit under Option 2........................................................ 259

8.3.3 The monopolist’s profit under Option 3........................................................ 263

8.3.4 Option 2 revisited............................................................................................ 274

8.4 A perfectly competitive insurance industry 276

8.5 Exercises 283

8.5.1 Exercises for Section 8.2: Two types of customers....................................... 283

8.5.2 Exercises for Section 8.3: The monopolist under asymmetric information 284

8.5.3 Exercises for Section 8.4: A perfectly competitive insurance industry . . 286

8.6 Solutions to Exercises 287

Part IV: Asymmetric Information: Signaling

9 Signaling..................................................................................................... 297

9.1 Earnings and education 297

9.2 Signaling in the job market 299

9.2.1 Signaling equilibrium....................................................................................... 299

9.2.2 Pareto inefficiency.......................................................................................... 301

9.2.3 Alternative interpretation of a signaling equilibrium.................................. 302

Indices versus signals	305
More than two types	309
A more general analysis	311
Signaling in other markets	322
Market for used cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	322
Advertising as a signal of quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	323
Other markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	324
Exercises	325
Exercises for Section 9.2: Signaling in the job market . . . . . . . . . . . . . . . .	325
Exercises for Section 9.3: Indices versus signals . . . . . . . . . . . . . . . . . . . . .	328
Exercises for Section 9.4: More than two types . . . . . . . . . . . . . . . . . . . . .	329
Exercises for Section 9.5: A more general analysis . . . . . . . . . . . . . . . . . .	330
Exercises for Section 9.6: Signaling in other markets . . . . . . . . . . . . . . . .	330
Solutions to Exercises	331
Part V: Moral Hazard
Moral Hazard in Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	343
Moral hazard or hidden action	343
Moral hazard and insurance	344
Two levels of unobserved effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	345
The reservation utility locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	348
The profit-maximizing contract for a monopolist . . . . . . . . . . . . . . . . . . .	354
Exercises	359
Exercises for Section 10.2.1: Two levels of unobserved effort . . . . . . . . .	359
Exercises for Section 10.2.2: The reservation utility locus . . . . . . . . . . . . .	361
Exercises for Section 10.2.3: The profit-maximizing contract . . . . . . . . . .	362
Solutions to Exercises	363
Moral Hazard in Principal-Agent . . . . . . . . . . . . . . . . . . . . . . .	369
Moral hazard in Principal-Agent relationships	369
Risk sharing under moral hazard	370
The case with two outcomes and two levels of effort	374
The case with more than two outcomes	388
Exercises	393
Exercises for Section 11.2: Risk sharing under moral hazard . . . . . . . . . .	393
Exercises for Section 11.3: Two outcomes and two levels of effort . . . . .	394
Exercises for Section 11.4: The case with more than two outcomes . . .	398
Solutions to Exercises	400