Risk, Uncertainty and the Agricultural Firm Book

Preface vii

1 Introduction 1

1.1 Formulating the Risk Problem 3

1.2 Decision Criteria 6

1.3 Decision Making Under Risk: Fact and Fiction 8

2 Probability Theory - A Mathematical Basis for Making Decisions Under Risk and Uncertainty 10

2.1 Set Theory and Probability 13

2.2 Random Variables 15

2.3 Conditional Probability and Independence 17

2.4 Some Useful Distribution Functions 20

2.5 Expected Value, Moments, and the Moment Generating Function 28

2.6 Estimating Probability Functions 35

2.6.1 Eliciting Subjective Probability Functions 36

2.6.2 Objective Estimation of Probability Functions 37

2.6.3 Production Functions and Trends 44

2.7 Martingales and Random Walks 52

2.8 Summary 56

3 Expected Utility - The Economic Basis of Decision Making Under Risk 57

3.1 Consumption and Utility 58

3.2 Expected Utility 63

3.2.1 Intuition behind Von Neumann and Morgenstern Proof 65

3.2.2 Conceptual Structure of the Axiomatic Treatment of Numerical Utilities 67

3.2.3 Analytical Examples of Expected Utility 69

3.3 Expected Value - Variance and Expected Utility Models 73

3.4 Problems with Expected Utility 76

3.5 Summary 82

4 Risk Aversion in the Large and Small 83

4.1 Arrow-Pratt Risk Aversion Coefficient 88

4.1.1 Local Risk Aversion 88

4.1.2 Transformations of Scale for the Arrow-Pratt Absolute Risk Aversion Coefficient 90

4.2 Eliciting Risk Aversion Coefficients 95

4.2.1 Direct Elicitation of Utility Functions 95

4.2.2 Equally Likely Risky Prospect and Finding Its Certainty Equivalent (ELCE) 97

4.3 Summary 99

5 Portfolio Theory and Decision Making Under Risk 101

5.1 The Expected Value - Variance Frontier 106

5.2 A Simple Portfolio 108

5.3 A Graphical Depiction of the Expected Value-Variance Frontier 109

5.4 Mean-Variance versus Direct Utility Maximization 109

5.5 Derivation of the Expected Value-Variance Frontier 112

5.5.1 Derivation without a Risk-Free Asset 112

5.5.2 Derivation with a Risk-Free Asset 119

5.6 Summary 122

6 Whole Farm-Planning Models 124

6.1 Farm Portfolio Models 124

6.1.1 Gains to Diversification Using Certainty Equivalence 128

6.1.2 Extension to a Multiperiod Portfolio 129

6.2 Minimize Total Absolute Deviation 131

6.3 Focus-Loss 136

6.4 Target MOTAD 138

6.5 Direct Utility Maximization 139

6.6 Discrete Sequential Stochastic Programming 141

6.7 Chance-Constrained Programming 143

6.8 Interpreting Shadow Values from Risk Programming Models 146

6.9 Summary 148

7 Risk Efficiency Approaches - Stochastic Dominance 149

7.1 Stochastic Dominance 150

7.1.1 The Concept of an Efficiency Criteria 151

7.1.1.1 First-Degree Stochastic Dominance 152

7.1.1.2 Second-Degree Stochastic Dominance 152

7.1.2 Increasing Risk 154

7.1.2.1 Definition Based on Unanimous Preference 157

7.1.2.2 Mean Preserving Spread 157

7.1.2.3 Risk Aversion with Respect to a Function 162

7.2 Applications of Stochastic Dominance 166

7.3 Summary 172

8 Dynamic Decision Rules and the Value of Information 173

8.1 Decision Making and Bayesian Probabilities 173

8.2 Concepts of Information 177

8.3 A Model of Information 180

8.4 Summary 182

9 Market Models of Decision Making under Risk 183

9.1 Risk Equilibrium from the Consumer's Point of View 185

9.2 The Role of the Riskless Asset 189

9.3 Risk Equilibrium from the Firm's Perspective 190

9.3.1 Deriving the Security Market Line 191

9.3.2 Supply of Stocks from the Firm 193

9.3.3 Tests of the CAPM 194

9.3.4 Incorporating Risk using CAPM 194

9.4 Arbitrage Pricing Theorem 196

9.4.1 Single-Factor Model 198

9.4.2 Two-Factor Model 200

9.5 Empirical Applications of Capital Market Models 202

9.5.1 Capital Asset Pricing Models 202

9.5.2 Tests for CAPM Efficiency 204

9.5.3 Cross-Section Regression 205

9.5.4 Arbitrage Pricing Model 206

9.6 Summary 208

10 Option Pricing Approaches to Risk 209

10.1 Introduction to Options and Futures 209

10.1.1 Futures and the Hedge 209

10.1.2 Options 211

10.1.3 Option Pricing using Black-Scholes 215

10.2 Real Option Valuation 219

10.2.1 Derivation of the Value of Waiting 220

10.2.2 Application to Citrus 223

10.3 Crop Insurance 223

10.4 Summary 226

11 State Contingent Production Model: The Stochastic Production Set 227

11.1 Depicting Risk and Input Decisions in the Production Function 227

11.1.1 Estimation of Stochastic Production Functions using Quantile Regression 228

11.1.2 Developing the State-Space Representation 234

11.2 State Production Set and Input Requirement Set 237

11.3 Distance Functions and Risk Aversion 238

11.3.1 Defining the Distance Function in State-Contingency Space 238

11.3.2 Risk Aversion and Valuing States 240

11.3.3 Duality, Benefit, and Distance Functions 241

11.3.4 Defining Risk Aversion Graphically 243

11.3.5 Constant Relative and Absolute Risk Aversion 244

11.3.5.1 Risk Premium 244

11.3.5.2 Derivation of the Effort Function 246

11.4 Summary 249

12 Risk, Uncertainty, and the Agricultural Firm - A Summary and Outlook 250

Appendix A Measure Theory and the Justification of Random Variables 253

Appendix B Derivation of the Moments of the Inverse Hyperbolic Sine Distribution 257

Appendix C Numerical Techniques for Applied Optimization and Solution of Nonlinear Systems of Equations 265

Appendix D An Axiomatic Development of Expected Utility 268

Appendix E A GAMS Program to Select Optimal Portfolios 276

Appendix F R Program to Derive Optimum Portfolio with and without a Risk-Free Asset 278

Appendix G Program to Compute the Efficient Frontier with and without a Risk-Free Asset 281

Appendix H GAMS Program for the Portfolio Problem 283

Bibliography 285

Index 291