U NIVERSIT E´ DE T OULOUSE 1 – S CIENCES S OCIALES M IDI -P YR E´ N E´ ES S CIENCES E´ CONOMIQUES

T H E` SE ´ Pour le Doctorat es Sciences Economiques

´ GESTION ET REGLEMENTATION DES INFRASTRUCTURES

Pr´esent´ee et soutenue publiquement le lundi 9 d´ecembre 2002 par C YRIL HARITON

sous la direction de ´ JACQUES CR EMER

M EMBRES DU J URY

´ Bernard CAILLAUD, Professeur a` l’Ecole Nationale des Ponts et Chauss´ees (rapporteur) ´ Jacques CREMER, Directeur de recherche C NRS a` l’Universit´e Toulouse 1 G´erard GAUDET, Professeur a` l’Universit´e de Montr´eal (rapporteur) Bruno JULLIEN, Directeur de Recherche C NRS a` l’Universit´e Toulouse 1 Patrick REY, Professeur a` l’Universit´e Toulouse 1

L’Universit´e n’entend ni approuver ni d´esapprouver les opinions particuli`eres du candidat.

Table des mati`eres D´eroulement de la th`ese

i

Remerciements

v

Table des mati`eres

xvi

Table des figures

xvii

Liste des tableaux

xix

Principaux th`emes abord´es

1

Th´ematiques de la th`ese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Charges d’acc`es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

R´eglementation et biens indispensables a` la production . . . . . . . . . . . . . . .

5

Allocation de l’acc`es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Organisation des chapitres et principaux r´esultats . . . . . . . . . . . . . . . . . . . . . 10 Chapitre 1 : Nature des charges d’acc`es . . . . . . . . . . . . . . . . . . . . . . . 10 Chapitre 2 : R´eglementation, facilit´e essentielle et extraction de rente . . . . . . . 13 Chapitre 3 : Gestion d’infrastructure dans un contexte dynamique . . . . . . . . . 15 Chapitre 4 : Ench`ere optimale en pr´esence d’un coˆut exog`ene des transferts . . . . 18 1

The nature of access charge

21

(Choix de la structure de charge d’acc`es) 1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

xi

` TABLE DES MATIERES

xii 1.2

Tradeoff in public economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3

Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4

Symmetric oligopoly with homogeneous good . . . . . . . . . . . . . . . . . . . . . 30 1.4.1

Symmetric Generalized Cournot . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4.2

Symmetric free entry oligopoly . . . . . . . . . . . . . . . . . . . . . . . . 32

1.5

The programs of the regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6

Pure ad valorem access charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.7

Optimal prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.8

Unrestricted regulatory tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.9

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.A.1 Proof of lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.A.2 Proof of proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.A.3 Proof of proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.A.4 Proof of proposition 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.A.5 Proof of lemma 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2

Regulation, essential facility and rent extraction

53

(R´eglementation, facilit´e essentielle et extraction de rente) 2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2

Regulation with exogenous cost of public funds . . . . . . . . . . . . . . . . . . . . 59

2.3

Benchmark cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4

2.3.1

Regulation of both firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3.2

No regulation at all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Socially harmful regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4.1

Strategy of the regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4.2

Strategy of the upstream firm . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.5

Endogenous decision to regulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.6

Unregulated national upstream firm . . . . . . . . . . . . . . . . . . . . . . . . . . 69

` TABLE DES MATIERES

2.7

2.8

2.9

xiii

Shared bargaining power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.7.1

Bargaining procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.7.2

Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.7.3

General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Asymmetric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.8.1

Model with asymmetric information . . . . . . . . . . . . . . . . . . . . . . 79

2.8.2

Analysis of contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Regulation under budget constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.9.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.9.2

No regulation of market 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.9.3

Regulation of firm D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.9.4

Choosing whether or not to regulate . . . . . . . . . . . . . . . . . . . . . . 89

2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.A Annex: Asymmetric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.A.1 Benchmark cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Regulation of both firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 No regulation of the downstream firm . . . . . . . . . . . . . . . . . . . . . 95 2.A.2 Socially harmful regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Strategy of the regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Strategy of the upstream firm . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.A.3 Endogenous decision to regulate . . . . . . . . . . . . . . . . . . . . . . . . 97 2.B Annex: Ramsey-Boˆıteux framework . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.C Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.C.1

Proof of lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.C.2

Proof of lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.C.3

Proof of proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

2.C.4

Proof of lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.C.5

Proof of lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.C.6

Proof of lemma 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

` TABLE DES MATIERES

xiv

2.C.7

Proof of lemma 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.C.8

Proof of lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.C.9

Proof of lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.C.10 Proof of lemma 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.C.11 Proof of lemma 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.C.12 Proof of lemma 2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3

Dynamic allocation of access

117

(Gestion de l’acc`es a` une infrastructure dans un contexte dynamique) 3.1

3.2

3.3

3.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.1.1

Internet access pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.1.2

Related access pricing concerns in other industries . . . . . . . . . . . . . . 122

3.1.3

A general model of congestion . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.1.4

Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.1.5

Roadmap of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Smart market mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2.1

A story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.2

Strategy of the customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.2.3

The profit of the seller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A two types case of the general model . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.3.1

Model with discrete types . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.3.2

Problem of the seller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.3.3

Optimal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.3.4

An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A continuous version of the general model . . . . . . . . . . . . . . . . . . . . . . . 143 3.4.1

Model with continuous types . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.4.2

Problem of the auctioneer . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.4.3

Maximization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.4.4

Perfect information case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

` TABLE DES MATIERES

3.4.5 3.5

3.6

xv

Asymmetric information case . . . . . . . . . . . . . . . . . . . . . . . . . 149

Properties of the allocation mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 151 3.5.1

Efficient mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3.5.2

Profit maximizing mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.5.3

Short vs Long term buyers: Perfect information case . . . . . . . . . . . . . 156

3.5.4

Short vs Long term buyers: Asymmetric information case . . . . . . . . . . 159

3.5.5

Efficient vs Profit maximizing allocation mechanisms . . . . . . . . . . . . . 161

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

3.A Appendix: “Smart market” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.A.1 A note on the solution to equation (3.6) . . . . . . . . . . . . . . . . . . . . 166 3.A.2 Proof of proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.B Appendix: Two types case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.B.1

Proof of lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.B.2

Proof of lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.B.3

Proof of lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.B.4

Proof of lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.B.5

Proof of lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.B.6

Proof of proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.C Appendix: Continuous model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4

3.C.1

Proof of lemma 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.C.2

Proof of lemma 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.C.3

Proof of lemma 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.C.4

Proof of lemma 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.C.5

Proof of lemma 3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

3.C.6

Proof of proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Optimal auction with exogenous transfer costs

179

ˆ exog`ene des transferts mon´etaires) (Ench`ere optimale en pr´esence d’un cout 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

` TABLE DES MATIERES

xvi

4.2

4.3

Allocation framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.2.1

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.2.2

Optimization program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.2.3

Perfect information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Optimal and efficient mechanisms under transfer costs . . . . . . . . . . . . . . . . 187 4.3.1

Equivalent optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 187

4.3.2

Allocation rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.3.3

Auctioneer vs firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.4

Example: Symmetric bidders with uniform distribution . . . . . . . . . . . . . . . . 196

4.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

4.B Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.B.1

Constraint on total (expected) transfers rather than individual (expected) transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

4.B.2

Proof of lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

4.B.3

Proof of lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

4.B.4

Proof of proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4.B.5

Proof of proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.B.6

Proof of lemma 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Conclusion g´en´erale

209

Bibliographie

213

Table des figures 1.1

Relative positions of specific taxes yielding either fiscal revenues R∗ or Ramsey price p∗1 and the impact in term of downstream profits . . . . . . . . . . . . . . . . . . . . 44

2.1

The industry vertical structure with exogenous cost of public funds: without (left) or with (right) regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2

Timing of the game: regulation vs no production . . . . . . . . . . . . . . . . . . . 62

2.3

Timing of the game: regulation vs vertical contract . . . . . . . . . . . . . . . . . . 67

2.4

Thresholds for social welfare and firm U’s profit relative to bargaining powers of firm U with respect to the regulator and firm D . . . . . . . . . . . . . . . . . . . . 76

2.5

The industry vertical structure under budget constraint: without (left) or with (right) regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.6

Thresholds for social welfare and firm U’s profit, when firm U’s unregulated profit do not benefit public funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.1

Demand for access in the analysis of “smart market” . . . . . . . . . . . . . . . . . 131

3.2

Bidding behavior of agents in a “smart market”. . . . . . . . . . . . . . . . . . . . . 132

3.3

Stationary bidding function in its first period auction for any agent. . . . . . . . . . . 133

3.4

Demand for access in the two-types model. . . . . . . . . . . . . . . . . . . . . . . 137

3.5

Optimal stationary profit with perfect information . . . . . . . . . . . . . . . . . . . 149

xvii

Liste des tableaux 2.1

Results on the opportunity of regulating the downstream firm . . . . . . . . . . . . . 68

2.2

Results on the opportunity of regulating the downstream firm, with a domestic upstream firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.3

Results on the opportunity of regulating the downstream firm, with asymmetric information on its cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.4

Influence of cost cross-derivatives on equilibrium outcomes . . . . . . . . . . . . . . 114

3.1

Number of hosts advertised in the Domain Name System. . . . . . . . . . . . . . . . 118

3.2

Fragmentation of Internet players (1994 – 2002). . . . . . . . . . . . . . . . . . . . 119

3.3

Example of connection pricing schemes (U UNET T3 connection, 1998). . . . . . . . 121

3.4

Possible optimal allocation probabilities in the two types case . . . . . . . . . . . . . 141

3.5

Numerical examples of parameters values inducing each optimal strategy (δ = 0.9) . 142

3.6

Optimal allocation strategies with identical per period valuations . . . . . . . . . . . 143

xix

Chapitre 4 ˆ Ench`ere optimale en pr´esence d’un cout exog`ene des transferts mon´etaires (Optimal auction with exogenous transfer costs)

4.1

Introduction

How should an auctioneer optimally allocate one good when, first, it faces bidders with independent private valuations for the good and, second, it costs him λ euro to demand 1 euro to any bidder? In particular, how different are the allocation strategies of a profit maximizing firm and a social welfare maximizing regulator in this framework? This chapter studies this problem and shows the following results. First, a profit maximizing seller does not modify its allocation process when it faces transfer cost. Second, a social welfare maximizing regulator finds it beneficial to change its allocation rule. For some value of the transfer cost, it may still screen potential bidders but, for large values, it may well prefer to abandon screening and allocate the good randomly among buyers. In this later case, the auction process is not based on bidders’ willingness to pay. More generally, a social welfare maximizing regulator chooses to screen among bidders’ types whenever the profit maximizing firm prefers not to do so, while the firm finds it profitable to screen whenever the regulator chooses not to do so. It should be emphasized that these results are not related to any technicalities (meaning that

179

180

CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

without the cost of transfer but keeping other parameters identical, the social planner could perfectly screen bidders) but are rather induced by the nature of transfer cost. Optimal auctions.

Optimal auctions have been extensively studied since Myerson (1981) and

Riley and Samuelson (1981) and still are a very active research area in economic theory.1 In his seminal paper, Myerson exhibits how one can reduce the allocation problem of a profit maximizing auctioneer in presence of asymmetric information with respect to buyers’ willingness to pay to a simple comparison of virtual valuations. These later correspond to the valuation vi agent i attaches to the good minus the cost for the auctioneer induced by the need to avoid that this buyer vi pretends to be another (less valued) one. Thus, the main problem remains to give the right incentives for buyers to reveal their private information. Afterwards, the comparison is straightforward as far as virtual valuations are non decreasing in the buyer’s own valuation vi : the auctioneer can perfectly screen agents regarding their willingness to pay. When this is not the case, the problem becomes (a little bit) more technical and some bunching appears. Bunching corresponds to an area of valuations in which the auctioneer does not screen buyers, that is when all buyers’s valuations lie in this area, the auctioneer does allocate the good randomly among them. One of the main lesson brought by Myerson in this paper is that usual distribution functions of valuation (uniform e.g.) do not generate any bunching technicality so that a profit maximizing auctioneer bases its allocation procedure upon truthfully revealed buyers’ valuations. The same technique can be used to design the allocation procedure of a social welfare maximizing social planner. In this later case, transfers are settled down for the only purpose of inducing revelation of willingness to pay and the good is allocated to the agent with the highest valuation. Thus, transfers do not interfere with the objective function of the social planner which designs its allocation procedure only on the basis of bidders’ valuations. These results are by now rather standard. Nevertheless, one has to remind that when transfers are costless in an allocation procedure, the social planner suffers no cost to build incentives for bidders to reveal truthfully their willingness to pay, contrary to a profit maximizing auctioneer. 1 Please

auctions.

refer to Klemperer (1999) for a survey of the auction literature and, more specifically, section 4 on optimal

4.1. INTRODUCTION

Regulation.

181

This caveat has been recognized by the regulation literature. sIn the stylized set-

ting popularized by Laffont and Tirole (1993),2 the social planner has to finance a natural monopoly, with unknown marginal cost, but does not benefit from costless lump-sum transfers. Taking 1 euro from consumers to allocate them to the natural monopoly costs to the society (1 + λ) euros, i.e. there is a pure economic loss of λ euro as a consequence of the transfer. The rationale for this cost of public funds λ is the distortions induced by the use of an imperfect tax system to finance the State budget. The more funds are required, the more distortions are induced by this imperfection. One of the main lesson of this literature is that, in the presence of this cost of public funds, even if the regulator decides (usually)3 to screen among the monopoly cost-types, it cannot achieve first best allocation but has to trade off between rent extraction by the natural monopoly (which has to be financed by the State) and production efficiency. Cost of transfers. Thus, the absence of costless transfers does have an influence on the allocation process and one can wonder what is the consequence of “some kind” of transfer cost on the optimal and on the efficient allocation mechanisms of one good to independent buyers. This is the subject of the present chapter which studies a classical problem of allocation, in a framework a` la Myerson, but in the presence of costly (money) transfers. When any buyer pays 1 euro to the auctioneer, this later only receives (1 − λ) euro, the difference being lost. This depicts a second-best world where inefficiency arises by assumption (as long as λ > 0). Even though this cost mimics in a way the one used in the regulation literature, one should think of the cost of transfer as a different shortcut: there is a priori no imperfect taxation story in the case studied in this chapter.4,5 2 Please

refer to Laffont and Tirole (1993) for further references and details, especially assumption 8 in the introductive chapter, p. 38 and chapter 3.9, pp. 194–199. 3 As in the optimal auction problem, the screening result relies on some technical assumptions: monotonicity of the hazard rate (assumption 1.2, p. 66 – please refer to Bagnoli and Bergstr¨om (1989) for more on the subject). This is the equivalent, in their setting, to the increasing virtual valuation assumption in the optimal auction framework with independent private values. 4 In Laffont and Tirole’s (1993) chapter 7 on auctioning incentive contracts, the authors compute the optimal auction in a regulatory/procurement context in the presence of costly public funds. Nevertheless, the basic trade-off faced by the auctioneer remains the same: it prefers to get 1 euro from the firms than from consumers because it limits the distortions related to raising funds from consumers through a distortive tax scheme. 5 Dana and Spier (1994), for example, use the cost of public funds in an optimal auction where the auctioneer selects both the (downstream) market structure and granted buyer(s). The story in their work is directly related to the Laffont and Tirole’s (1993) one, that is the regulator prefers to raise money from firms through payments for the auctioned good than from consumers through distorting taxation. Thus, any euro taken from firms is valued as 1 + λ euro taken from consumers.

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First, it can of course result from the administrative cost associated with the treatment of transfers. Alternatively, it can also fit the case where the allocation process is done by an independent regulator that will not benefit from the whole budget raised by the auction, the regulator thinking at this lost budget as lost for the economy (corruption with accounts abroad e.g.). Second and more interestingly, the buyer may anticipate renegotiation of the payment after the allocation took place. In the presence of lobbying, for example, a regulator may anticipate that pressure groups would first secure the allocation of a procurement contract and, then, renegotiate the amount selected during the selection procedure. Another example is related to future shocks. When the good sold is a durable input, the value of the good for the bidder is the expected future gains attached to its use. When buyers have limited liability, a shock in these expected benefits may leave the buyer unable to sustain the price agreed upon during the auction process. This has arisen in the recent public sale of spectrum rights in the telecom sector, when several major players in this industry, which owned spectrum rights, felt (or nearly felt) in bankruptcy.6 Thus, a cost of transfers can arise in presence of limited liability and shocks, lobbying or uncertainty in judicial processes. One main difference must be emphasized between the two costs: the cost of transfers is a “oneway” effect while the cost of public funds in the regulatory literature is “two-ways”. “One-way” means that this effect takes place only when money is given by bidders to the auctioneer. Were the auctioneer to subsidize one bidder with negative transfers by 1 euro, then this same auctioneer would value the transfer as 1/1 − λ > 1 euros. This is equivalent to the creation of money in the economy. Such a behavior is restricted by the assumption of non negative transfers.7 This assumption does not arise in the regulatory literature. Indeed, the revenue requirement of the State, that gives rise to the distortions related to an imperfect tax system, exists outside the regulatory concern. Regulation will either increase or decrease the need for public funds. On the one hand, raising the monopoly rent by 1 euro requires to increase the State budget by 1 + λ euros. On the other hand, extracting 1 euro of rent from the monopoly, if this later has positive profits, allows to decrease the State budget by 1 + λ 6 This

example is not fully satisfactory as it is probably better described by a common value environment. one could only assume that globally, transfers do not create money, i.e. the sum of all transfers is non negative. This assumption does not modify the nature of the results exhibited in section 4.3. Please refer to section 4.B.1 in the appendix, p. 201. 7 Alternatively,

4.2. ALLOCATION FRAMEWORK

183

euros. Thus, the equivalence between 1 euro from the firm and 1 + λ euros from consumers is true for negative and positive transfers in the regulatory literature. The chapter is organized as follows. Section 4.2 presents the allocation framework, a` la Myerson, and introduces notations as well as the optimization problems to be solved by either a benevolent regulator or a profit maximizing firm. Their respective solutions are detailed in section 4.3. Section 4.4 studies a simple instance of the general problem with uniformly distributed valuations and symmetric bidders, in order to better understand the type of solution that arises in this framework and the main lessons one can get from the consequences of the presence of transfer cost.

4.2 4.2.1

Allocation framework Notations

In this setting, an auctioneer wishes to allocate one good among I potential buyers. Each bidder i ∈ {1, . . . , I} attaches a positive value vi ∈ [vi ; vi ] to the consumption of this good. The notation v stands for {v1 , . . . , vI } and v−i for v\vi . The willingness to pay vi is a private information only known by agent i, while the cumulative distribution function of valuations F i (vi ) : [vi ; vi ] → [0; 1], with associated density f i (vi ), is common knowledge among all agents and the auctioneer. The probability distribution of each agent i is assumed to be strictly increasing over [vi ; vi ], so that its −1

inverse function F i (vi ) : [0; 1] → [vi ; vi ] exists and is strictly increasing. The objective of the auctioneer is to identify a mechanism {pi (v) , t i (v)}i=1,...,I , where pi (vi , v−i ) is the probability of agent i to be allocated the object given other bidders proposed valuations v−i and t i (vi ) is its expected paiement, that fulfills its objective function. This later is the expectation on all bidders’ valuation of the transfers it gets from bidders plus their weighted utility, with a weight α.   The expected probability for agent i to get the good is noted qi (vi ) = Ev−i pi (v) and the expected utility faced by any bidder i is U i (vi ) = qi (vi ) vi − t i (vi ). When the weight α is equal to 0, the auctioneer maximizes only net transfers. This corresponds to a profit maximizing firm. On the contrary, when α = 1, then the auctioneer maximizes the sum of bidders’ utility and net transfers,

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

i.e. social welfare. It acts as a benevolent regulator. As these names will repeatedly be used in the reminding of the text, let give the following definition. Definition 4.1. For an arbitrary value of the parameter α, the seller is called an “auctioneer”. When α = 1, it is called a (social welfare maximizing) “regulator”. When α = 0, it is a (profit maximizing) “firm”. Introducing the parameter α allows to treat both the firm and the regulator cases at the same time, but this parameter has other reasons to be. For example, when buyers are firms, the regulator can have redistributive objectives that induce undervaluation of firm’s profits. Or, alternatively, buyers can also be foreign undertakings whose utility does not entirely benefit the auctioneer’s economy, inducing a discount on their utility to fit their real impact. The main difference between this analysis and the traditional one of an optimal single-unit auction, as studied by Myerson (1981), lies in the presence of a cost associated to transfers between the potential buyers and the auctioneer. As next sections will exhibit, this rather harmless departure from the standard theory emphasizes bunching issues in the allocation rule.

4.2.2

Optimization program

Assume first that there is no asymmetry of information on the side of the auctioneer i.e. this later gets free of charge the valuations given by agents to the good to be allocated. The objective function of the firm corresponds to its expected profits made of expected payments of each agent i at   period t, Evi t i (vi ) , minus a fraction λ of each one of these expected payments which is lost in the transaction. For the regulator, these benefits are made of current expected payments of agent i minus   the transfer cost associated, i.e. (1 − λ) Evi t i (vi ) , as well as the expected utility that the buyer i     associates with consumption of the good, Evi U i (vi ) = Evi qi (vi ) vi − t i (vi ) . Thus, when the auctioneer values bidders’ utility with a weight α with respect to transfers, it ends with an objective function, for each agent i, of Evi [(1 − λ)t i (vi ) + αU i (vi )] = Evi [αqi (vi ) vi + (1 − λ − α)t i (vi )]. Assume now that there is an asymmetry of information between the mechanism designer and the agents in the sense that each bidder’s valuation vi is a private information for which a prior distribution, F i (vi ), is common knowledge. The characterization of the optimal mechanism for the

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4.2. ALLOCATION FRAMEWORK

auctioneer can be achieved by using the Revelation Principle which states that the mechanism can be described by functions {pi (v) , t i (v)}i=1,...,I provided agents have an incentive to reveal truthfully their valuation. Given this truthful revelation, which is secured by incentive compatibility constraints (ICi ), the expressions of the expected benefits remain the same.  Thus, the problem of the auctioneer is to find pi (v) , t i (v) i=1,...,I that is solution of the following problem π=

I

max

∑ Evi

{pi (.), t i (.)}i i=1

  αvi qi (vi ) + (1 − λ − α)t i (vi )

(P 1)

under the following constraints    ∀i, ∀vi ,       ∀i, ∀vi ,    ∀i, ∀vi ,      ∀i, ∀v,       ∀v,

U i (vi ) = qi (vi ) vi − t i (vi ) > 0  vi = arg maxv˜i qi (v˜i ) vi − t i (v˜i )

(IRi )

t i (vi ) > 0

(Ti )

pi (v) ∈ [0; 1]

(Pi )

∑i pi (v) 6 1

(P0 )

(ICi )

These constraints are standard. Under asymmetric information, feasible mechanisms are those that satisfy, for each type i of agents, incentive compatibility (ICi ), individual rationality (IRi ), with an outside opportunity normalized to zero, and allocate the good in a feasible way, i.e. the functions pi cannot allocate more than one time the good (P0 ) and have to respect the definition of a probability (Pi ). Eventually, transfers must stay positive. For a firm, this is a clear consequence of its objective function and for a regulator, this is needed to avoid the creation of money through the subsidization of bidders (1 − λ euro given by the regulator becoming, with the agreed convention on λ, 1 euro for the consumer). Thus, this must be the case for any value of α.

4.2.3 Perfect information When the auctioneer has a perfect knowledge of bidders’ willingness to pay, the problem turns to the one described by (P 1) without the incentive compatibility constraint. Two cases occur, depending on the sign of (1 − λ − α). If 1 − λ − α < 0, transfers are too costly and are avoided, and the auctioneer

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

has to solve for I

max

∑ E vi

{pi (.)}i i=1

  αvi qi (vi ) .

When 1 − λ − α > 0, agent i’s transfers are limited by its individual rationality constraint, which is binding, and the auctioneer’s program becomes I

max

∑ E vi

{pi (.)}i i=1

  (1 − λ) vi qi (vi ) .

In both cases, the allocation procedure ends to be exactly the same as without transfer costs, that is the good is allocated to the bidder exhibiting the highest (positive) valuation. Thus, under perfect information, any auctioneer can screen among potential bidders. When α = 0, only the second case is plausible as λ < 1. Thus, screening is possible in all the parameter space. The main purpose of the perfect information case lies in the emphasis it puts on the sign of [(1 − λ) − α], which will be key to the reminding analysis even if it does not play any role in the perfect information case. It compares (1 − λ), the value the auctioneer attaches to bidders’ transfers, and α, the value the auctioneer accounts for bidders’ utility. If, for example, 1−λ < α, the auctioneer values at a higher stake bidders’ utility than transfers, so that any euro more to the bidders is better than to the transfers. As bidder’s utility itself includes transfers, the auctioneer becomes a priori reluctant to demand transfers from bidders. Definition 4.2. An auctioneer characterized by 1 − λ < α is called “utility-biased”. An auctioneer characterized by 1 − λ > α is named “transfer-biased”. Thus, one of the key element to the analysis of optimal auction with transfer costs is the relative weight for the auctioneer of bidders utility with respect to secured transfers. Recalling definition 4.1, notice that a firm is an example of transfer-biased auctioneer and a regulator corresponds to a utility-based auctioneer.

4.3. OPTIMAL AND EFFICIENT MECHANISMS UNDER TRANSFER COSTS

4.3

187

Optimal and efficient mechanisms under transfer costs

4.3.1 Equivalent optimization problem Using Myerson’s (1981) methodology, the following lemmas exhibit, as a first step, equivalent conditions to incentive compatibility and individual rationality constraints on one hand, and to constraints on transfers on the other hand. Lemma 4.1. A mechanism is incentive compatible and individual rational if and only if   ∀i,       

∀i, ∀vi , ∀v˜i ,

U i (vi ) > 0   (vi − v˜i ) qi (vi ) − qi (v˜i ) > 0

∀i, ∀vi ,

U i (vi ) = U i (vi ) +

(IR0i ) (ICi0 )

R vi i v q (x) dx i

(ICi00 )

Lemma 4.2. An incentive compatible and individual rational mechanism satisfies the positive transfers constraint (Ti ) if and only if

∀i,

vi qi (vi ) > U i (vi )

(Ti0 )

Three elements are worth being noticed. First, qi (vi ), the expected probability of agent i to be allocated the good must be increasing in its own valuation vi . This condition is directly related to incentives for agent i to reveal its true willingness to pay and will be key to the reminding analysis. Second, bidder i’s expected utility must be non-decreasing in its valuation vi . Third, the utility of the least efficient agent i is bounded. On the one hand, it must be larger than 0, the normalized outside option, in order to induce the agents’ participation. On the other hand, the auctioneer cannot subsidize agents and this limits the amount an agent can save from the transaction. Using lemmas 4.1 and 4.2, one can rewrite the optimization program (P 1) in the following way. Lemma 4.3. Let {pi∗ (.)}i=1,...,I be solution of the following problem π = max

I

∑ Ev

{pi (.)}i i=1

   αvi + (1 − λ − α) J i (vi ) pi (v) − (1 − λ − α)U i (vi )

(P 2)

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

under the constraints   ∀i, ∀vi , ∀v˜i ,    ∀v, ∀i,     ∀v,

  (vi − v˜i ) qi (vi ) − qi (v˜i ) > 0

(ICi0 )

pi (v) ∈ [0, 1]

(Pi )

∑i pi (v) 6 1

(P0 )

and {t i∗ (.)}i=1,...,I be given by ∀vi , ∀i,

 Z i∗ t (vi ) = Ev−i vi p (v) − i∗

vi

vi

i∗



p (x, v−i ) dx −U i (vi )

with   1 − λ > α : U i (v ) = 0, i  1 − λ < α : U i (v ) = E v pi∗ (v , v ), v−i i i i −i Then {pi∗ (.) , t i∗ (.)}i=1,...,I solves program (P 1). Lemma 4.3 exhibits a new program to be solved in pi (vi ) only that gives a solution to program (P 1). Recalling definitions 4.1 and 4.2, the firm (α = 0) is a transfer-biased auctioneer (1 − λ > α) and one gets back to the standard result that it is costly and worthless to leave agents with the lowest valuation with any positive utility: U i (vi ) = 0 for all i. The remaining program shrinks to the standard one except that the firm keeps only 1 − λ of what it got without transfer cost. The regulator’s objective function (α = 1) becomes the maximization of ∑i Ev [[vi −λJ i (vi )]pi (v)+ λU i (vi )]. Two elements show up in this formulation. First, when the virtual valuation J i (vi ) is increasing, it may well be the case that vi − λJ i (vi ) is decreasing. Loosely speaking, this means that there might exist situations where the firm chooses to screen bidders when the regulator prefers to bunch them. Second, allocating the good to agent vi is a priori valuable for the auctioneer. Note that these two elements are of opposite sign, meaning qualitatively that they should occur in different parameter spaces. In general, the basic trade-off faced by the auctioneer depends on its type. A transfer-biased auctioneer (1 − λ > α) faces the same situation as the firm, i.e. costly information gathering which requires to leave some rents to agents with the highest willingness to pay. For a utility-biased auc-

4.3. OPTIMAL AND EFFICIENT MECHANISMS UNDER TRANSFER COSTS

189

tioneer (1 − λ < α), the situation mimics the regulator’s one with a discontinuity, at the low bound, in the valuation attached to bidders and bunching issues potentially arising in different parameter ranges than the firm.

4.3.2

Allocation rule

Exhibiting the solution of (P 2) requires the introduction of a few more notations. Moreover, it will be useful for further reference to discuss separately the case of the profit maximizing firm (α = 0). Let ignore, for the moment, the second term in the brackets of (P 2) related to U i (vi ). Given lemma 4.3, the maximization programs in the firm and in the auctioneer cases both sum up to the maximization of the expected sum of, respectively, J i (vi ) and K i (vi ) with   J i (vi ) = vi − 1−Fi i (vi ) , f (vi )  K i (v ) = αv + (1 − λ − α) J i (v ) = α 1−F i (vi ) + (1 − λ) J i (v ). i

i

i

f i (vi )

i

Following Myerson’s (1981) methodology, let define the following functions  −1   hiJ (q) = J i (F i (q)),      −1 −1   hiK (q) = K i (F i (q)) = αF i (q) + (1 − λ − α) hiJ (q), R   i (q) = q hi (x) dx,  H  J 0 J    R R   H i (q) = q hi (x) dx = α q F i−1 (x) dx + (1 − λ − α) H i (q). J K 0 0 K

The functions hij (q) are re-scaling of the virtual valuations to the parameter space of valuations probability. The functions H ij (q) are the primitive of the functions hij (q). In intervals where virtual valuation ( j = J or j = K) is increasing (decreasing), hij (q) is increasing (decreasing) and H ij (q) is convex (concave). The convexity of HKi (q), in turn, implies that its derivative, hiK (q), is increasing and that the modified virtual valuation K i (q) is increasing. Thus, if one faces a non monotone (but continuous) modified virtual valuation, the way to secure the incentive compatibility requirements of increasing expected allocation probability is to “convexify” function HKi (q). Let call conv [H (q)]

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

the convex hull of function H (q) and define the following functions   H˜ i = conv H i , J J   H˜ i = conv H i , K

K

  J˜i (vi ) = H˜ i0 (F i (vi )), J  K˜ i (v ) = H˜ i0 (F i (v )). i

i

K

The unctions H˜ ij (q) are convex hulls of the functions H ij (q) which allow us to build “equivalent” virtual valuations J˜i (vi ) and K˜ i (vi ). These later functions are increasing in intervals where H ij (q) is convex and flat in others where screening is not compatible with the incentive compatibility requirement. In these intervals, bunching occurs, the auctioneer does not screen among bidders’ types and value them at the average modified virtual valuation over this bunching range. Let now consider the second term in the brackets of (P 2) related to U i (vi ) when 1 − λ < α.8 Then, low type bidders’ utility is given a positive weight in the utility-biased auctioneer’s objective function. The way problem (P 2) is written suggests that this additional term induces a discontinuity in the valuation the auctioneer attaches to bidder vi if vi > 0. Indeed, if the auctioneer screens agents i for small vi , it attaches a value   1 − F i (vi ) αvi + (α + λ − 1) vi − J i (vi ) = αvi + (α + λ − 1) f i (vi ) to bidder vi and α (vi + ε) − (α + λ − 1) J i (vi + ε) =α (vi + ε) + (α + λ − 1)

1 − F i (vi + ε) f i (vi + ε)

− (α + λ − 1) (vi + ε) to bidder vi + ε (taking α = 1 in both cases). The inverse of the distribution function is continuous and (α + λ − 1) (vi + ε) is positive for a utility-biased auctioneer. Thus, if vi > 0, there always exists ε sufficiently low such that bidder vi is worth a higher value than bidder vi + ε. Recall that this term comes from the constraint that transfers must stay positive when one takes into account the consequences of both incentive compatibility and individual rationality constraints (cf. lemma 4.2). More precisely, constraint (ICi0 ) tells that any bidder of type i ends with at least the 8 When

1 − λ > α, U i (vi ) is optimally set to 0 and the situation is as described in the beginning of this section.

4.3. OPTIMAL AND EFFICIENT MECHANISMS UNDER TRANSFER COSTS

191

minimum level U i (vi ). If the good is allocated to bidder vi , then incentive constraints demand the good to be allocated to any bidder i. But the value the auctioneer attaches to bidder vi + ε is lower than the one it attaches to bidder vi . Thus, one ends with the same kind of difficulty than with virtual valuations where incentive compatibility concerns may interfere with the candidate solution to select the largest positive virtual valuation. Here, leaving least valued agents with any positive utility may impede the auctioneer to satisfy the incentive compatibility requirements. Consequently, the extra value associated with bidder vi is of some use if the auctioneer’s optimal strategy is not to screen among bidders with valuations in the neighborhood of vi , that is if there exists q∗ > 0 such that the auctioneer bunches bidders in the range [0; q∗ ]. On the contrary, the auctioneer gives up the extra value of bidder vi if its optimal answer sets q∗ = 0. It turns out that this discontinuity at a discrete point cannot be exploited by the auctioneer. Basically, this is related to two elements: first, the initial decrease in the valuation potentially violates the incentive compatibility constraint; second, there is a zero probability to face precisely bidder vi . Thus, bunching at low valuation does not occur because of the presence of this extra value associated to bidders vi (it can still occur for other reasons) and the equivalent valuation function that is to be used by the auctioneer corresponds to K i . Proposition 4.1. A utility-biased auctioneer does not gain by considering the term U i (vi ) in its objective function P 2. In other words, when it designs its allocation procedure, the auctioneer cannot take into account the extra benefit induced by bidder vi being allocated the good. Thus, the auctioneer has to maximize    ∑Ii=1 Ev αvi + (1 − λ − α) J i (vi ) pi (v) , over pi (v), under the incentive compatibility constraint (ICi0 ) and constraints related to the probability nature of functions pi (v). Let call N (v) the set of bidders for whom K˜ i (vi ) is maximal among all bidders  N (v) =

i| K˜ i (vi ) = max K˜ j v j




j=1,...,I

and, by a slight abuse of notation, N (v) also denotes its cardinality. Combining lemma 4.3 and proposition 4.1, the good should be allocated among the bidders belonging to N (v).

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

Proposition 4.2. Let {pi∗∗ (.)}i=1,...,I and {t i∗∗ (.)}i=1,...,I satisfy

∀vi , ∀i, ∀vi , ∀i,

  1/N (v) if i ∈ N (v), pi∗∗ (vi ) =  0 if i ∈ / N (v),   Z vi i∗∗ i∗∗ i∗∗ t (vi ) = Ev−i vi p (v) − p (x, v−i ) dx −U i (vi ) vi

with   1 − λ > α : U i (v ) = 0, i  1 − λ < α : U i (v ) = E v pi∗∗ (v , v ). v−i i i i −i Then {pi∗∗ (.) , t i∗∗ (.)}i=1,...,I is a solution to (P 1). This allocation mechanism has the same kind of properties than the classical one, without transfer cost. It excludes low types bidders, it may not be ex post efficient and it can even sell the good to a bidder whose willingness to pay is not the highest.9

4.3.3 Auctioneer vs firm Our prime interest is to study, for a given transfer cost λ, the relative allocation procedure of a profit maximizing firm and any auctioneer characterized by a parameter α > 0, i.e. bunching and non bunching intervals of their respective strategies. A first element of comparison is the occurrence of negative modified virtual valuation relative to negative virtual valuation. The derivative of the modified virtual valuation attached by the auctioneer to any agent i is negative whenever   α  1 − λ > α : J i0 (vi ) 6 − 1−λ−α < 0,   1 − λ < α : J i0 (vi ) > + α > 0. 1−λ−α Thus, the range of bidders’ willingness to pay where the (auctioneer’s) modified virtual valuation is negative can be easily compared to the sign of the (firm’s) virtual valuation. 9 See

Myerson (1981, p. 67) for some examples in this area.

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4.3. OPTIMAL AND EFFICIENT MECHANISMS UNDER TRANSFER COSTS

Lemma 4.4. The incentive compatibility concerns faced by a transfer-biased auctioneer occur in a narrower range of bidders’ valuation than the ones faced by the firm. On the contrary, a utilitybiased auctioneer faces these same concerns in intervals of buyers’ willingness to pay where the firm has no such concerns. Nevertheless, the correspondence between negative (modified) virtual valuations and bunching is not direct. Thus, one has to turn to the comparison of convex hulls of HJi (q) and HKi (q). This is eased by the two following equalities which are directly deduced from H ij (q) definition  R −1   HKi (q) − (1 − λ − α) HJi (q) = α 0q F i (x) dx, h i   H i (q) − (1 − λ) H i (q) = α R q 1−F i (vi ) ◦ F i−1 (x) dx. K J 0 f i (v ) i

−1

Regarding the first equality, function F i (v) is assumed to be strictly increasing, thus F i (q) is also strictly increasing and

R q i−1 (x) dx is strictly convex in q. Regarding the second one, the 0 F

right-hand side function is convex (concave) if and only if its second derivative relative to q is positive (negative), that is if 1 − F i (v) / f i (v) is increasing (decreasing) in v. This is equivalent to 1 − F i (v) being log-convex (log-concave).10 Applying corollary 3 of Bagnoli and Bergstr¨om (1989), it directly follows that the right-hand side of the second equality is convex whenever 1 −F i (v) is logconvex and concave whenever f i (v) is monotone increasing.11 Note also that f i (v) being monotone increasing is a sufficient condition for virtual valuation J i (v) to be monotone increasing.12 If H ij (q) is strictly convex over an interval [q1 ; q2 ], then it should verify the property ∀β ∈ ]0; 1[

βH ij (q1 ) + (1 − β) H ij (q2 ) − H ij (βq1 + (1 − β) q2 ) > 0.

In such areas, the allocation process is perfectly screening among potential bidders. On the contrary, when the allocation process cannot screen bidders in the range ]q1 ; q2 [, i.e. HJi (q) > H˜ Ji (q) for all q     i (v) f i (v) 0 00 i standard manipulations yield: 1−F ↑⇔ 1−F 1 − F i (v) > 0. i (v) ↓⇔ − ln 1 − F (v) ↓⇔ ln f i (v) 11 Bagnoli and Bergstr¨ om (1989) exhibit, p. 29, the log-concavity properties of some common distributions. Columns “density” ( f i (v)) and “reliability” (1 − F i (v)) are of special interest regarding the current exercise. It appears that most of commonly used distributions have log-concave density. This table also shows that a log-convex density function can generate either a log-concave or a log-convex reliability function. This is what prevents a refinement for the logconvexity condition of 1 − F i (v).
12 The derivative of J i (v) with respect to v is equal to [2[ f i (v)]2 + 1 − F i (v) f i0 (v)]/[ f i (v)]2 . It is positive if f i0 (v) i is positive. 10 Indeed,

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CHAPITRE 4. OPTIMAL AUCTION WITH EXOGENOUS TRANSFER COSTS

in [q1 ; q2 ] and HJi (q) = H˜ Ji (q) at both q1 and q2 , then the function HJi (q) is such that ∀β ∈ [0; 1]

βH ij (q1 ) + (1 − β) H ij (q2 ) − H ij (βq1 + (1 − β) q2 ) 6 0.

Thus, comparing convex hulls is equivalent to comparing weighted average values taken by both functions at extreme points to their values at the weighted average point. From the properties discussed in previous paragraph, it is easy to see that, for any q1 , q2 and β strictly between 0 and 1  i  βHK (q1 ) + (1 − β) HKi (q2 ) − HKi (βq1 + (1 − β) q2 )   − (1 − λ − α) βHJi (q1 ) + (1 − β) HJi (q2 ) − HJi (βq1 + (1 − β) q2 ) > 0 because

R q i−1 F (x) dx is strictly convex on [0; 1], and 0

 i  βHK (q1 ) + (1 − β) HKi (q2 ) − HKi (βq1 + (1 − β) q2 )   − (1 − λ) βHJi (q1 ) + (1 − β) HJi (q2 ) − HJi (βq1 + (1 − β) q2 )

  >0 