Costly Price Discrimination

Transcription

1 Costly Prie Disrimination Peter T. Leeson and Russell S. Sobel Department of Eonomis, West Virginia University February 16, 26 Abstrat In standard miroeonomi theory, perfet prie disrimination is soially e ient. However, this theory neglets that enating prie disrimination is ostly to rms. We prove that when this ostliness is aounted for, perfet prie disrimination is often soially ine ient. Under linear demand and marginal ost funtions, for pure monopolists, perfet prie disrimination is sometimes soially ine ient. For monopolisti ompetitors it is always soially ine ient. Under both market strutures, when perfet prie disrimination is soially ine ient, it is more ine ient than uniform priing. Our model is ompletely general. It applies to all prie-searher market strutures, with and without xed entry osts. 1

2 1 Introdution Is perfet prie disrimination soially e ient? A survey of eonomists, or glane through any eonomis textbook, would universally suggest the answer is yes. Few readers beginning this paper would disagree. The perfetly prie disriminating monopolist resolves the deadweight loss by expanding output, reating a soial gain, and transfers onsumer surplus to himself, whih is soially neutral. The net result is inreased soial welfare. While this reasoning seems sensible, by the end of this paper you will be onvined that it is often wrong. The obviousness of perfet prie disrimination s soial e ieny has preluded examinations of its auray. 1 The rst, and only, serious hallenge to this onventional wisdom is Bhaskar and To (24) who prove that under monopolisti ompetition with xed entry osts, perfet prie disrimination an be soially ine ient. Beause ompetitive pressures in these markets fully dissipate pro ts and onsumer surplus is fully extrated, there is no soial surplus generated to o set the xed entry osts to soiety. 2 Building on their insight, we prove that perfet prie disrimination is soially ine ient in a muh broader lass of ases. We show that under linear demand and marginal ost funtions, for pure monopolists, perfet prie disrimination is sometimes soially ine ient. For monopolisti ompetitors it is always soially ine ient. Our theory is ompletely general. In our model perfet prie disrimination requires neither monopolisti ompetition nor xed entry osts to be welfare reduing. Even under pure monopoly with zero xed entry osts, perfet prie disrimination an be soially ine ient. 1 A sizeable literature examines the welfare properties of seond- and third-degree prie disrimination (see for instane, Varian 1985, Shmalensee 1981, Katz 1984, 1987, Borenstein 1985, Chen 1999, Holmes 1989, Shwartz 199, Corts 1998). On nonlinear priing generally see, for instane, Spene 1976, 1977, Katz 1983, Roberts 1979, Chiang and Spatt See also, Stole (25). 2

3 We show this ours beause rms fae osts of enating prie disrimination. 3 These osts, whih inlude segmenting onsumers, identifying elastiities, and preventing resale, are signi ant in all industries. 4 This, of ourse, is the reason not all rms enat this priing strategy. The omission of these transations osts from existing theories of prie disrimination is important beause, as Varian has pointed out: A full welfare analysis of attempts to engage in [perfet] prie disrimination annot neglet the transations osts involved in the negotiation itself (1989: 64). The intuition behind our model of how ostly perfet prie disrimination an lead to ine ieny is straightforward. To enat this priing strategy a rm would be willing to expend resoures up to the amount of its gain from doing so: the sum of the deadweight loss and onsumer surplus, both of whih are transferred to the monopolist under this prie regime. However, the soial gain of suh a move is only the value of solving the deadweight loss. The private bene t of perfet prie disrimination therefore exeeds the soial bene t. Although this divergene is not a problem when, as previous theory has assumed, the ost of prie disriminating is zero, it beomes a problem if prie disrimination is ostly. Expenditures to enat prie disrimination beyond the value of the deadweight loss are soially wasteful but privately pro table. The rm is expending sare resoures to seure a transfer of onsumer surplus that yields no soial gain. Departing from the ubiquitous assumption that prie disrimination is ostless to the rm, we derive the general and spei onditions under whih perfet prie disrimination is soially ine ient in the presene of these osts. Setion 2 establishes generally (for all prie searhers) when enating perfet prie disrimination is 3 While the ostliness of prie disrimination has been reognized in passing by some (see for instane Tirole 21 and Posner 1975), no one has reonsidered the theory of prie disrimination in light of these osts. 4 In fat, there is an entire industry known as yield management that harges rms for speialized onsulting and software to help them implement prie disrimination. These servies an be extremely ostly, with the software alone osting upwards of $1 million per year for a single large rm (see Lieberman 1991, 1993). 3

4 privately pro table but soially ine ient. Setion 3 does this spei ally for the ase of pure monopoly, while Setion 4 does so for the ase of monopolisti ompetition. Setion 5 onludes. 2 Costly Perfet Prie Disrimination and Soial E ieny: The General Case Figure 1 depits the traditional welfare analysis for a single-prie prie searher. The area of triangle de is the deadweight loss to soiety assoiated with the prie searher s market power. Consumer surplus is given by the area of triangle ab. [Figure 1 about here.] Suppose this produer wants to onvert his priing strategy to one of perfet prie disrimination. For reasons disussed above, doing so is ostly. Let C be the prie searher s ost of enating perfet prie disrimination. 5 For ease of exposition, here we assume the ost of perfetly prie disriminating is a xed ost of adopting the new priing strategy. However, the results we derive apply equally when there are variable osts of perfetly prie disriminating, though the interpretation hanges slightly (see the Appendix). We also assume that the ost of implementing perfet prie disriminating is industry spei. To determine the pro tability of implementing perfet prie disrimination, the prie searher must weigh the bene ts of enating perfet prie disrimination against the ost, C. His bene t of perfetly prie disriminating onsists of two omponents. The rst onsists of transforming the area of the deadweight loss triangle de into produer surplus. Formally, this area an be written as [P (Q) MC(Q)], whih represents the lost gains from trade under a single-priing strategy. In the traditional 5 This assumes ostless uniform priing. If uniform priing is also ostly, C refers to the additional ost of adopting the more omplex priing strategy of perfet prie disrimination. 4

5 perfet prie disrimination analysis, this portion of the prie searher s bene t is generally the only bene t onsidered. Equally important to the prie searher, however, is the seond omponent of his bene t from implementing perfet prie disrimination. This is the triangle ab. Qm R The area of this triangle, [P (Q) P m ]; onstitutes the onsumer surplus enjoyed by onsumers under uniform priing, whih is transferred to the prie searher by enating perfet prie disrimination. This seond portion of the prie searher s bene t from perfetly prie disriminating, the transferred onsumer surplus, does not ontribute to soial welfare as does the rst portion of his bene t, the transformed deadweight loss. There is no e ieny gain from transferring onsumer surplus under a single-prie strategy to produer surplus under a strategy of perfet prie disrimination; there is only a transfer. Thus, the transferred onsumer surplus represents a private bene t that is not a soial bene t. When the prie searher s total private bene t of perfetly prie disriminating, Qm R [P (Q) P m ]+ [P (Q) MC(Q)] (transferred onsumer surplus plus transformed deadweight loss), is greater than his ost of enating perfet prie disrimination, C, he will do so. In ontrast, where his total private bene t of this priing strategy is less than its ost, he will hoose not to enat perfet prie disrimination. Two possibilities, with ontraditory soial welfare e ets, arise when the prie searher nds perfet prie disrimination privately pro table. In the rst ase, the ost of perfet prie disrimination is less than or equal to the deadweight loss of uniform priing, C [P (Q) MC(Q)]. When he implements perfet prie disrimination soiety gains. The deadweight loss of uniform priing is removed, and it is removed at a ost less than the soial bene t of doing so. In the seond ase, the ost of enating perfet prie disrimination is higher, and exeeds the deadweight loss of maintaining uniform priing. Sine the prie searher s 5

6 Qm R bene t from pursuing perfet prie disrimination equals [P (Q) P m ]+ [P (Q) MC(Q)], he is willing to expend resoures up to this point to implement perfet prie disrimination. Here, however, a problem emerges. While it is privately pro table for the prie searher to pursue perfet prie disrimination in this ase, it is soial welfare reduing. Every dollar spent by the prie searher beyond [P (Q) MC(Q)] to implement perfet prie disrimination generates a bene t to him by transforming would-be onsumer surplus into produer surplus. But from the perspetive of soial welfare, these dollars are a waste resoures expended only to a et the transfer of other resoures. The e et is like that originally desribed by Tullok (1967) on the soial losses of rent seeking. The prie searher s private ost/bene t alulus an lead him to pursue prie disrimination when it is soially ine ient. This leads us to our rst proposition. Proposition 1 For all prie searhers, perfet prie disrimination is privately profitable but soially ine ient when the ost of enating perfet prie disrimination exeeds the deadweight loss generated under uniform priing, but is less than or equal to the sum of transformed deadweight loss and transferred onsumer surplus. Proof. From above, the prie searher s private bene t from enating perfet prie disrimination is: R Qm [P (Q) P m ] + [P (Q) MC(Q)] The soial bene t from the prie searher enating perfet prie disrimination is only: [P (Q) MC(Q)] Qm R So, when [P (Q) P m ] + [P (Q) MC(Q)] C > [P (Q) MC(Q)]: Qm R (A) [P (Q) P m ] + [P (Q) MC(Q)] C ; enating perfet prie disrimination is pro table for the prie searher, and 6

7 (B) [P (Q) MC(Q)] C < ; enating perfet prie disrimination is soially ine ient. In the analysis that follows we onsider only those ases for whih enating perfet prie disrimination is privately pro table. Thus, we assume that C always satis es (A). Let us all (B) the soial ine ieny ondition. This ondition implies an important fat about the relative ine ieny of perfet prie disrimination and uniform priing. Proposition 2 When perfet prie disrimination is soially ine ient, it is always more soially ine ient than uniform priing. Proof. From Proposition 1 we know that enating perfet prie disrimination is privately pro table and soially ine ient when: [P (Q) P m ] + [P (Q) MC(Q)] Qm R C > [P (Q) MC(Q)]: We also know that the soial ine ieny of maintaining uniform priing is: [P (Q) MC(Q)]: From this, it immediately follows that whenever C satis es the soial ine ieny ondition, it is also more soially ine ient than maintaining uniform priing. 3 Costly Perfet Prie Disrimination Under Pure Monopoly A pure monopoly may be the result of government-ereted barriers to entry a speial grant of monopoly privilege, or may emerge naturally in the market when there are signi ant eonomies of sale in an industry. Pure monopolists, like all prie searhers, have a hoie about the kind of priing sheme they will follow, and also like all prie searhers, they fae a ost of perfetly prie disriminating. Beause they do not fae ompetition, however, pure monopolists an earn positive eonomi 7

8 pro t. There are no fores tending to push this pro t to zero. Thus, here we onsider only pure monopolists earning positive eonomi pro t. 6 To understand when enating perfet prie disrimination is soially ine ient, it is useful to rst think about the issue intuitively. Consider Figure 2. [Figure 2 about here.] MC depits the situation of onstant marginal prodution osts. Under this marginal ost funtion, the threshold of soially ine ient perfet prie disrimination is given by the triangle ab the deadweight loss transformed into produer surplus by enating perfet prie disrimination. One the ost of enating perfet prie disrimination exeeds this area, it beomes soially ine ient. Now onsider MC, the ase in whih marginal prodution osts are inreasing. As MC rotates up to MC, the deadweight loss eliminated by enating perfet prie disrimination beomes smaller. To see this, ompare the size of the deadweight loss when the prie searher faes onstant marginal prodution ost (MC) triangle ab onsidered before, to the size of the deadweight loss when the prie searher faes rapidly inreasing marginal prodution osts (MC ) triangle abd. Crossing the ine ieny threshold in this ase therefore requires only a relatively small ost of enating perfet prie disrimination. The reverse happens when marginal prodution osts are dereasing in output, as they are for MC. Here, the ine ieny threshold of enating perfet prie disrimination grows substantially larger than under onstant marginal ost, MC. The deadweight loss eliminated by perfet prie disrimination enlarges to the triangle abe. Enating perfet prie disrimination needs be quite ostly before its implementation is soially ine ient. The growing (shrinking) deadweight loss solved by enating perfet prie disrimination as marginal prodution osts move from steeply inreasing (dereasing) to steeply dereasing (inreasing) suggests that for a given 6 A pure monopolist ould earn exatly zero eonomi pro t, in whih ase he formally resembles a monopolisti ompetitor, whih we onsider in the subsequent setion. 8

9 ost of enating perfet prie disrimination, C, the likelihood that enating perfet prie disrimination is soially ine ient dereases (inreases) as marginal prodution osts go from steeply inreasing (dereasing) to steeply dereasing (inreasing). With this in mind we are ready for our rst proposition regarding pure monopoly. Proposition 3 For a pure monopolist who faes linear demand and marginal prodution osts, enating perfet prie disrimination is soially ine ient if and only if the slope of marginal prodution ost is greater than the produt of the (absolute value of the) slope of demand and the prie searher s rate of return on his investment in enating perfet prie disrimination minus one. Proof. Let C be the share of the potential bene ts a prie searher reeives by perfetly prie disriminating that he must spend to suessfully implement perfet prie disrimination. So, C = C R [P (Q) P m ] + [P (Q) Qm MC(Q)] Sine enating perfet prie disrimination is ostly, C > : Further, beause the pure monopolist we onsider earns positive eonomi pro t, C < 1. Thus, < C < 1: From the soial ine ieny ondition we know that enating perfet prie disrimination is soially ine ient if and only if: C > [P (Q) MC(Q)]. Rewriting this in terms of C gives us: C R Qm [P (Q) P m ] + [P (Q) MC(Q)]! > [P (Q) MC(Q)] Therefore, to determine when enating perfet prie disrimination is soially ine - ient we need to establish when this inequality holds. To do this, onsider a pure monopolist who faes a linear demand funtion of the form: P = Q, where ; > and is the (absolute value of the) slope of 9

10 demand. His total prodution ost under uniform priing is given by the funtion: T C = +Q+Q 2, whih results in a linear marginal ost funtion, where ;, may be positive (in the ase linearly inreasing marginal prodution osts), negative (in the ase of linearly dereasing marginal prodution osts), or zero (in the ase of onstant marginal prodution osts), and 2 is the slope of marginal prodution ost: His total ost when he enats perfet prie disrimination is therefore: T C = + Q + Q 2 + C, where C is the xed ost of perfet prie disrimination. The monopolist s problem is as follows: First, determine his pro t-maximizing quantity ( ) and prie (P m ): Q max = P (Q)Q T C(Q) = ( Q)Q ( + Q + Q 2 + C) His F.O.C. is thus = ( ) 2( + )Q = Solving for : = 2( + ) Substituting and solving for P m gives: P m = = ( + ) To alulate : R Qm [P (Q) P m ] and [P (Q) MC[ ] = MC(Q)], we need to evaluate MC at + + We must also alulate the ompetitive level of output, Q : To do this set P = MC: 1

11 Q = + 2Q: And solving for Q : Q = 2 + Now, alulate R Qm [P (Q) P m ] : R Qm [P (Q) P m ] = ( )2 8( + ) 2 And [P (Q) MC(Q)] : [P (Q) MC(Q)] = 2 ( ) 2 8( + ) 2 (2 + ) Now, substitute bak into C MC(Q)], whih yields: R Qm [P (Q) P m ] + [P (Q) MC(Q)]! > [P (Q) ( ) C 2 8( + ) + 2 ( ) 2 > 2 8( + ) 2 + (2 + ) 2 ( ) 2 8( + ) 2 + (2 + ) Solving for 2 (the slope of MC): 2 > C Note that sine C = C Qm R Q R [P (Q) P m]+ [P (Q) MC(Q)] Qm 1 C, C is equal to the monopolist s rate of return on his investment in enating perfet prie disrimination, whih we all r: So, this expression may be re-written in terms of the monopolist s rate of return on investment in enating perfet prie disrimination as: 2 > (r 1) 11

12 4 Costly Perfet Prie Disrimination Under Monopolisti Competition The foregoing disussion established under what onditions enating perfet prie disrimination, for a pure monopolist who found this priing strategy pro table, is soially ine ient. This setion establishes the same for monopolisti ompetitors. The ritial di erene here is that monopolisti ompetitors fae ompetitive pressures from their rivals that tend toward the elimination of eonomi pro t in equilibrium. In ontrast to pure monopolists, eonomi pro t is equal to zero for these prie searhers, even on their prie disriminating ativities. This ompetitive pressure an be thought of in two ways. On the one hand, the ompetition of monopolisti ompetitors for onsumers in a partiular industry tends to drive down the demand that eah rm faes in the industry, reduing eah rm s eonomi pro t to zero in equilibrium. Alternatively, the ompetitive pressure of rival prie searhers for resoures inluding those used to suessfully enat perfet prie disrimination tends to drive up the prie of these resoures suh that in equilibrium eah monopolistially ompetitive prie searher is expending resoures on perfetly prie disriminating and other ativities equal to the bene t of these resoures in generating him revenue. Eonomi pro t is again driven to zero. The zero eonomi pro t ondition for monopolisti ompetitors in equilibrium therefore means that for eah prie searher in an industry, the ost of implementing perfet prie disrimination is equal to the prie searher s private bene t of doing Qm R so: C = [P (Q) P m ] + [P (Q) MC(Q)]: So, under monopolisti ompetition, C = 1, or equivalently, r = : One other important hange needed to modify the model from pure monopoly to 12

13 monopolisti ompetition deals with the range of potential marginal prodution ost funtions that prie searhers under monopolisti ompetition may fae. Although pure monopolists may fae dereasing marginal prodution osts, monopolisti ompetitors annot. If they did, the situation would result in a natural monopoly, meaning there is only one prie searher in the industry, as in the ase we onsidered in the previous setion. Under monopolisti ompetition the slope of the marginal prodution ost urve must be greater than or equal to zero, so 2. With this in mind, onsider Proposition 4. Proposition 4 For monopolisti ompetitors who fae linear demand and marginal prodution osts, enating perfet prie disrimination is always soially ine ient. Proof. To prove this proposition we simply follow the proedure for proving Proposition 3. From the soial ine ieny ondition we know that for perfet prie disrimination to be soially ine ient, the ost of enating it must satisfy: C > [P (Q) MC(Q)]. Re-written in terms of C this was: C R Qm [P (Q) P m ] + [P (Q) MC(Q)]! > [P (Q) MC(Q)] Using substitution: ( ) C 2 8( + ) + 2 ( ) 2 > 2 8( + ) 2 + (2 + ) 2 ( ) 2 8( + ) 2 + (2 + ) Solving for 2 (the slope of MC): 2 > C 13

14 Under monopolisti ompetition, zero eonomi pro t implies C = 1, so this gives: 2 > Sine under monopolisti ompetition 2, this is always satis ed. Note that this ondition for soial ine ieny applies even when =, implying no xed ost of market entry for the rm. Thus, unlike previous analyses of monopolisti ompetition that require xed entry osts for perfet prie disrimination to be ine ient, our theory does not. 5 Conlusion We have demonstrated how under plausible onditions prie searhers are led to pursue too muh perfet prie disrimination, generating welfare losses even when perfet prie disrimination is used. This ours beause prie disrimination is ostly to implement and the rm will invest resoures both to transform the deadweight loss, whih reates a soial bene t, and transfer existing onsumer surplus to itself, whih does not. For pure monopolists, enating perfet prie disrimination is sometimes soially ine ient. For monopolisti ompetitors, enating perfet prie disrimination is always soially ine ient. For both market strutures, when perfet prie disrimination is soially ine ient, it is always more ine ient than if uniform priing were maintained. The straightforwardness of our argument, one it has been explained, makes it suseptible to the laim that it is obvious. However, the obviousness of our result is rivaled only by the alleged obviousness of perfet prie disrimination s soial e ieny. Of ourse, both annot simultaneously be true. The proposition that perfet prie disrimination is always soially e ient remained unquestioned until Bhaskar and To (24). Their important work showed 14

15 that under monopolisti ompetition with xed entry osts, perfet prie disrimination an be soially ine ient. Here we build on their onlusion to provide a general theory of perfet prie disrimination s soial ine ieny, whih also holds for pure monopolists and in ases with zero xed entry osts. The widely-held belief that perfet prie disrimination is soially e ient is onditional on ignoring all osts of enating prie disrimination. However, these osts are generally substantial and always exist. One they are inorporated into the analysis, the soial e ieny of perfet prie disrimination is no longer obvious. While some authors have mentioned the osts of enating prie disrimination in passing, our analysis is the rst to model these osts and derive their impliations. We examined the ine ieny of ostly perfet prie disrimination only under the relatively simple ases of linear demand and linear marginal ost funtions. Future analyses should investigate whether the propositions regarding the ine ieny of perfet prie disrimination derived in this paper also hold under less simplifying assumptions about demand and prodution osts. We also irumsribed our disussion to the ase of perfet prie disrimination. But there is good reason to think that the general problem of ine ieny we desribed also applies to seond- and third-degree prie disrimination. Here too, if prie disrimination is ostly, sine the prie searher s bene ts of prie disrimination involve not only the transformed deadweight loss under uniform priing but also the transferred onsumer surplus, in a subset of ases prie disrimination will prove privately pro table but be soially ine ient. 15

16 Referenes [1] Bhaskar, V. and Ted To (24). Is Perfet Prie Disrimination Really E ient? An Analysis of Free Entry, RAND Journal of Eonomis, [2] Borenstein, S. (1985). Prie Disrimination in Free-Entry Markets, RAND Journal of Eonomis, [3] Chen, Y. (1999). Oligopoly Prie Disrimination and Resale Prie Maintenane, RAND Journal of Eonomis, [4] Chiang, Raymond and Chester Spatt (1982). Imperfet Prie Disrimination and Welfare, Review of Eonomi Studies, [5] Cowling, Keith and Dennis Mueller (1978). The Soial Costs of Monopoly Power, Eonomi Journal, [6] Corts, Kenneth (1998). Third-Degree Prie Disrimination in Oligopoly: All- Out Competition and Strategi Commitment, RAND Journal of Eonomis, [7] Holmes, T. (1989). The E ets of Third-Degree Prie Disrimination in Oligopoly, Amerian Eonomi Review, [8] Katz, Mihael (1983). Non-Uniform Priing, Output and Welfare Under Monopoly, Review of Eonomi Studies, [9] Katz, Mihael (1984). Prie Disrimination and Monopolisti Competition, Eonometria, [1] Katz, Mihael (1987). The Welfare E ets of Third-Degree Prie Disrimination in Intermediate Good Markets, Amerian Eonomi Review,

17 [11] Lieberman, Warren (1991). Making Yield Management Work for You: Ten Steps to Enhaned Revenues, in Arthur D. Little In., Travel, Tourism and Hospitality Bulletin. [12] Lieberman, Warren (1993). Debunking the Myths of Yield Management, The Cornell H.R.A. Quarterly, [13] Roberts, Kevin (1979). Welfare Considerations of Nonlinear Priing, Eonomi Journal, [14] Shmalensee, Rihard (1981). Output and Welfare Impliations of Monopolisti Third-Degree Prie Disrimination, Amerian Eonomi Review, [15] Shwartz, M. (199). Third-Degree Prie Disrimination and Output: Generalizing a Welfare Result, Amerian Eonomi Review, [16] Spene, Mihael (1976). Produt Seletion, Fixed Costs, and Monopolisti Competition, Review of Eonomi Studies, [17] Spene, Mihael (1977). Nonlinear Pries and Welfare, Journal of Publi Eonomis, [18] Stole, Lars (25). Prie Disrimination and Imperfet Competition, in M. Armstrong and R. Porter eds. Handbook of Industrial Organization Vol. III. Amsterdam: North-Holland. [19] Tirole, Jean (21). The Theory of Industrial Organization. Cambridge: MIT Press. [2] Tullok, Gordon (1967). The Welfare Costs of Tari s, Monopoly and Theft, Western Eonomi Journal, [21] Varian, Hal (1985). Prie Disrimination and Soial Welfare, Amerian Eonomi Review,

18 [22] Varian, Hal (1989). Prie Disrimination, in R. Shmalensee and R. Willig eds., Handbook of Industrial Organization Vol. I. Amsterdam: North-Holland. 18

19 A Appendix Here we onsider the ase when the prie searher s ost of pratiing perfet prie disrimination inludes variable ost omponents (with respet to the rm s output), either alone, or in addition to xed ost omponents. Let a more generalized form of the ost of perfet prie disrimination to the rm, C, be: C = + 1 Q + 2 Q 2 where is the xed ost of enating perfet prie disrimination. The variable ost omponent is determined by 1 and 2. When 2 =, and 1 >, then 1 measures the onstant per unit ost of perfetly prie disrimination (a ost per unit of preventing resale, for example). When 1 > and 2 >, the marginal ost (per additional unit of output) of perfetly prie = Q, whih an be either an inreasing (when 2 > ) or dereasing (when 2 < ) amount per unit as additional units of output are produed. In the presene of 1 osts alone, the interept of the rm s marginal ost urve is inreased, resulting in a parallel upward shift in the marginal ost urve. When 2 osts are present, the slope of the rm s marginal ost urve is also a eted. This an be seen more learly in the rm s new total ost funtion: T C = +Q+Q 2 +C = +Q+Q 2 +( + 1 Q+ 2 Q 2 ) = (+ )+(+ 1 )Q+(+ 2 )Q 2 whih produes a marginal ost funtion of: MC = ( + 1 ) + 2( + 2 )Q 19

20 These two equations may be re-written using = +, = + 1, and = + 2 as: T C = + Q + Q 2 and MC = + 2 Q Thus, inluding variable osts of perfet prie disrimination in our analysis simply results in all appearanes of and being replaed by and (note that is already used in our analysis beause we inlude both, whih an be zero or positive, and C, our equivalent of the xed ost of enating perfet prie disrimination ): Beause, like, falls out of the equations prior to the nal solutions in all of our propositions, onstant per unit variable osts of the 1 variety do not hange our results. However, when the variable ost of perfetly prie disriminating either inreases or dereases with output beause of 2 osts, altering the slope of the rm s marginal ost urve, the interpretation of our propositions involving hanges slightly. In Proposition 3, where the soial ine ieny ondition is 2 > (r 1), this now beomes 2 > (r 1), or 2( + 2 ) > (r 1): The intuition remains the same in that we are still referring to the slope of the rm s marginal ost urve (2), but this slope is now the slope of marginal ost inlusive of any slope-altering marginal osts related to enating perfet prie disrimination. Proposition 4, where the soial ine ieny ondition is 2 >, is similarly a eted to beome 2( + 2 ) >. 2

21 Figure 1. The Prie Searher s Private Benefit from Perfet Prie Disrimination P D a P m b P d e MC MR Q D Q 21

22 Figure 2. Marginal Cost and the Soial Ineffiieny of Perfet Prie Disrimination P MC D P m a d b MC e MC MR D Q Q Q Q 22