## Nonlinear PricingWhat do phone rates, frequent flyer programs, and railroad tariffs all have in common? They are all examples of nonlinear pricing. Pricing is nonlinear when it is not strictly proportional to the quantity purchased. The Electric Power Research Institute has commissioned Robert Wilson to review the various facets of nonlinear pricing. The work starts with a general non-mathematical discussion, followed by a more technical presentation intended for readers with a fairly advanced background. Thorough and detailed, this study has ample examples of case studies from a variety of industries. |

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### Contents

Introduction | 3 |

11 Motivations for Nonlinear Pricing | 5 |

12 Practical Uses for Nonlinear Pricing | 7 |

13 Feasibility of Nonlinear Pricing | 10 |

The Electric Power Industry | 15 |

15 Overview of the Chapters | 17 |

16 Summary | 23 |

FUNDAMENTALS OF NONLINEAR PRICING | 25 |

85 A General DemandProfile Formulation | 200 |

86 Summary | 206 |

MULTIDIMENSIONAL TARIFFS | 209 |

Multidimensional Pricing | 211 |

91 A Multidimensional Formulation | 212 |

92 Design of Multidimensional Tariffs | 219 |

93 Examples | 223 |

94 An ItemAssignment Formulation | 228 |

Illustrations of Nonlinear Pricing | 27 |

21 Time and Newsweeks Advertising Rates | 28 |

22 EDFs Tariffs for Electric Power | 30 |

23 PGEs Curtailable Service | 36 |

24 MCIs and ATTs Telephone Tariffs | 39 |

25 Federal Express Mail Rates | 41 |

27 Summary | 44 |

Models and Data Sources | 46 |

31 Descriptions of Demand Behavior | 49 |

32 Estimation of the Demand Profile | 57 |

33 Welfare Considerations | 62 |

34 Cautions and Caveats | 66 |

35 Standard Assumptions | 67 |

36 Summary | 71 |

Tariff Design | 73 |

42 Extensions and Qualifications | 81 |

43 The Bundling Interpretation | 88 |

44 Fixed Costs and Fixed Fees | 89 |

45 Multipart Tariffs | 91 |

46 Summary | 96 |

Ramsey Pricing | 98 |

51 The Price Schedule of a Regulated Firm | 99 |

52 ParetoImproving Tariffs | 108 |

53 LongDistance Telephone Tariffs | 118 |

54 Summary | 121 |

DISAGGREGATED DEMAND MODELS | 123 |

SingleParameter Disaggregated Models | 125 |

61 A Model with Discrete Types | 126 |

62 Models with OneDimensional Types | 132 |

63 TwoPart Tariffs | 136 |

64 Multipart Tariffs | 141 |

65 Nonlinear Tariffs | 150 |

66 Some Examples | 153 |

67 Fixed Costs and Fixed Fees | 158 |

68 Summary | 163 |

INCOME EFFECTS | 164 |

72 Characterization of the Optimal Tariff | 165 |

73 Examples | 166 |

74 Fixed Fees | 173 |

75 Summary | 175 |

Technical Amendments | 177 |

82 Derivations of the Optimal Tariff | 185 |

83 Approximate Optimality of Multipart Tariffs | 190 |

84 Multiple Dimensions of Customers Types | 194 |

95 Summary | 235 |

Priority Pricing | 236 |

Priority Pricing | 237 |

102 Ramsey Pricing of Priority Service | 244 |

103 Nonlinear Pricing in a Priority Queuing System | 253 |

104 Summary | 257 |

Capacity Pricing | 259 |

112 Cost Recovery of LoadDuration Demands | 261 |

113 Pricing LoadDuration Demands | 266 |

114 Summary | 276 |

MULTIPRODUCT TARIFFS | 279 |

Multiproduct Products and Competitive Tariffs | 281 |

121 Separable Multiproduct Tariffs | 282 |

122 Monopolistic Competition | 301 |

123 Oligopoly | 303 |

124 Nonlinear Supply Schedules | 309 |

125 Summary | 311 |

Multiproduct Pricing | 313 |

131 Special Features of Multiproduct Tariffs | 314 |

132 Formulation | 317 |

133 A Reprise of Priority Service | 325 |

134 Single Type Parameters | 327 |

135 Multiple Type Parameters | 332 |

136 Construction of Multiproduct Prices | 344 |

137 Summary | 348 |

Multiproduct Tariffs | 350 |

141 Formulation | 351 |

142 The Optimal Multiproduct Tariff | 354 |

143 Numerical Examples | 359 |

LoadFactor Tariffs | 363 |

145 An Alternative Formulation | 368 |

146 Summary | 371 |

SUPPLEMENT | 373 |

Other Applications of Nonlinear Pricing | 375 |

152 Incentives in Markets with Private Information | 382 |

153 Incentives in Labor Contracting | 390 |

154 Taxation and Regulatory Policies | 395 |

155 Summary | 398 |

Bibliography | 401 |

162 Chapter References | 406 |

References | 411 |

423 | |

### Common terms and phrases

aggregate aggregate demand analysis applications assignment assume assumption benefit function boundary bundle q capacity chapter constraint construction consumer's surplus customer of type customers purchasing demand charges demand function demand profile depends derived described distribution function efficient energy charges example Figure firm firm's profit fixed cost fixed fee formulation g-th increment g-th unit income effects increasing indicates industries interpreted ironing procedure Lagrange multiplier marginal charges marginal cost marginal price schedule maximize measures menu multipart tariffs multiple multiproduct tariffs nonlinear pricing nonlinear tariff number of customers obtained offered optimal price schedule optimal tariff optimality condition options price differential price discrimination price elasticity priority service producer's surplus profit contribution profit margin provides purchase q purchase set purchase sizes q-th quantity discounts quantity q Ramsey number Ramsey pricing reliability represents Section selected specified substitution effects supply two-part tariff uniform price usage utility values volume bands Wright tariff zero