Pricing to Preclude Sabotage in Regulated Industries

Transcription

1 Pricing to Preclude Sabotage in Regulated Industries by Arup Bose*, Debashis Pal**, and David E. M. Sappington*** Abstract We characterize the optimal access price and retail price for a vertically-integrated incumbent supplier (V ) that faces limited competition from a new entrant in the retail sector. The optimal prices provide V with a relatively high wholesale pro t margin and a relatively low retail pro t margin. Consequently, V has no incentive to raise the costs of its retail rival. Keywords: sabotage, raising rivals costs, wholesale and retail price regulation September 206 * Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata 70008, India (bosearu@gmail.com). ** Department of Economics, University of Cincinnati, Cincinnati, Ohio 4522 USA (debashis.pal@uc.edu). *** Department of Economics, University of Florida, PO Box 740, Gainesville, FL 326 USA (sapping@u.edu). We thank the editor, Yossi Spiegel, two anonymous referees, David Brown, Mark Jamison, and Ted Kury for very helpful comments and suggestions.

2 Introduction The Telecommunications Act of 996 requires incumbent suppliers of telecommunications services in the U.S. to make their networks available to retail competitors on reasonable terms and conditions. This Act, like its counterparts throughout the world, re ects the belief that mandated access of this sort can foster industry competition by enabling new suppliers to compete for retail customers without having to build their own ubiquitous networks (Cave, 2006; Bourreau et al., 200). One potential impediment to such competition is that a vertically-integrated incumbent supplier may have an incentive to raise the costs of its retail rivals in order to diminish their competitive impact and thereby increase the incumbent s retail pro t (Economides, 998; Beard et al., 200; La ont and Tirole, 200; Bustos and Galetovic, 2009). On the other hand, limiting the success of retail rivals can reduce their purchase of network access and thereby reduce the incumbent supplier s wholesale pro t (Sibley and Weisman, 998a,b; Mandy, 2000; Weisman and Kang, 200; Kondaurova and Weisman, 2003). The strength of each of these countervailing incentives varies with the prevailing regulatory policy. In particular, a relatively high retail pro t margin can encourage the incumbent supplier to disadvantage its retail rivals whereas a relatively high wholesale margin can discourage such sabotage. These well-known conclusions suggest that the optimal regulated retail and wholesale pro t margins are likely to vary with such industry features as the ability of the incumbent supplier to sabotage the activities of retail rival without regulatory detection and punishment. Speci cally, one might suspect that the wholesale pro t margin should be increased and the retail pro t margin reduced in settings where sabotage is di cult to detect and preclude, ceteris paribus. This seemingly sensible policy prescription, though, turns out to be incorrect. We identify conditions under which the optimal regulatory policy is completely insensitive to the incumbent supplier s ability to raise its rival s cost. The same policy that is optimal 47 U.S.C. 25.

3 when the incumbent has no ability whatsoever to raise its rival s cost remains optimal when the incumbent is readily able to sabotage its retail rival. This is the case because, regardless of the incumbent s ability to disadvantage its rival, the optimal regulatory policy establishes a relatively high price for access to the incumbent s network and a relatively low price for the incumbent s retail product. The resulting relatively high wholesale pro t margin for the incumbent eliminates its incentive to sabotage its retail rival. Therefore, even when it is particularly adept at raising its rival s cost, the regulated incumbent supplier will refrain from cost-increasing sabotage under the optimal regulatory policy. Consequently, this optimal policy does not change as the incumbent s ability to raise the costs of a retail rival changes. We demonstrate the optimality of a relatively high wholesale pro t margin in a setting where a regulator sets both the price of access (w) and the retail price (p 0 ) charged by a vertically-integrated incumbent supplier (V ). Regulators often set both wholesale and retail prices in practice in settings where retail competition is emerging but has not developed to the point where it alone can fully discipline the incumbent supplier s retail operations. 2 To understand why a regulator optimally sets a relatively high wholesale pro t margin in such settings, suppose to the contrary that the regulator a orded V a relatively high retail pro t margin. Then the regulator could increase consumer surplus without reducing V s pro t by reducing V s retail pro t margin (by reducing p 0 ) and increasing its wholesale margin (by increasing w). This is the case because the reduction in p 0 increases consumer surplus at the same rate that it reduces V s retail pro t. In addition, the increase in w increases V s wholesale pro t. Furthermore, the resulting increase in V s retail output and the reduction in its wholesale output increase V s pro t on balance when its retail pro t margin exceeds its wholesale pro t margin. Consequently, if V s retail pro t margin ever exceeded its wholesale pro t margin, the regulator could increase consumer surplus by reducing p 0 and increasing w without altering V s aggregate pro t. This nding raises questions about the common regulatory practice of setting wholesale 2 For instance, regulators set both retail and wholesale prices for many years after retail competition was introduced in the telecommunications and electricity sectors. Section 2 provides additional detail. 2

4 prices to re ect production costs in order to avoid increases in input prices that might limit or distort downstream competition. As we explain in the concluding discussion, this policy may have considerable merit in settings where retail competition is robust and where the regulator s information about prevailing industry conditions is limited. More generally, though, elevated wholesale pro t margins can increase consumer welfare while eliminating incentives for cost-increasing sabotage. We illustrate the resulting welfare gains in Section 4 where we compare the performance of the optimal regulatory policy with policies that set the access price equal to V s cost of supplying access. Our nding that the optimal regulatory policy precludes incentives for sabotage by a vertically-integrated supplier is reminiscent of La ont and Tirole s (200) observation that global price cap regulation can eliminate incentives for sabotage under certain conditions. 3 Our analysis di ers from La ont and Tirole s analysis in part by characterizing the optimal regulatory policy in a setting where retail competition is imperfect and by demonstrating that the optimal policy eliminates incentives for cost-increasing sabotage. 4 The ensuing analysis proceeds as follows. Section 2 describes the key elements of our formal model. Section 3 characterizes the optimal regulatory policy and explains its key features. Section 4 contrasts the performance of the optimal policy with policies that set the price of access equal to V s cost of supplying access. Section 5 concludes and suggests directions for future research. The Appendix presents the proofs of the formal conclusions that are not included in the main text. 3 Under global price cap regulation, all of the services supplied by a vertically-integrated incumbent supplier (S) are placed in the same basket of services. A weighted average of the prices charged for these services is then precluded from exceeding a speci ed value. La ont and Tirole (200, pp ) show that when the weight on each service price re ects S s equilibrium supply of the service and when S faces a competitive fringe, S will not nd it pro table to raise the fringe s operating cost marginally. See La ont and Tirole (996, 2000) for additional analyses of global price cap regulation. 4 Under global price cap regulation, the regulated rm may have an incentive to implement a substantial level of cost-increasing sabotage even though the rm may not bene t from small levels of sabotage. See La ont and Tirole (200, p. 76). 3

5 2 The Model We consider a setting in which a vertically-integrated incumbent supplier (V ) produces a critical input access to its network infrastructure and supplies a retail product. A rival supplier (E for entrant ) produces a di erentiated retail product. E must secure access to V s infrastructure in order to produce the retail product. One unit of access is required to produce each unit of the retail product. E s cost of producing x units of its retail product is F + [ w + c ] x, where F is a xed cost of production, w is the unit price of access to V s network, and c 2 [ c; c ] is an additional (downstream) unit production cost. 5 V s cost of supplying x units of access to E and supplying x 0 units of its own retail product is F 0 + c u x + [ c u + c d ] x 0. F 0 is a xed cost of operation for V, c u is V s unit cost of supplying access (both to itself and to E), and c d is V s incremental (downstream) unit cost of supplying its retail product. E s downstream unit production cost, c, is the realization of a random variable. V can in uence the distribution of this random variable by undertaking an anticompetitive action a 2 [ 0; a ]. This action can represent a variety of activities. For instance, V might require E to employ expensive complementary inputs that are truly unnecessary, but are alleged to be essential requirements for ensuring reliable, secure access to V s network. To capture V s potential in uence on E s operating costs, let G(c j a) denote the distribution function for c, given action a. We assume G a (cja) < 0 for all a 2 [ 0; a ], so higher values of a make the higher realizations of c systematically more likely (and the lower realizations of c systematically less likely) in the sense of rst-order stochastic dominance. 6 K(a) 0 will denote V s cost of pursuing action a, where K 0 (a) 0 for all a 2 [ 0; a ]. This cost might include, for example, the resources that V must employ to convince the regulator to require E to comply with the (unnecessarily stringent) conditions that V imposes 5 This additional downstream cost includes any costs associated with employing access to produce the retail product. 6 The subscript a denotes the partial derivative with respect to a. We will say that V refrains from all anticompetitive activity when it undertakes action a = 0. 4

6 as prerequisites for access to its network. K(a) might also include the expected regulatory penalties and/or court-awarded damages that V risks by engaging in anticompetitive activities. The regulator sets both the price of access (w) and the retail price of the incumbent s product (p 0 ). As is the case in many regulated industries characterized by emerging competition, the regulator does not regulate the price of E s retail product. E chooses its preferred retail price (p ) after observing the established access price, the incumbent s retail price, and E s realized downstream unit cost of production. Retail competition between V and E is captured in standard Hotelling fashion. V is located at point 0 and E is located at point, so it will sometimes be convenient to refer to V as supplier 0 and to E as supplier. Potential retail customers are distributed uniformly on the [0; ] line segment between the two suppliers. Unless otherwise noted, the mass of potential customers, N, is normalized to, for expositional ease. Each consumer derives value v j from one unit of supplier j s product (j 2 f0; g). A consumer incurs unit cost t j in travelling to purchase the product from supplier j. 7 Therefore, a consumer located at distance d from supplier j will derive net value v j p j t j d when he purchases a unit of supplier j s retail product at price p j. We assume v 0 and v are su ciently large that every consumer purchases one unit of the retail product in equilibrium. A consumer will purchase the product from the supplier that o ers the highest net value. Our model is designed to capture settings where retail competition is emerging but has not yet developed to the point where it alone can fully discipline the incumbent supplier s retail operations. 8 Although regulators tend to replace retail price regulation with wholesale price regulation after robust retail competition has developed, they often impose both wholesale and retail price regulation for considerable periods of time as retail competition 7 t might exceed t 0, for example, if limited experience with the new industry entrant leads consumers to favor ongoing interaction with the (familiar) incumbent supplier. 8 Limited competitive discipline can arise in our model, for instance, when t is substantially larger than t 0. 5

7 develops. 9 This has been the case in the U.S. telecommunications industry, for instance. The Telecommunications Act of 996 required incumbent local exchange carriers (ILECs) in the U.S. to unbundle key elements of their networks and sell these elements to retail competitors. Even as they speci ed prices for these wholesale services, regulators commonly restricted the prices the ILECs charged for retail telecommunications services. In their 2003 review of state regulatory policy in the U.S. telecommunications industry, Tardi and Taylor (2003, p. 348) report that they are not aware of any regulator that has fully deregulated retail services. A subsequent review identi ed only two states (Oklahoma and South Dakota) that had fully deregulated the prices of all retail telecommunications services more than a decade after the passage of the 996 Act. 0 Corresponding experience prevails in the electricity sector. Retail customers have had a choice among retail suppliers for many years. However, in many jurisdictions, regulators continue to regulate the retail prices that incumbent suppliers charge for electricity. This is the case in the state of Pennsylvania (in the U.S.) and the province of Alberta (in Canada), for example. 2 A review of competition in the retail electricity sector observes that in the large majority of cases market prices co-exist with regulated tari s (Defeuilley, 2009, p. 377). 3 To specify the regulator s formal problem in settings like these, let x j (w; p 0 ; c) denote supplier j s equilibrium retail output when the access price is w, the price of V s retail product is p 0, and E s downstream marginal cost is c. Let 0 (w; p 0 ; c) denote V s corresponding 9 Hortaçsu et al. (205) report that regulated incumbent suppliers tend to retain market power for considerable periods of time following the introduction of retail competition both because many customers lack full information about competitors operations and because even well-informed customers often prefer to purchase from familiar incumbent suppliers. 0 See Table 3 in Tardi (2007, p. 2). Similarly, it was not until 2006 that Ofcom deregulated the prices of most retail telecommunications services in the UK, although it had been regulating wholesale ( network access ) prices throughout the preceding decade (Ofcom, 2006). Regulators also set the compensation that vertically-integrated incumbent suppliers receive when their transmission assets are employed to deliver the electricity that competing suppliers sell to retail customers. 2 See Alberta Government (2005) and Pennsylvania Public Utility Commission (2007). 3 The retail prices that incumbent suppliers charge for electricity have been largely deregulated in some jurisdictions in recent years, including Texas and the UK (Defeuilley, 2009). 6

8 pro t. Then V s expected pro t when it undertakes action a, given w and p 0, is: 4 E f 0 (w; p 0 ; c) j ag = [ w c u ] Z c x (w; p 0 ; c) dg(cja) + [ p 0 c u c d ] c Z c x 0 (w; p 0 ; c) dg(cja) F 0 K(a). () c The regulator seeks to maximize expected consumer surplus while ensuring that V secures at least its reservation level of expected pro t, which is normalized to 0. 5 Let S(w; p 0 ; c) denote equilibrium consumer surplus given w, p 0, and c. This surplus is the net value that all consumers receive in equilibrium. The regulator s formal problem in this setting, [RP], is: subject to: Maximize p 0 0; w 0 E f 0 (w; p 0 ; c) j ag E f S(w; p 0 ; c) j ag Z c 0, where a 2 arg max ea c S(w; p 0 ; c) dg(cja) f E f 0 (w; p 0 ; c) j eag g. (2) The timing in the model is as follows. First, the regulator sets w and p 0 to maximize expected consumer surplus, anticipating the action that V will undertake. Second, V determines its preferred action (a 2 [ 0; a ]). Third, E s downstream unit cost of production (c) is determined and E chooses its retail price (p ). Fourth, consumers choose their preferred supplier of the retail product. 3 Findings Let ex 2 [0; ] denote the location of the consumer who is indi erent between purchasing the retail product from V and E, given their respective prices p 0 and p. Then: v 0 p 0 t 0 ex = v p t [ ex ]. (3) 4 Here and throughout the ensuing analysis, Efg denotes expectations with respect to c. 5 As Carlton (2007) observes, The Department of Justice and the Federal Trade Commission, the two federal antitrust agencies [in the United States], often state that their focus is on consumers, which seems to imply a focus on consumer surplus. As we explain in section 5, our key qualitative conclusions persist when the regulator places some value on industry pro t. 7

9 Equation (3) implies that when V and E both serve retail customers in equilibrium, their respective outputs (given prices p 0 and p ) are: x 0 (p 0 ; p ) = v 0 v + p p 0 + t t 0 + t ; x (p 0 ; p ) = v v 0 + p 0 p + t 0 t 0 + t. (4) Equation (4) implies that, given the regulated access price (w), V s retail price (p 0 ), and downstream unit cost c, E will choose p to: v v 0 + p 0 p + t 0 Maximize [ p w c ] p t 0 + t ) p = 2 [ v v 0 + p 0 + t 0 + c + w ]. (5) Straightforward calculations then provide the characterization of equilibrium outputs identi- ed in Lemma. The lemma refers to Assumption, which ensures that the rms production costs, rm-speci c transportation costs, and customer valuations of the rms products are all su ciently similar that both rms serve retail customers in equilibrium. This assumption is presumed to hold throughout the ensuing analysis. 6 Assumption. p 0 w 2 ( v 0 v + c t 0 ; v 0 v + 2 t + t 0 + c ). Lemma. The equilibrium outputs of V and E, given p 0, w, and c are, respectively: x 0 (w; p 0 ; c) = v 0 v + 2 t + t 0 + c ; x (w; p 0 ; c) = v v 0 + t 0 + c. (6) The expressions for the equilibrium outputs identi ed in Lemma are readily employed to derive a corresponding expression for V s pro t. Lemma 2. 0 (w; p 0 ; c) = [ c d ] x 0 (w; p 0 ; c) + w c u F 0. 0 (w;p 0 s = c d. Lemma 2 implies that V s pro t increases as E s downstream unit cost (c) declines if: 6 Assumption includes endogenous variables. Assumption 2 below, which entails only exogenous parameters, ensures that Assumption holds at the solution to [RP]. 8

10 c d < 0, w > p 0 c d, w c u > p 0 c u c d. (7) Inequality (7) holds if V s wholesale pro t margin (w c u ) exceeds its retail pro t margin (p 0 c u c d ). In this event, V s overall pro t increases as retail customers switch their purchases from V to E. 7 As E s downstream unit cost (c) declines, E will reduce its equilibrium price, and thereby attract more customers from V. Therefore, V s pro t increases as c declines if inequality (7) holds. Not surprisingly, equilibrium consumer surplus also increases as c declines for any speci- ed values of p 0 and w. This is the case because E reduces its retail price as its cost declines, and the reduced price increases consumer surplus. Lemma 3. S(w; p 0 ; c) is a strictly decreasing function of c. We now proceed to demonstrate that, regardless of the functional form of K(), inequality (7) holds under the optimal regulatory policy, so V will refrain from all anticompetitive activity. This demonstration is facilitated by considering the hypothetical setting where it is common knowledge that V s action is xed exogenously at some ba 2 [ 0; a ]. Let E f 0 (w; p 0 )jbag E f 0 (w; p 0 ; c)jbag K(ba) denote V s expected pro t in this setting. The regulator s problem in this setting, [RP- ba], is: subject to: Maximize p 0 0; w 0 E f 0 (w; p 0 )jbag E fs(w; p 0 ; c)jbag = Z c Z c c S(w; p 0 ; c) dg(cjba) 0 (w; p 0 ; c) dg(cjba) K(ba) 0. (8) c Proposition characterizes the solution to [RP- ba]. The proposition refers to Assumption 2, which is assumed to hold throughout the ensuing analysis. This assumption ensures that V and E both serve retail customers at the solution to [RP-ba] for all ba 2 [ a; a ] because 7 In the present model of Hotelling competition with full market coverage, every increase in E s retail output produces an identical reduction in V s retail output. Under alternative forms of retail competition in which the diversion ratio is less than unity, V s pro t would increase even more rapidly as E s retail output increases when < c d. 9

11 the rms product valuations and downstream production costs are not too dissimilar (so jv v 0 j, jc d c j, and jc d c j are su ciently small) and transportation costs are intermediate in magnitude. Assumption 2. max 2 [ v v 0 c + t 0 ] ; v 0 v t [ 3 c c ] < c d < v 0 v + 2 t t + c. Proposition. At the solution to [RP- ba]: E f 0 (w; p 0 )jbag = 0, w = c u + F 0 + K(ba) and [ v 0 v + E fcjbag t 0 c d ] [ v 0 v + E fcjbag + 2 t t c d ] 9 [ t 0 + t ] > 0, (9) p 0 w = 3 [ v 0 v t 0 + E fcjbag + 2 c d ] 2 (0; c d ). (0) Corollary. V s pro t is a strictly decreasing function of c at the solution to [RP- ba]. Proof. The conclusion follows directly from Lemma 2 because < c d, as stated in (0) and demonstrated in the proof of Proposition. Recall that the smaller values of c become systematically more likely as V reduces its anticompetitive activity (a). Therefore, Corollary implies that under the regulatory policy that constitutes the solution to [RP- ba], V s expected pro t would increase if V s anticompetitive action were to decline below ba. This conclusion is recorded formally as Lemma 4. The lemma refers to E f 0 (ea j ba)g, which is V s expected pro t when it undertakes action ea under the regulatory policy that constitutes the solution to [RP- ba]. Lemma 4. E f 0 (ea j ba)g is a strictly decreasing function of ea for all ea; ba 2 [ 0; a ]. Proof. The proof follows from Corollary and the fact that G a (cja) < 0 for all c 2 (c; c). Lemma 4 implies that when V operates under the regulatory policy that constitutes the solution to [RP-ba], the value of a that maximizes V s expected pro t is a = 0. By refraining 0

12 from all anticompetitive activity, V maximizes the likelihood that E will secure a relatively low downstream unit cost and so will attract a relatively large number of retail customers by setting a low price for its product. E s relatively large retail sales induce a relatively large demand for access to V s network, which generates relatively substantial wholesale pro t for V. This conclusion is recorded formally as Lemma 5. Lemma 5. For any ba 2 [0; a], V will set a = 0 and secure nonnegative expected pro t under the regulatory policy that constitutes the solution to [RP- ba]. Proof. Lemma 4 implies that E f 0 (ea j ba)g is maximized at ea = 0. Furthermore, Proposition implies that E f 0 (ba j ba)g = 0. Therefore, E f 0 (a j ba)g 0 for all a 2 [ 0; ba]. The regulator shares V s preference for an absence of anticompetitive activity. This absence increases the likelihood that E s downstream unit production cost will be relatively low, and so E will set a relatively low price for its product. Recall from Lemma 3 that consumer surplus increases as c declines, so expected consumer surplus increases as a declines. Lemma 6. E fs(w; p 0 ; c) j a)g is a strictly decreasing function of a for all a 2 [ 0; a ]. Proof. The proof follows immediately from Lemma 3 and the fact that G a (cja) < 0 for all c 2 (c; c). Lemma 5 holds for all values of ba 2 [ 0; a]. Therefore, whenever the regulator implements the policy that maximizes expected consumer surplus while ignoring the impact of the policy on V s incentive to raise its rival s cost, V will refrain from all anticompetitive activity. Consequently, the regulator will optimally implement the policy that maximizes expected consumer surplus while ensuring zero expected pro t for V when V refrains from all anticompetitive activity. V will indeed set a = 0 under the policy, regardless of the magnitude of the costs it would need to incur in order to raise its rival s cost. This conclusion is stated formally as Proposition 2.

13 Proposition 2. The solution to [RP] is the solution to [RP- 0]. Therefore, at the solution to [RP], a = 0 and w and p 0 are as speci ed in (9) and (0), with ba = 0. Proof. The proof follows immediately from Proposition and Lemmas 5 and 6. V has no incentive to raise its rival s cost under the optimal regulatory policy because the policy provides a relatively high wholesale pro t margin for V. Consequently, V welcomes expanded demand for access to its network, which is secured by refraining from any activity that would raise E s cost and thereby reduce E s retail output, and so reduce E s demand for access to V s network. Observation helps to explain why the regulator optimally ensures that V s wholesale pro t margin (w c u ) exceeds its retail pro t margin (p 0 c u c d ), so < c d. 8 = x 2 0 = x 0 2 [ p0 c u c d (w c u ) if c d. To interpret Observation, consider a setting where c d, so V s retail pro t margin (weakly) exceeds its wholesale pro t margin. Observation implies that the regulator can increase consumer surplus (S) in this setting without reducing V s pro t by reducing p 0 w. To explain this conclusion, suppose the regulator increases w and reduces p 0 at the same rate when c d. Doing so leaves E s price unchanged because the price reduction induced by the decline in V s retail price is o set by the price increase induced by the higher cost of access. 9 Consequently, the net impact of simultaneously increasing w and reducing p 0 at the same rate is to increase S at the rate x 0, which re ects the rate at which the expenditures of V s customers decline due to the reduction in p When c d, V s pro t ( 0 ) declines more slowly than S increases as w is increased 8 Recall inequality = 0 from equation (5). 20 See equation (4) in the Appendix. Because consumers act to maximize their welfare when choosing their preferred supplier, the changes in x 0 and x induced by the identi ed changes in w and p 0 have no rst-order impact on consumer surplus. 2

14 and p 0 is reduced at the same rate. This is the case because the reduction in 0 has three components. First, V s retail pro t declines at the rate x 0 as p 0 declines because retail customers pay less for V s product. This is precisely the rate at which S increases as p 0 declines. Second, V s wholesale pro t increases at the rate x as w increases because E pays more for the access it secures from V. Third, V s pro t increases at the rate p 0 c u c d (w c u ) 0 as the reduction in p 0 expands V s retail output and reduces its wholesale output at the same rate. The increased pro t rate here arises because V s retail pro t margin exceeds its wholesale pro t margin (because c d, by assumption). 2 Because S increases more rapidly than 0 declines as declines whenever c d, the regulator can secure a higher level of consumer surplus by reducing in a manner that does not reduce V s pro t. Therefore, the regulator will optimally set below c d, which ensures that V s wholesale pro t margin exceeds its retail pro t margin. Consequently, V will refrain from any activity that would increase E s costs and thereby reduce E s demand for V s relatively lucrative wholesale service. 4 Comparison with Cost-Based Pricing Policies Before concluding, we brie y contrast the performance of the optimal regulatory policy identi ed in Proposition 2 with (common) cost-based policies that link the access price to measures of V s cost of providing access. Speci cally, suppose the regulator sets the access h F price equal to c u + 0 x 0 +x i, which is the sum of V s marginal cost of suppling access (c u ) and a fraction ( 2 [0; ]) of a measure of V s average xed cost of production. 22 might be viewed as the fraction of V s xed cost of production (F 0 ) that is allocated to its upstream operation when calculating V s average cost of supplying access. In addition to setting the access price at the identi ed measure of upstream average cost, the regulator sets p 0 to maximize expected consumer surplus while ensuring nonnegative (expected) pro t for V. 2 If V s retail pro t margin exceeds its wholesale pro t margin, then V s pro t will increase whenever its retail output increases at least as rapidly as its wholesale output declines. Therefore, the fact that this third component of the change in 0 is positive is not an artifact of the presumed full market coverage under Hotelling competition. 22 Recall that, in equilibrium, x 0 + x = N, the mass of potential retail customers. 3

15 To illustrate the di erent outcomes that can arise under cost-based pricing policies and the optimal policy identi ed in Proposition 2, consider a setting with the following features. There are ; 000 potential customers (so N = ; 000). In addition, v 0 = v = 2, so consumers value the vertical dimension of the suppliers products symmetrically. However, t 0 = 6 and t = 8, so in this sense consumers nd it more convenient to purchase the product from V than from E. V s xed cost of production (F 0 ) is 2; 500 and E s xed cost (F ) is 50. V s upstream marginal cost (c u ) is and its downstream marginal cost (c d ) is 2. One-half of the identi ed measure of V s xed cost is allocated to its upstream operations when calculating s V s upstream average cost of production, so = 0:5. E s marginal cost (c) is a random variable on [0; 6], with piecewise-linear density function g(cja) = + a [6 2c] 6 36 that peaks at c = 3 for a 2 (0; ]. Under this density function, the higher cost realizations (c > 3) become more likely and the smaller cost realizations (c < 3) become less likely as V s sabotage (a) increases. Sabotage is costless for V, so K(a) = 0 for all a 2 [0; ]. Table presents the outcomes that arise in this simple setting under both the optimal pricing policy and the cost-based pricing policy. The table reports the access price (w), the retail prices of V and E (p 0 and p, respectively), the fraction of customers that purchase the retail product from V in equilibrium (x 0 =N), the expected pro ts of V and E (Ef 0 g and Ef g, respectively), and the expected consumer surplus (EfSg) that arise under the two policies. Pricing Policy w p 0 p x 0 =N Ef 0 g Ef g EfSg Optimal 4:97 5:30 9:63 0: :98 3; 824:40 Cost-Based 2:25 5:8 9:03 0: :68 3; 484:5 Table. Outcomes under the Optimal and Cost-Based Pricing Policies In the setting considered in Table, the access price is substantially lower under the costbased pricing policy than under the optimal policy. The resulting relatively low upstream revenue requires an increase in p 0 to ensure nonnegative pro t for V. Despite the increase in 4

16 V s retail price, E sets a lower retail price under the cost-based pricing policy because of the lower access price it faces. The lower price that E sets increases the number of customers it serves in equilibrium. These increased sales and the reduced cost of access generate increased pro t for E. The relatively low wholesale pro t margin (w c u = :25) and the relatively high retail pro t margin (p 0 c u c d = 2:8) that V faces under the cost-based pricing policy induces V to attempt to increase E s cost of production. V undertakes sabotage a =, which increases E s expected cost and thus E s expected price. The regulator anticipates V s anticompetitive activity and reduces p 0 to eliminate the increase in pro t that V secures from its sabotage. 23 Relative to the identi ed cost-based pricing policy, the optimal pricing policy secures a 9:8% increase in consumer surplus in the setting of Table. This increase re ects in part the reduction in E s expected downstream marginal cost of production that arises when the optimal policy eliminates V s incentive to raise its rival s cost Conclusions We have shown that consumer welfare is maximized when a vertically-integrated supplier (V ) is a orded a relatively high wholesale pro t margin and a relatively low retail pro t margin. V has no incentive to raise the cost of its retail rival (E) under this policy because doing so would reduce E s demand for the relatively pro table wholesale service that V supplies. We have derived this conclusion in the context of a general Hotelling framework that allows production costs, transportation costs, and product valuations to di er across suppli- 23 If the regulator failed to account for V s sabotage (by simplying assuming that a = 0) when setting p 0, she would set p 0 = 5:89, which would enable V to secure pro t Ef 0 g = 58:52 in equilibrium. The resulting consumer surplus would be EfSg = 3; 43: The corresponding increase in consumer surplus secured under the optimal policy would be higher (lower) if, ceteris paribus: (i) the potential impact of V s anticompetitive action on E s downstream marginal cost were more (less) pronounced; (ii) or F 0 were smaller (larger), so the value of w under the cost-based pricing policy were further below (closer to) its value under the optimal pricing policy; or (iii) t t 0 were smaller (larger), so the increase in E s equilibrium output under the cost-based pricing policy were associated with a smaller (larger) increase in equilibrium customer transportation costs. 5

17 ers. 25 We conjecture that the conclusion is robust to alternative models of retail competition, although conclusive analytic results have proved to be elusive. Numerical solutions reveal that the conclusion generally holds when, for example, the retail rivals face linear demand curves for their di erentiated products. 26 Our conclusion may seem inconsistent with the substantial e ort that regulators have devoted to detecting and limiting sabotage in the telecommunications industry (Wood and Sappington, 2004) and with empirical studies that identify discrimination against una l- iated retail rivals in regulated industries (Rei en et al., 2000; Rei en and Ward, 2002). However, sabotage that has arisen in practice might re ect activities that reduce the incumbent supplier s operating cost while raising rivals costs. For instance, the sabotage might entail a reduction in the incumbent supplier s (costly) e ort to ensure seamless, high quality interconnection with its retail rivals (Rei en, 998). If the cost savings from such sabotage substantially exceed the expected regulatory nes and other costs associated with sabotage, then V might nd the sabotage to be pro table. Alternatively, the sabotage observed in practice might re ect regulated prices that di er from the optimal prices identi ed above. Indeed, wholesale prices in the telecommunications industry often are set to approximate e cient unit production costs (FCC, 200, p. 4). Such prices can reduce an incumbent supplier s wholesale pro t margin below its retail pro t margin, thereby creating incentives for the supplier to raise the costs of its retail rivals. 27 Regulatory policies that focus on keeping wholesale prices close to cost can serve con- 25 We have followed most studies in the literature by considering the incentives to raise rivals costs rather than reduce the demand for rivals products. Mandy and Sappington (2007) show that incentives for such demand reduction often do not arise even when incentives to raise rivals costs prevail. However, La ont and Tirole (200, p. 77) show that global price cap regulation can introduce incentives for demand-reducing sabotage. 26 In particular, we have been unable to identify parameter values for which the key qualitative conclusion reported above fails to hold when the demand for supplier j s product is a b p j + d j p k for j; k 2 f0; g (k 6= j), where a > 0; b > 0; and d j 2 (0; b) are parameters. 27 Bose et al. (206a) analyze a setting where the downstream interaction between a vertically-integrated provider (VIP) and a downstream (Cournot) rival is unregulated, but the regulator can limit the VIP s upstream pro t by regulating the price of access to its network. The authors demonstrate that the VIP often has an incentive to raise its rival s cost in this setting, and examine the optimal means to mitigate this incentive. 6

18 sumers well when retail competition is robust and when the regulator s knowledge of downstream industry conditions is limited. 28 However, we have shown that when retail competition is more limited and the regulator is well informed about industry conditions, consumers can be well served when a vertically-integrated incumbent supplier is permitted to recover its costs via a relatively large wholesale pro t margin and a smaller retail pro t margin. Potential concerns about imposing an inappropriate regulatory price squeeze on more e - cient retail competitors do not arise when the regulator is well informed about the operating costs of all suppliers. 29 However, in settings where: (i) industry suppliers are well informed about prevailing demand and supply conditions but the regulator is not similarly informed; and (ii) retail competitors are able to impose strong discipline on the incumbent supplier, consumers may be better served by policies like global price cap regulation (La ont and Tirole, 996, 2000, 200) than by policies in which a regulator dictates speci c prices for each of the incumbent supplier s retail and wholesale services. We have focused on the setting where the regulator seeks to maximize expected consumer surplus (E fsg). However, it can be shown that our key qualitative conclusions persist more generally. Speci cally, suppose the regulator seeks to maximize E fsg + 0 E f 0 g + E f g, where i 2 [0; ) denotes the weight the regulator places on the pro t of rm i 2 f0; g. It can be shown that as long as < 2 in this setting, the regulator will continue to set V s wholesale pro t margin above its retail pro t margin, thereby precluding incentives for sabotage. 30 However, when the regulator cares su ciently about E s pro t (i.e., when ), she may reduce the access charge (to lower E s production costs) and/or increase 2 V s retail price (to increase the demand for E s product) to the point that V s retail pro t margin exceeds its wholesale pro t margin. 28 Mandy (2009) demonstrates the merits of pricing inputs to re ect marginal costs of production in settings with limited information about industry demand. 29 Tardi and Taylor (2003, p. 347) observe that unduly lowering the incumbents regulated retail prices will squeeze both the incumbents because expected pro ts are lower and the entrants because the margin between wholesale and retail prices leaves less room to compete. 30 See Bose et al. (206b). 7

19 Future research should explicitly account for the limited information about industry conditions that typically prevails in practice. Future research might also consider alternative forms of retail competition and di erent forms of sabotage. For instance, rather than raise a rival s cost, sabotage might reduce the quality of a rival s product and thereby reduce consumer demand for the product. The foregoing analysis suggests that under an optimally designed regulatory policy, an incumbent supplier likely will refrain from this form of sabotage also. 3 3 This conjecture is supported by Mandy and Sappington (2007) s nding that even when an incumbent supplier su ers no reduction in wholesale pro t when its sales of wholesale access decline, the supplier may prefer not to reduce the demand for its rival s product. This is the case because the reduced demand for its product can induce the rival supplier to reduce the price of its retail product, which can reduce the incumbent supplier s retail pro t. 8

20 Proof of Lemma Given p 0 and w, E chooses p to: (4) and () provide: Proof of Lemma 2 Appendix Maximize p [ p c w ] v v 0 + p 0 p + t 0 t 0 + t ) v v 0 + p 0 p + t 0 ( p c w ) = 0 ) p (p 0 ) = 2 [ v v 0 + p 0 + t 0 + c + w ] ) p (p 0 ) p 0 = 2 [ v v 0 p 0 + t 0 + c + w ]. () x 0 (w; p 0 ; c) = 2 [ v 0 v + t ] + v v 0 + p 0 + t 0 + c + w = v 0 v + 2 t + t 0 p 0 + c + w x (w; p 0 ; c) = 2 [ v v 0 + t 0 ] = v v 0 + t 0 + p 0 (c + w) Since x 0 + x =, (2) and (3) imply: + v 0 v + p 0 t 0 c w, and (2). (3) 0 (w; p 0 ; c) = [ p 0 c d c u ] x 0 + [ w c u ] x F 0 (4) = [ p 0 c d ] x 0 + w x c u F 0 = = [ p 0 c d ] [ v 0 v + 2 t + t 0 p 0 + c + w ] + w [ v v 0 + t 0 + p 0 (c + w) ] c u F 0 [ p 0 c d ] [ v 0 v + 2 t + t 0 p 0 + c + w ] + w [ v v 0 + t 0 + p 0 (c + w) 2 ( t 0 + t ) + 2 ( t 0 + t ) ] c u F 0 9

21 = = = = [ p 0 c d ] [ v 0 v + 2 t + t 0 p 0 + c + w ] + w [ v v 0 t 0 2 t + p 0 (c + w) ] + w c u F 0 [ p 0 c d ] [ v 0 v + 2 t + t 0 (p 0 w) + c ] w [ v 0 v + t t (p 0 w) + c ] + w c u F 0 [ p 0 w c d ] [ v 0 v + 2 t + t 0 (p 0 w) + c ] + w c u F 0 [ c d ] [ v 0 v + 2 t + t 0 + c ] + w c u F 0 (5) = [ c d ] x 0 (w; p 0 ) + w c u F 0 0(w; p 0 ; s = c d. Proof of Lemma 3 (2) and (3) imply that aggregate transportation costs are: Z x 0 Z x t 0 x dx + t x dx = t0 (x 0 ) 2 + t (x ) = 2 " 0 [ v 0 v + 2 t + t 0 p 0 + c + w ] 2 [ v v 0 + t 0 + p 0 (c + w) ] 2 t 0 ( ) 2 + t ( ) 2 # = t 0 [ v 0 v + 2 t + t 0 p 0 + c + w ] 2 + t [ v v 0 + t 0 + p 0 (c + w) ] 2 8 [ t 0 + t ] 2. (6) (5), (2), (3), and (6) imply: S (w; p 0 ; c) = x [ v p ] + [ v 0 p 0 ] x 0 t0 (x 0 ) 2 + t (x ) 2 (7) 2 = v v 0 + t 0 + p 0 (c + w) v 2 [ v v 0 + p 0 + t 0 + c + w ] + [ v 0 p 0 ] v 0 v + 2 t + t 0 p 0 + c + w t 0 [ v 0 v + 2 t + t 0 p 0 + c + w ] 2 + t [ v v 0 + t 0 + p 0 (c + w) ] 2 8 [ t 0 + t ] 2 (8) 20

22 = v v 0 + t 0 + p 0 w c v 2 (v v 0 + p 0 w + t 0 + c + 2 w) + [ v 0 p 0 + w w ] v 0 v + 2 t + t 0 p 0 + c + w t 0 [ v 0 v + 2 t + t 0 (p 0 w) + c ] 2 + t [ v v 0 + t 0 + p 0 w c ] 2 8 [ t 0 + t ] 2 = v v 0 + t 0 + p 0 w c v 2 (v v 0 + p 0 w + t 0 + c) v v 0 + t 0 + p 0 w c w v0 v + 2 t + t 0 p 0 + c + w + [ v 0 (p 0 w) ] v0 v + 2 t + t 0 (p 0 w) + c w t 0 [ v 0 v + 2 t + t 0 (p 0 w) + c ] 2 + t [ v v 0 + t 0 + p 0 w c ] 2 = w + v v 0 + t 0 + c + [ v 0 ] v 0 v + 2 t + t 0 + c 8 [ t 0 + t ] 2 v 2 ( v v t 0 + c ) t 0 [ v 0 v + 2 t + t 0 + c ] 2 + t [ v v 0 + t 0 + c ] 2 8 [ t 0 + t ] 2 = v w + v v 0 + t 0 + c [ v v t 0 + c ] [ v v 0 + t ( t 0 + t ) 4 ( t 0 + t ) c ] v0 + v 0 v + 2 t + t 0 + c 2 ( t 0 + t ) + 2 ( t 0 + t ) 2 ( t 0 + t ) v0 v + 2 t + t 0 + c 2 ( t 0 + t ) t 0 [ v 0 v + 2 t + t 0 + c ] 2 + t [ v v 0 + t 0 + c ] 2 8 [ t 0 + t ] 2 2

23 = v w + v v 0 + t 0 + c [ v v t 0 + c ] [ v v 0 + t ( t 0 + t ) 4 ( t 0 + t ) c ] v0 + v 0 v t 0 + c v0 v + 2 t + t 0 + c + v 0 2 ( t 0 + t ) 2 ( t 0 + t ) t 0 [ v 0 v + 2 t + t 0 + c ] 2 + t [ v v 0 + t 0 + c ] 2 8 [ t 0 + t ] 2 v v 0 + t 0 + c = w + v 0 + [ v v 0 ] 2 ( t 0 + t ) [ v v t 0 + c ] [ v v 0 + t 0 + c ] 4 ( t 0 + t ) t 0 [ v 0 v + 2 t + t 0 + c ] 2 t [ v v 0 + t 0 + c ] 2 8 [ t 0 + t ] 2 8 [ t 0 + t ] 2 v v 0 + t 0 + c = w + v 0 + [ v v 0 ] 2 ( t 0 + t ) v0 v + 2 t + t 0 + c 2 ( t 0 + t ) v0 v + 2 t + t 0 + c 2 ( t 0 + t ) [ v v t 0 ] 2 c 2 4 [ t 0 + t ] t 0 ( v0 v + 2 t + t 0 ) c ( v 0 v + 2 t + t 0 ) + c 2 8 [ t 0 + t ] 2 t (v v 0 + t 0 + ) 2 2 c ( v v 0 + t 0 + ) + c 2 8 [ t 0 + t ] 2 (9) p 0; = v v c 4 [ t 0 + t ] 2 t 0 [ v 0 v + 2 t + t 0 ] 8 [ t 0 + t ] 2 2 t 0 c 8 [ t 0 + t ] t [ v v 0 + t 0 + ] 2 t c 8 [ t 0 + t ] 2 8 [ t 0 + t ] 2 = 8 [ t 0 + t ] 2 f 2 t [ v v 0 + t 0 + ] 2 c [ t 0 + t ] 2 t 0 [ v 0 v + 2 t + t 0 ] 4 [ t 0 + t ] [ v v 0 ] + 4 c [ t 0 + t ] 4 [ t 0 + t ] g s = [ v v 0 ] [ 2 t t 4 (t 0 + t ) ] + 2 t 0 t 4 t 0 t 2 t [ 2 t + 2 t 0 4 (t 0 + t ) ] + c [ 2 (t 0 + t ) + 4 (t 0 + t ) ] = [ v 0 v t 0 p 0 + w + c ] < 0. (20) 22

24 The inequality in (20) re ects Assumption. Proof of Proposition Let denote the Lagrange multiplier associated with constraint (8) and let denote the Lagrange multiplier associated with the constraint w We begin by proving: Lemma A. The Lagrangian function associated with [RP-ba] is: L = E fs(w; p 0 ; c)jbag + E f 0 (w; p 0 )jbag = w + B0 + B w 8 [ t 0 + t ] + w + c d 2 ( t 0 + t ) ( v 0 v + 2 t + t 0 + Efcjbag ) c u F 0 K(ba) where B 0 is comprised of all terms that do not involve w or and where, (2) B 4 [ v 0 v + 2 t + 2 t 0 + Efc j bag ] t t 0 + t [ 2 ( v v 0 + t 0 ) 2 Efc j bag ] + t 0 t 0 + t [ 2 ( v 0 v + 2 t + t 0 ) + 2 Efc j bag ] = 2 [ v v 0 ] 6 t 0 8 t 2 Efcjba g. (22) Proof. The lemma follows from (5), provided: E fs(w; p 0 ; c)jbag = w + B0 + B + 2. (23) 8 [ t 0 + t ] (9) implies: v v 0 + t 0 + Efcjbag E fs(w; p 0 ; c)jbag = w + v 0 + [ v v 0 ] 2 ( t 0 + t ) [ v v t 0 ] 2 Efc 2 jbag 4 [ t 0 + t ] v0 v + 2 t + t 0 + Efcjbag 2 ( t 0 + t ) t 0 ( v0 v + 2 t + t 0 ) Efcjbag [ v 0 v + 2 t + t 0 ] + Efc 2 jbag 8 [ t 0 + t ] 2 t ( v v 0 + t 0 + ) 2 2 Efcjbag ( v v 0 + t 0 + ) + Efc 2 jba g 8 [ t 0 + t ] 2. (24) 32 If w were negative, E could purchase (and then not employ) an in nite number of units of access. We will demonstrate below that p 0 0 at the solution to [RP - ba ]. Therefore, it is not necessary to explicitly constrain p 0 to be nonnegative. 23

25 (24) implies that the coe cient on w in E fs(w; p 0 ; c)jbag is. Furthermore, the coe cient on 2 in E fs(w; p 0 ; c)jbag is: z 2 = 4 [ t 0 + t ] + t 0 8 [ t 0 + t ] 2 t 8 [ t 0 + t ] 2 = 4 [ t 0 + t ] + 8 [ t 0 + t ] = 8 [ t 0 + t ]. In addition, the coe cient on in E fs(w; p 0 ; c)jbag is: z = v v 0 2 [ t 0 + v v 0 ] 4 [ t 0 + t ] v 0 v + 2 t + t 0 + Efcjba g = t 0 [ 2 ( v 0 v + 2 t + t 0 ) 2 Efcjba g ] t [ 2 ( v v 0 + t 0 ) 2 Efcjba g ] 8 [ t 0 + t ] 2 8 [ t 0 + t ] 2 8 [ t 0 + t ] f 4 [ v v 0 ] 4 [ t 0 + v v 0 ] 4 [ v 0 v + 2 t + t 0 + Efcjba g ] t 0 [ 2 ( v 0 v + 2 t + t 0 ) 2 Efcjba g ] t 0 + t t [ 2 (v v 0 + t 0 ) 2 Efcjba g ] t 0 + t g. (24) implies that the remaining terms in E fs(w; p 0 ; c)jbag are independent of w and. Let w and denote the values of w and identi ed in (9) and (0). Then the proposition follows from Lemma A if Assumption holds at w @ = 0, w = 0, (25) = w = w w = w w + [ c d ] [ v 0 v + 2 t + t 0 + E fcjbag ] w + [ c d ] [ v 0 v + 2 t + t 0 + E fcjbag ] c u F 0 K(ba) 0, (26) c u F 0 K(ba) = 0, (27) w 0, w = 0, 0; and 0. (28) We will show that the relations in (25) (28) hold when =, = 0, and w are as speci ed in (9) and (0), respectively. From (2), at a solution to @ = = + 0 ; w [ + ] = 0 ; and (29) B [ t 0 + t ] + v0 v + 2 t + t Efcjba g + c d 2 ( t 0 + t ) = 0. (30) 24

26 Observe that: t 0 [ ( v 0 v + 2 t + t 0 ) E fcjbag ] t 0 + t + t [ ( v v 0 + t 0 ) E fcjbag ] t 0 + t = E fcjbag [ t 0 + t ] + [ v v 0 ] [ t 0 + t ] t 0 [ 2 t + t 0 ] + t 0 t t 0 + t E fcjbag [ t 0 + t ] + [ v v 0 ] [ t 0 + t ] t 0 [ t 0 + t ] = t 0 + t = v v 0 E fcjbag t 0. (3) (30) implies that if = ; then: B [ t 0 + t ] + v 0 v + 2 t + t E fcjbag + c d = 0, B [ v 0 v + 2 t + t E fcjbag + c d ] = 0, 6 + B + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] = 0. (32) The value of that solves the equation in (32) is as speci ed in (0). From (0) and (22): > 0, B + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] > 0, 2 [ v v 0 ] 2 E fcjbag 6 t 0 8 t + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] > 0, 2 [ v 0 v ] + 2 E fcjbag 2 t c d > 0, c d > 2 [ v v 0 E fcjbag + t 0 ]. (33) Assumption 2 ensures that the last inequality in (33) holds. From (0) and (22): < c d, 6 [ B + 4 ( v 0 v + 2 t + t 0 + E fcjbag + c d ) ] < c d, B + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] < 6 c d, 2 [ v v 0 ] 2 E fcjbag 6 t 0 8 t + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] < 6 c d, 2 [ v v 0 ] 2 t E fcjbag < 2 c d, v 0 v t 0 + E fcjbag < c d. (34) Because E fcjbag < 2 [ 3 c (34) holds. E fcjbag ], Assumption 2 ensures that the last inequality in 25

27 (0) and (27) imply that when = : w = c u +F 0 +K(ba) [ c d ] [ v 0 v + 2 t + t 0 + E fcjbag ] = c u + F 0 + K(ba) v 0 3 [ v 0 v + E fcjbag t 0 c d ] v + 2 t + t 0 + E fcjbag 3 (v 0 v + E fcjbag t c d ) = c u + F 0 + K(ba) 3 [ v 0 v + E fcjbag t 0 c d ] 3 [ 2 ( v 0 v ) + 2 E fcjbag + 6 t + 4 t 0 2 c d ] = c u + F 0 + K(ba) [ v 0 v + E fcjbag t 0 c d ] 9 [ t 0 + t ] [ v 0 v + E fcjbag + 2 t t c d ] > 0. (35) The inequality in (35) re ects Assumption 2, since E fcjbag < 2 [ 3 c E fcjbag ]. (28) and (29) imply that = 0 and =, since w > 0. Therefore, E f 0 (w; p 0 )jbag = 0. To verify that Assumption holds, rst observe from (0) and (22) that: > v 0 v + c t 0, 6 [ B + 4 ( v 0 v + 2 t + t 0 + E fcjbag + c d ) ] > v 0 v + c t 0, B + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] > 6 [ v 0 v + c t 0 ], 2 [ v v 0 ] 2 E fcjbag 6 t 0 8 t + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] > 6 [ v 0 v + c t 0 ], 2 [ v 0 v ] + 2 E fcjbag 2 t c d > 6 [ v 0 v + c t 0 ] (36), 4 c d > 4 [ v 0 v t 0 ] + 6 c 2 E fcjbag, c d > v 0 v t 0 + [ 3 c E fcjbag ]. (37) 2 Now observe from (22) and (36) that: < v 0 v + c + 2 t + t 0 26

28 , 6 [ B + 4 ( v 0 v + 2 t + t 0 + E fcjbag + c d ) ] < v 0 v + c + 2 t + t 0, B + 4 [ v 0 v + 2 t + t 0 + E fcjbag + c d ] < 6 [ v 0 v + c + 2 t + t 0 ], 2 [ v 0 v ] + 2 E fcjbag 2 t c d < 6 [ v 0 v + c + 2 t + t 0 ], 4 c d < 4 [ v 0 v ] + 8 t t + 6 c 2 E fcjbag, c d < v 0 v + 2 t t + [ 3 c E fcjbag ]. (38) 2 Assumption 2 ensures that the inequalities in (37) and (38) hold. Finally, observe that p 0 > 0 at the solution to [RP-ba]. This is the case because > 0 and w > 0, so p 0 = w + > 0. Proof of Observation From = [ v p + [ v 0 p 0 0 x 0 t 0 x 0 t = [ v p ) x + [ v 0 p 0 0 = x 0 + [ v p t x ] + [ v 0 p 0 t 0 x 0 t 0 = x 0 + [ v p t x (v 0 p 0 t 0 x 0 ) ] (39) (40) = x 0. (4) The equality in (39) holds 0 ( x = 0, from (5). The equality in (40) ( x (42) The equality in (4) holds because the marginal consumer is indi erent between purchasing from V and from E. From 0 = [ w c u + x 0 + [ p 0 c u c d 0, and 27