Equation Chapter 1 Section 1

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1 Equation Capter Section Ladder Pricing A New Form of Wolesale Price Discrimination Ian McQuin Dobbs Newcastle University Business Scool 5, Barrack Road Newcastle upon Tyne NE 4SE, UK. ianmdobbs@btinternet.com

2 Abstract Wolesale ladder pricing involves setting te wolesale price a retailer faces as a nonlinear (generally increasing) function of te price cosen by tat retailer. Te special case were te ladder pricing contract is linear is sown to be equivalent to a form of revenue saring. Optimal profit maximizing ladder pricing/revenue saring is examined, given tat retailers are privately informed of teir demands and costs, and ave control over weter tey participate, and if so, wat retail price tey set. Te solution is compared wit te alternative of wolesale quantity discounting as relative level of retailer demand/cost eterogeneity is varied; ladder pricing/revenue saring tends to outperform quantity discounting by an increasing amount te greater retailer demand eterogeneity is relative to cost eterogeneity. Ladder pricing as recently been implemented for 8 and related calls in UK Telecoms and as been subject to extended legal dispute; te case and issues involved are discussed. JEL Classification: C6, D4, L4. Keywords: Ladder pricing, Quantity discounts, Revenue Saring, Wolesale pricing, Nonlinear pricing, Francising.

3 . Introduction Tis paper seeks to bring attention to a type of wolesale pricing contract ereafter referred to as ladder or tiered wolesale pricing. Tis form of wolesale pricing contract does not seem to ave been discussed in te wolesale pricing literature, so it seems useful to explain its form, examine its scope, and make comparisons wit more familiar forms of wolesale contract. A ladder wolesale pricing contract is a contract in wic te wolesale price levied for eac participating retailer is contingent on te downstream price tat retailer subsequently cooses. Recent examples in UK telecoms practice ave featured linear and/or non-linear, continuous and/or step functions in wic te wolesale price is set as an increasing function of retail price, ence te terms ladder or tiered for te wolesale pricing contract. Te paper (i) examines te extent to wic ladder pricing can elp to coordinate supply cannels, (ii) discusses te linkage between ladder pricing and revenue saring and sows tat a linear ladder pricing contract is, in some circumstances, equivalent to revenue saring in conjunction wit uniform pricing, (iii) examines te relative performance of ladder pricing vis a vis quantity discounting, sowing ow relative performance is positively related to te extent of eterogeneity in retailer demands vis a vis teir costs, (iv) discusses te pros and cons of coosing a step function versus a continuous function for te ladder pricing contract and (v) discusses recent and possible legal and regulatory attitudes to tis form of pricing. UK Telecom fixed line network providers ave recently introduced ladder pricing contracts as a means of revenue saring for 8/845/87 and related call numbers (often referred to as NGNs, or non-geograpic numbers). Tese numbers are used in te UK by organizations, bot public and private, for elp lines etc. Te 8 number ranges ave been typically free to use by tose wo make a fixed line call. By contrast, mobile network operators (MNOs) ave often made significant carges for suc calls. Suc calls to service providers are typically terminated by te fixed line networks involved, and tey ave tried to implement wolesale ladder pricing in order to revenue sare wit te MNOs on tese number ranges. Te mobile network operators appealed te imposition of suc wolesale pricing, and te issue as gone before te regulator Ofcom and tence to te Competition Appeal Tribunal, te Court of Appeal, and ten te Supreme Court. Te Supreme Court made a final ruling (Supreme Court, 4) in July 4 tat suc a form of pricing sould indeed be permitted. Te economics of tis case revolved around weter te proposed wolesale carges 3

4 actually gave mobile operators an incentive to reduce teir retail prices for tese social calls. Clearly, if te wolesale carge rises in some sense fast enoug wit te coice of retail price, retailers will ave an incentive to reduce prices, and tat is likely to be beneficial to final consumers. Te wolesale pricing problem examined is one in wic a profit maximizing monopoly wolesaler faces a significant population of potential downstream profit maximizing retailers, knows te distribution function for teir type, a joint distribution including bot demand and cost caracteristics, but is unable to distinguis tem by type. Te wolesaler posts a contract, and te retailers ten coose weter to participate, and if so, wat price tey will carge at te retail level. Because of te timing of decisions, tis format is equivalent to a Stackelberg game in wic te wolesaler is te leader and retailers are followers, an important class of game used in past literature to model entry decisions by firms (e.g. Dixit, 98). In tis setting, if tere is just one retailer, it is well known tat quantity discounting scemes, including a simple two part tariff, can be used to fully coordinate te distribution cannel (e.g. Moorty, 987) and may perform well even in more complex settings (e.g. Raut et al., 8). However, wen tere are multiple eterogeneous retailers, and participation is a retailer coice variable, a two part tariff cannot be used to coordinate te distribution cannel and te wolesaler can do better profit wise by deviating from full coordination (Ingene and Parry, 995; Ingene et al., ). Basically, wen constrained to set te same fixed carge to all participating retailers, a profit maximizing wolesaler as an incentive to set a iger tan marginal cost price, and tis implies a reduction in overall cannel profitability. Using a more general quantity discounting contract will improve performance, but tere is still a fall away from full coordination joint profitability. If te wolesaler is able to monitor ex post te prices te participating retailers coose to set, ten setting ex ante a standard contract wic ties wolesale price carged to te price cosen downstream by te retailer can usually be designed to increase wolesaler profitability. Tis contract typically features a iger wolesale price, te iger te price subsequently cosen by a retailer, ence te term ladder pricing or tiered pricing. Tis paper sows tat if tere is eterogeneity over retail demands but not (marginal) costs, ladder pricing can acieve full cannel coordination - but tis is not attainable if tere is any eterogeneity in cost structures across retailers. In terms of wolesale profit performance, it 4

5 is sown tat ladder pricing outperforms quantity discounting by an amount tat tends to increase te greater te extent of eterogeneity in retailers demands vis a vis teir costs. Te optimal tariff will typically be non-linear (just as in te case of quantity discounting second degree price discrimination), altoug a linear ladder pricing tariff will often perform reasonably well. Interestingly, a linear ladder pricing contract (wolesale price a linear function of retail price) is sown to be formally equivalent to a revenue saring contract (revenue saring in conjunction wit setting a uniform price per unit). Revenue saring contracts used in practice often ave a simple linear structure, wit te same form of contract being offered to all agents/retailers (see Battacaryya and Lafontaine, 995). Altoug tere is a supply cain literature on revenue saring contracts (see e.g. Cacon and Lariviere, 5; Bernstein, Cen and Federgruen, 6; Gercak and Wang, 4; Pan et al., ), te focus is usually on a single or fixed number of supplier/ retailer cannels. To te autor s knowledge, te optimization problem of designing revenue saring or ladder pricing contracts, wen downstream retailers ave not only a decision on wat price to set, but also on weter to participate at all, as not been addressed. In tis paper, it is sown tat modeling te participation decision as a significant impact on te nature of te optimal contract; te optimal contract will often feature positive revenue saring in conjunction wit a uniform price tat is below wolesale marginal cost and quite possibly negative (altoug clearly, te overall payment from eac participating retailer to te wolesaler is non-negative). Contracts in practice do not usually feature suc low uniform prices, and te offering of a contract wit a very low or negative uniform price migt seem problematic in practice; from a PR perspective, tis suggests a possible preference for presenting suc contracts in a ladder pricing format. Wolesale ladder pricing is operational under te same conditions as tose required for revenue saring, and tese conditions are rater more restrictive tan tose for wolesale second degree price discrimination. Bot quantity discounting and ladder pricing contracts require tat downstream retailers cannot freely arbitrage or self-supply, but ladder pricing additionally requires tat prices set by individual retailers wo coose to participate can be subsequently monitored by te wolesaler. Altoug more restrictive, revenue saring is already manifest in a range of supply cain situations. DVD/video rentals is a classic example, but it is also manifest in transfer pricing environments and professional services (for example, letting agency fees, legal fees, accountancy fees). Professional fees often seem to be tied more to firm turnover or profitability tan to te actual amount of work done. Profit 5

6 or revenue saring, alongside francise fees, are also traditional components in francising and licensing (see e.g. Matewson and Winter, 985). Te assumption tat te wolesaler offers a standard ladder price contract is peraps wort discussing furter. Given te announced wolesale tariff, retailers wo coose to participate and ten coose a retail price in so doing reveal information about teir type to te wolesaler. If te wolesaler knew te type of eac retailer ex ante, it could make a take it or leave it bespoke deal wit eac retailer. Te setting examined ere is one in wic tis is not possible. Tat is, te wolesaler does not know te retailers type ex ante. Tis is not unrealistic. Even in settings were te wolesaler knows te name of eac retailer, te retailers may face demand uncertainty suc tat, period by period, tey ave varying type (in terms of level of demand, and possibly operating cost). Tere may also be oter considerations in play fairness is an important consideration for umans, and a publised ladder pricing contract features te same fairness attribute as quantity discounting namely tat te contract on offer can be seen to be te same for all retailers wo coose to participate. Finally, tere is te legal and regulatory environment to consider. Offering firm specific bespoke contracts to retailers is clearly more likely to encounter Robinson Patman Act litigation someting wic is clearly reduced wen te same contract is offered to all firms wo coose to participate. Te regulatory environment may also restrict firms to offering te same publised tariff to all wo wis to participate (as in te UK telecoms case). From te wolesaler s perspective, te benefits of ladder pricing (and revenue saring as a special case) over quantity discounting need to outweig te additional monitoring costs involved, altoug, as a general observation, monitoring costs may tend to fall as tecnology advances, so te scope for ladder pricing may tend to increase over time. From a marketing perspective, it is wort noting tat, precisely because tis form of contract is relatively unusual, te opportunity to utilize ladder pricing may be missed tat is, it may not be considered even were it is feasible. Tis is certainly te case in te Telecoms case discussed above; ladder pricing as been tecnically feasible in Telecoms for some time, but it is only recently tat it as been discovered as a tariff possibility. Making comparisons of te profit performance of quantity discounting and ladder pricing is difficult because tese two forms of nonlinear pricing occur in a setting were downstream agent type space is at least bivariate. Tat is, retailers will typically ave different cost structures and also face differing levels of demand. Computational difficulties are severe 6

7 wenever te type and/or product space is multidimensional (a point made clearly by e.g. Wilson, 996; Armstrong, 996). Altoug significant progress as been made on mecanism design problems wen bot type and product space is multidimensional (see e.g. Wilson, 999; Rocet and Stole, 3), te solution procedures are complex and te determination of optimal tariffs generally callenging. Making a performance comparison is even more callenging because tis requires not only tat optimal tariffing solutions are found, but in addition, teir associated profitability. In te polar cases were te type space is univariate, it is possible to make a fairly general statement; specifically, it can be sown tat if tere is solely retail demand eterogeneity, ladder pricing clearly outperforms quantity discounting. By contrast, wen tere is solely cost eterogeneity, te tariffs ave identical performance. Tis suggests tat te intermediate case, were te type space is (at least) bivariate, tis is likely to feature an intermediate level of relative performance. Section 3 of te paper illustrates tis by examining a special case were retailers ave constant but eterogeneous marginal costs, no fixed costs, and eterogeneous linear retail demand, wit te distribution for retailer marginal costs and for demand bot uniformly distributed. In tis setting, te wolesaler is unable to observe individual retailer demand or retailer marginal cost but is assumed to know te joint distribution of tese variables across retailers. Under tese assumptions, te screening problem is well beaved and it is possible to not only compute optimal tariff solutions, but also te profit te wolesaler can earn from tem. As conjectured from te results in te polar cases, it is ten sown tat te greater te demand eterogeneity across retailers, relative to cost eterogeneity, te better te performance of ladder pricing relative to quantity discounting. Te results sow tat ladder pricing can offer significant increases in performance, and clearly suggest tat tis is likely to be greater to te extent tat demand eterogeneity is greater tan cost eterogeneity (someting tat seems likely in most practical applications).. Relative performance of Ladder Pricing vis a vis Quantity Discounting Tis section discusses two polar cases (i) were tere is solely retail demand eterogeneity, and (ii) were tere is solely retail (marginal) cost eterogeneity. Te analysis of ladder pricing solutions in tese cases is reasonably straigtforward, and provides a range of elpful insigts regarding te nature of te solution. 7

8 . Demand eterogeneity Suppose a wolesaler sells a good to a continuum of downstream retailers wo operate in exclusive territories. Retailers are parameterized via a single type variable [, ] were. If a retailer cooses a retail price p ( p ), it faces a level of demand qp (, ) l were q ( p, ) (demand curves ave downward slope) and q (, ) p (demand is strictly increasing in te type variable). 3 Te wolesaler as constant marginal cost m, and retailers ave te same constant marginal cost, and tere are no fixed costs. Eac retailer knows its marginal cost and also te demand scedule it faces; te wolesaler is assumed to know te structure of te retailers demand functions but is unable to identify retailer type ex ante, and so is unable to offer bespoke contracts. It is well known tat in tis situation, wolesaler profitability under second degree price discrimination (quantity discounting) can improve on tat under a uniform wolesale price contract, but it is inefficient in tat it fails to maximise total cain profits across te population of retailers tat coose to participate. For example, Robert Wilson (997, pp. 57-8) illustrates tis point in a numerical example in wic demands are linear and te type distribution is uniform. By contrast, it is straigtforward to sow tat ladder pricing yields a fully efficient solution, and one in wic te wolesaler can appropriate all of tis joint profit. To see tis, consider te problem of maximizing wole cain profitability. Wole cain profitability for te wolesaler and a type retailer is denoted w r(subscripts w for wolesaler, r for retailer) and is given as w r ( p m) q( p, ). () Te coice of retail price tat maximizes wole cain profit must satisfy te first order condition tat l / ( ) (, ) (, ), () wr p p m q p q p and te participation constraint tat w r p m and q( p, ). (3) 8

9 Now suppose te wolesaler offers a wolesale price contract in wic wolesale price wp ( ) is a smoot function of retail price, and te retailers make independent decisions on wat downstream price to set. Te type retailer profit r is now r p w( p) q( p, ), (4) and te retailer will coose p to satisfy te first order condition / ( ) (, ) ( ) (, ), (5) r p p w p q p w p q p wit participation defined by r p w( p) and q( p, ). (6) Te problem is to design a wolesale price scedule suc tat te solution in equations (5), (6) is identical to tat in equations (), (3). Putting () and (5) togeter, we ave q( p, ) p w( p) p m q ( p, ) w( p) (7) so tat m w ( p) w( p) p m p m, (8) an ordinary differential equation wic as general solution w( p) m ( p m). (9) were is an arbitrary constant. Tis ladder pricing contract induces exactly te same retailer price coices and participation as in te joint profit maximizing solution. Notice tat substituting te optimal ladder wolesale price function (9) into (4) gives r p m p m q p, p mq p, w r, () and te profit tis ladder price function yields to te wolesaler is 9

10 ( ) w w p m q p m q w r. () Tus te optimal ladder price scedule tat incentivizes all participating retailers to coose teir joint-profit maximizing retail prices is formally equivalent to a profit saring rule. Tis is wy te ladder pricing contract works it induces profit saring for all downstream retailers, and tis naturally induces tem to carge te rigt downstream price in order to maximize wole cain profitability. Wolesaler profit is clearly ten maximized by moving, equivalent to setting a ladder price of te form w( p) p, altoug tis puses participation to te limit since ten r p w( p) q p, () for all retailers (tat is, tey are all indifferent as to weter tey participate or not). 4 Tis (linear) ladder pricing solution is also equivalent to a revenue saring contract in wic tere is bot revenue saring alongside a uniform wolesale price. To see tis, note tat te overall wolesale carge under ladder pricing is w( p) q m p m q pq ( ) m q, (3) so it is equivalent to a sare of revenue and a uniform wolesale price of ( )m. Notice tat te uniform wolesale price is less tan wolesale marginal cost m and tat as te wolesaler aims to maximize profit (puses ), tis uniform wolesale price is negative (equal to, te negative of retail marginal cost). Tat is, te tariff tat maximizes wolesaler profit depends on retailers marginal cost, but not on te wolesaler s marginal cost. Revenue saring arrangements observed in practice (e.g. in video rentals, in francising) often feature uniform pricing in addition to a saring of revenue. However, tese contracts typically feature revenue saring in conjunction wit a positive uniform wolesale price eiter equal to or above wolesaler marginal cost. Suc a tariff (revenue sare plus positive wolesale uniform price) is clearly not profit maximizing for te wolesaler, nor is it efficient from te perspective of te wole cain. Tis observation tat an optimal revenue saring contract sould feature below (wolesale) marginal cost pricing seems to be an interesting observation in its own rigt. From a practical perspective, if it was in fact desirable to offer a revenue saring contract wit a negative uniform price, given tis migt appear somewat

11 unusual and peraps confusing to retailers, it would seem tat presenting te contract in a ladder pricing format as muc to recommend it. Given ladder pricing/revenue saring acieves maximum wole cain profitability, clearly it will outperform an optimized quantity discounting contract. To illustrate te difference in relative performance, it is useful to give a simple numerical example. Suppose ten retailers ave linear demands differentiated by a single type variable wic defines an inverse demand function of te form 5 p q, (4) were p is te retail price cosen by te retailer (assumed to be profit maximizing uniform price) and q is te amount purcased from te wolesaler and ten sold on to te final retail market. Te distribution for te retailer demand intercept is assumed uniform on an interval [, ] ( ). Te wolesaler as constant marginal cost m, and retailers all ave l l te same constant marginal cost, and tere are no fixed costs. Eac retailer knows its marginal cost and also te demand scedule it faces; te wolesaler is unable to distinguis tese caracteristics ex ante, and so is unable to offer bespoke contracts. Te quantity discounting solution to tis special case of te above model as been discussed by Robert Wilson (997, pp. 57-8); Wilson sows tat if retailers all ave te same marginal cost (and tis is common knowledge), and te wolesaler can select an arbitrary outlay scedule, te optimal scedule involves quantity discounting suc tat marginal price declines linearly wit te quantity retailers coose to purcase. Specifically, te optimal marginal price w*( q ) offered to all retailers takes te form w*( q) ( m ) q. (5) Notice tat not all retailers will participate; only tose wit type satisfying m ( ) (6) do so. Te Wilson example is actually a special case in wic in wic te parameters take values m,, (a bencmark case), and Wilson sows tat te maximum l attainable wolesaler profit, under quantity discounting is (7)

12 wit only alf te retailers participating (tose wit type ). By contrast, under te ladder pricing solution, wen m,,, te optimal solution is w( p) p, tere is no exclusion ( e ), and profit to te wolesaler is l ( ) ( ).833. (8) W w p m q p d d Tus te maximum wolesale profit under ladder pricing is exactly twice tat wic can be earned from quantity discounting (actually, te profit earned is exactly twice tat under quantity discounting watever te level of retailer marginal cost). Te fact tat optimal ladder pricing is equivalent to profit (or revenue) saring in te case were all retailers ave identical cost functions, wit constant marginal costs, sould not be surprising. Unlike wit second degree price discrimination, ladder pricing in tis case is equivalent to profit saring and as previously remarked, tis enables te wolesaler is able to incentivize all retailers to coose downstream prices so as to maximize wole cain profitability wilst at te same time te wolesaler takes all of tis wole cain profitability.. Cost Heterogeneity In te oter polar case, were all retailers face te same demand function, but ave eterogeneous marginal costs, te maximum profit performance for te wolesaler is te same weter a quantity discounting outlay scedule or a ladder pricing contract is offered. Tis follows because, in tis case (in contrast to te demand eterogeneity case), tere is a one-to-one mapping between prices and quantities. To see tis, suppose marginal retailer cost is, were tis retailer type variable as an arbitrary distribution over support, ( l ), wilst te retailer demand type variable is constant across retailers. Te profit a type retailer earns wen faced wit te ladder pricing contract is given by (4) as ( ) ( ) ( ) r p q p w p q p (omitting te demand type parameter since it is now constant across retailers), wilst for te quantity discounting case, if te wolesaler offers a revenue outlay scedule Rq ( ) (wit marginal price ( q) R( q) ), te type retailer profit is ( ) ( ) r p q p R q p. Tese profit functions are identical if R q( p) w( p) q( p). (9) l

13 Tat is, given te structure of qp ( ) is common knowledge, if wp ( ) was an optimal ladder pricing contract, ten tis is reproduced by a quantity discounting outlay scedule R (.) defined by (9), and equally, if R(.) denoted a profit maximizing coice for te revenue outlay scedule, tis can be reproduced via te ladder pricing contract defined as w( p) R q( p) q( p). 6 Intuitively, te fact tat ladder pricing and quantity discounting give te same performance in tis case is to be expected; were private information solely concerns retailer marginal cost, te wolesaler is no longer able to incentivize joint profit maximizing beavior for retailers, and te optimal ladder pricing solution is precisely equivalent to tat of optimal quantity discounting (because, as previously remarked, all retailers ave te same demand function in tis special case, tere is a one-to-one mapping between prices and quantities via tis single demand function). By contrast, wen all te eterogeneity lies in retail demands (all retailers aving te same marginal cost), ladder pricing is more effective at extracting profit because, unlike quantity discounting, it can be designed to provide incentives for all retailers to maximize wole cain profitability. Tis observation is suggestive of te idea tat te relative performance of te two contracts may depend on te relative extent of demand and cost eterogeneity wit te greater te former, te better te relative performance of ladder pricing vis a vis quantity discounting. Tis ypotesis is explored in te next section..3 Cost and Demand eterogeneity In tis section, te case were tere is bot demand and cost eterogeneity is examined. As previously remarked, it is examined in a simplified setting involving linear demand as in equation (4), were denotes te upper intercept of te inverse retailer demand curve, distributed uniformly on, (were ) wilst retailer marginal cost is distributed uniformly on te range [,]. In tis way, te bivariate uniform distribution on te rectangle,,, becomes a function solely of te parameter. As, ladder pricing and quantity discounting solutions can be expected to asymptote to te polar case were tere is purely retailer cost eterogeneity, wilst as, te solutions sould asymptote toward te case were tere is purely retail demand eterogeneity. Tus te expectation is tat wolesale profitability under ladder pricing will be closer to tat for 3

14 quantity discounting for low, but tat, as is increased (as demand eterogeneity becomes dominant relative to cost eterogeneity), so ladder pricing performance will improve toward te level obtained in section.. To reduce algebraic clutter, wolesaler marginal cost is set at m= at te outset. Given te rectangular uniform distribution, te optimal quantity discounting outlay scedule can be sown to be piecewise linear. One would expect a similar result for te ladder pricing, wit a linear scedule being close to optimal for larger. Given te complexity of obtaining a general non-linear ladder pricing solution, for relative simplicity, only te class of linear scedules is considered for ladder pricing. Tis provides a lower bound for profit performance for ladder pricing, and ence it suffices to sow te likely extent of out-performance. It also as merit in tat it provides a direct comparison of optimal revenue saring wit optimal non-linear quantity discounting in te presence of bot demand and cost eterogeneity. Te Ladder Pricing Solution Ladder price is assumed linear and increasing in retail price, taking te form w( p) p. () were [,] and is unrestricted in sign. Te problem is to maximize wolesaler profit by selecting,. A participating retailer earns profit r ( p w) q ( p w) p ( p p) p () and te retailer FONC is tat p p r / p ( p p) ( ) p ( ) ( ) ( ) p. ( ) ( ) Excluded retailers are tose tat cannot earn positive profit for any coice of q; marginal retailers are tose wo are only able to earn zero profit at most. Denote a marginal retailer as one wit parameter values e, e wo cooses quantity q e and sets retail price profit means tat p p () p e. Zero, (3) r e e e e and participation means tat te retailer FONC must old; tus from (), 4

15 p e e e. ( ) Using (3) and (4), a little algebra establises tat te solution is uniquely tat and tat / e e e e (4), (5) p e. (6) e Tis means tere is a set of marginal participating retailers, defined as (, ) : [,], [, ], e e e e e e and te set of all wo participate, denoted, conditional on te coices of,, is (, ) : [,], [, ],. (7) Te sape of te participation set depends on te wolesaler s coice of ladder price scedule (te values for, ) as illustrated in figures and. Figure te Participation set wen (or ) 5

16 Figure te Participation set wen (or ) Te participation region needs some careful andling - it turns on te following two key parameters: / ( ) / Tese values are sown in figures and, wic sow ow te participation region depends on te coice of,. Te grap of (8) (9) (in figure for [, ], in figure, for [, ] ) represents te set of marginal retailers; figure illustrates te case were, in wic case te participation region is triangular, wilst figure gives te case were, were it is a trapezoid. It is possible tat is negative (since is unrestricted in sign). Tis will affect te evaluation of wolesale profit as te participation region is ten truncated to te left at in bot figures. Tis is straigtforward to take into account (see below). Wolesaler profit is negative or positive, as follows: W depends on weter figure or figure applies, and also weter Case (a) If ( ) / ( ) : (as in figure ) Ten 6

17 w( p) q( p) / dd / p p dd W / p Max(,) p dd / ( d d. Max,) ( ) ( ) Notice te lower limit for -integration is Max(,) and tat () is used to replace p in line 3). Expanding tis and performing te integrations, tis can eventually (after a lot of routine algebra) be reduced to W 4 3 ( ) ( ) Max(,) Max (,) b (Note in tese equations te notation f ( x) f ( b) f ( a) ). Tis is a function of te a parameters, given tat, are determined by tese parameters, and so it can be optimized numerically.. (3) (3) Case (b) ( ) / ( ) (as in Figure ) Ten w( p) q( p) / dd / p p dd W / p pdd p pdd Max(,) ( 4 Max,) / 4 Max( dd,) ( ) ( ) dd ( ) ( ). (3) wic again can be eventually be simplified to obtain 7

18 W / ( ) 4 3 ( ) ( ) Max(,) ( (,) ) Max Again tis is a function of te parameters, and can be numerically optimized. Results are presented in section.3 below, were tey are compared wit tose obtained for te quantity discounting solution. (33) Te quantity discounting solution Te optimal solution can be obtained using te demand profile approac (Wilson, 993). Care needs to be taken to deal appropriately wit te participation region, as in te case above for ladder pricing. Because te analysis is bot lengty and intricate to present it is omitted. 7 Te profit maximizing wolesale quantity discounting outlay scedule tat results is: 3 w*( q) q q[, ) 4 3 w*( q) q q[, ] Te associated wolesale profit function is ten a polynomial function of ; it takes te form 3 Case (a) If : Wolesaler profit W is given by were z * 4 z * 4 z * 3 ze * ze * W e e e ze* 3 6 z 6z dz 5 3 z 7 z 9z dz 3 3 8

19 z *. e 3 Case (b) If : Wolesaler profit W is given by were and W z z 3z 4 z ze* 3 z 8 z 3 e e z 8z 7 z *. e 3 e e It is terefore straigtforward to evaluate profit performance as a function of te parameter. Numerical Results Having obtained te solutions, tis section reports results for ladder pricing and quantity discounting; Table below presents tese results for a range of values for ; it reveals tat ladder pricing always outperforms quantity discounting in tis setting, and tat te difference in profitability is strictly increasing in. Te relative profitability is also broadly increasing wit. Relative profitability is relatively static until exceeds.5. After tat it climbs quickly toward an asymptote of around %. Tis is consistent wit te results reported in section.; wit all te eterogeneity concentrated on retailer marginal cost, te profit ratio was % and wen all te eterogeneity was concentrated on retailer demand te profit ratio was %. 9

20 Table Wolesaler Profitability * * * ( LP ) W under Ladder Pricing * ( QD ) W under Quantity Discounting Profit Difference: * W ( LP) minus ( QD) * W % Ratio * W * W ( LP) ( QD)...5.4E-5 9.6E-6.4E E-3.6E-3.43E E-3 7.4E E- 3.75E Notice tat for small, te optimal ladder price function is simply w( p).5 p, and tis coincides wit tat for te polar case were tere is all cost eterogeneity and no demand eterogeneity. As te extent of demand eterogeneity relative to cost eterogeneity increases (as increases), so * converges to te value of unity obtained in te oter polar case, in section., were tere was solely demand eterogeneity. In tat polar case, te solution was w( p) p (were, te retailer marginal cost was in tat case a unique number, being te same for all retailers). In tis section, te marginal cost for retailers is distributed over te range [,], so te average marginal cost across all retailers is.5. Of course, te average marginal cost of participating retailers will be less tan.5 because, ceteris paribus, te iger te marginal cost, te more likely te retailer is priced out of te market. In fact, computing te participation percentage, tis is strictly increasing and rises monotonically toward % as is increased so te average marginal cost is in fact converging on.5. Hence it is plausible tat, as revealed in Table, * sould converge toward a figure of minus.5 as per te polar case.

21 3. Ladder Pricing Step Functions Te recent practice of ladder pricing in UK Telecoms as sometimes featured linear tariffs, but more commonly, step functions. Figure 3 below illustrates linear ladder pricing ( w ) non-linear ladder pricing ( w ) and step function ladder pricing ( w ) eac as functions of te B retail price p. Te first practical observation to make is tat step functions (suc as C A w C in figure 3) can be viewed as approximations to continuous functions and naturally a step function can be designed as an increasingly close approximation to any given continuous smoot wolesale price function, te smaller te step size cosen. Te first forms used in practice actually featured step functions, so it is of some interest to discuss te pros and cons of tis vis a vis te oter forms of contract. Practitioners ave argued tat step functions make te practice of ladder pricing operationally easier. Tis turns on te question of ow accurately te wolesaler is able to monitor retail prices. In te UK Telecom case, downstream retailers were mobile network operators operating relatively complex forms of retail pricing (menus of multi-part tariffs for example). Te wolesale price in tese cases was designed to be functionally related to te average downstream price. Given te complexity of te downstream offerings, and given likely variations over time in quantities sold, practitioners tended to argue tat te exact average downstream price migt fluctuate somewat and tat tere would be less need to adjust te wolesale price continuously over time if ladder pricing took te form of a step function (simply because even toug it migt fluctuate, te average price is more likely to locate on a single step, suc tat te wolesale price does not vary). It is clear tat, for a given wolesale pricing step function, on a step, wolesale price is constant, so te double marginalization incentive occurs (Spengler, 95). Tat is, for alternative coices of retail price tat induce te same wolesale price, it is as if te retailer is facing a uniform price. Naturally, tis also implies retailers will tend to want to pitc teir price nearer toward te foot of a step up on te wolesale price function altoug tey still ave te question of wic step to coose to locate on.

22 Figure 3 Ladder Pricing Contracts wp ( ) w A w B w C p 4. Legal and Regulatory Considerations Wolesalers wo price discriminate as between different retailers can fall foul of te Robinson Patman Act in te US. Te fact tat te same contract is offered to all wo coose to participate migt be tougt to guarantee fairness but tis is by no means clear; Moorty (987) points out tat quantity discount scedules, even if te same contract is offered to all retailers, will generally mean tat different retailers are carged different prices (bot marginal and average prices) contingent on te amount tey coose to purcase and tat tis can be grounds for claiming injury in tat firms competition wit oter retailers. 8 It migt seem tat similar arguments apply to te ladder pricing case; owever, te argument may be arder to make in tis case. Te point is tat, insofar as retailers are in competition downstream, and insofar as tey ave similar products, tey will need to carge similar prices, tey will face similar wolesale carges. However, insofar as retail products are eterogeneous, retailers may coose different retail prices and ence attract different wolesale prices (tat are not related to te underlying cost of wolesale supply), ence leading to possible action under te act. In tis context, it is interesting to note te parallels between ladder pricing and revenue saring, in tat te special case of a linear ladder pricing scedule is in fact equivalent to revenue saring plus a uniform unit wolesale price so te legal case is pretty muc te same for bot tese types of cannel arrangement. 9 Wilst in te US, Telecoms are regulated wit a relatively ligt touc, tis is not te case in te UK and Europe, were regulators quite commonly get involved in igly complex

23 economic assessments of Telecom practice. In te case of ladder wolesale pricing, Ofcom, te UK regulator as been involved in a dispute between wolesale operators and mobile network operators concerning wolesale ladder pricing proposals by te former companies (see Ofcom,, 3, for example). In tese cases, no issue was raised as to weter te proposed carges represented an abuse of a dominant position, and Ofcom considered tat it could be fair and reasonable for te wolesaler to introduce tiered wolesale carges. Ofcom s principal concern was weter te proposed carges would be likely to benefit mobile network customers troug inducing mobile network operators to reduce teir retail carges for calls to tese non-geograpic numbers (since in general, Ofcom took te view tat current retail prices were too ig). Ofcom s initial view was tat te wolesale tariffs did not guarantee consumer benefits, and so ruled against allowing te tariffs. Te wolesaler appealed, and te case was taken before te Competition Appeals Tribunal, were te Ofcom judgment was overturned. Te case was ten taken by te MNOs to te Court of Appeal, wic overturned te CAT judgment, and ten furter appealed by te network operator before te Supreme Court, wic finally ruled (July, 4) tat suc tariffs were indeed permissible and tat te onus was on tose wo objected to sow te probability tat te imposition of suc tariffs would result in material arm to callers. Tus, wolesale ladder pricing is back on te agenda for Telecom fixed line network operators. 5. Conclusions In tis paper, a new variant on wolesale price discrimination as been discussed. Ladder or tiered pricing involves linking te wolesale price to te price cosen downstream by retailers, generally by setting a iger wolesale price, te iger te coice of retail price downstream. Altoug revenue saring contracts ave found wide use in a range of settings, te generalization to non-linear ladder pricing appears to be new. Recent practice in UK Telecom markets appears to be te only example of non-linear ladder pricing in action (of wic te autor is aware); in tis case, several wolesale service providers ave offered a non-linear step function wolesale tariff and oters ave offered a linear or piecewise linear tariff. It seems useful to igligt te possibility and potential for introducing suc tariffs since, wenever revenue saring is feasible, so too is te more general form of (non-linear) ladder pricing. 3

24 Wen a wolesale ladder pricing or revenue saring contract is offered to all wo coose to participate, standard forms of revenue saring contract (in wic a uniform price is set at or above wolesaler marginal cost) are unlikely to be optimal. Profit maximizing ladder pricing, wen te ladder is restricted to te linear form, is equivalent to a revenue saring contract in wic revenue saring is combined wit setting a below wolesaler marginal cost and potentially negative uniform price (wilst of course te overall payment to te wolesaler remains positive for all retailers wo coose to participate). In particular, te greater te retailer demand eterogeneity relative to cost eterogeneity, te lower tis implied uniform price sould be in a revenue saring contract. As a practical concern, revenue saring contracts featuring a negative uniform price migt appear less tan attractive from a PR perspective as it is arder to explain or rationalize. Tat is, wen te wolesaler sets a positive uniform price (in conjunction wit revenue saring), te uniform price can be justified to retailers because it can be related to wolesale production costs. By contrast, if te wolesaler sets a revenue saring contract in wic tere is a negative uniform price, tis clearly is less easy to cost justify. No suc problem arises for te equivalent ladder pricing contract setting suc a pricing contract tus neatly sidesteps tis practical concern. It was also sown tat, witin te model, were tere is eterogeneity in bot retailer demands and costs, so long as tere is some demand side eterogeneity, profit maximizing ladder pricing always outperforms quantity discounting. Furter, ceteris paribus, te relative performance of ladder pricing vis a vis quantity discounting (broadly) tends to increase wit te extent of demand eterogeneity relative to marginal cost eterogeneity. Wilst tis insigt seems likely to old in more general setting, tis remains to be demonstrated (unfortunately, making performance comparisons wen tere are non-linear demands, non-linear cost functions, more general forms for te type distribution function etc. is seriously callenging). Te range of applications for wolesale ladder pricing is more restrictive tan tat for quantity discounting. All te conditions required for te latter must be present, and in addition, it must be possible for te wolesaler to monitor prices set at te individual retailer level. However, wilst te set of applications for wic ladder pricing is feasible is more restricted, it remains a significant set, since revenue saring is practiced in various environments (notably DVD/video rentals/ transfer pricing/francising/licensing) and ladder pricing as recently been used in UK telecom markets. 4

25 6. References Armstrong Mark, 996, Multiproduct nonlinear pricing, Econometrica, 64: Battacaryya Sugato and Lafontaine Francine, 995, Double sided moral azard and te nature of sare contracts, Rand Journal of Economics, 6: Bernstein Fernando G., Cen Fangruo, Federgruen Awi,6, Coordinating supply cains wit simple pricing scemes: te role of vendor-managed inventories, Management Science, 5: Cacon Gerard P. and Lariviere Martin A., 5, Supply cain coordination wit revenue saring contracts: strengts and limitations, Management Science, 5: Competition Appeals Tribunal,, Case Numbers 68/3/3/,69/3/3/,5/3/3/. Transcripts, including te final judgement, are available from te CAT website: ttp:// Dixit Avinas K., 98, Recent developments in Oligopoly teory, American Economic Review, 7: -7. Gercak Yigal and Wang Yunzeng., 4, Revenue saring versus wolesale price contracts in assembly systems wit random demand, Production and Operations Management, 3: Ingene Carles A. and Parry Mark E., 995, Coordination and manufacturer profit maximisation: te multiple retailer cannel, Journal of Retailing, 7: 9-5. Ingene Carles A., Taboubi Siem, and Zaccour Georges,, Game teoretic coordination mecanisms in distribution cannels: Integration and extensions for models witout competition, Journal of Retailing, 88: Matewson G. Frank and Winter Ralp A., 985, Te economics of francise contracts, Journal of Law and Economics, 8: Moorty Sridar, 987, Managing cannel profits: Comment, Marketing Science, 6: Ofcom,, Determination to resolve a dispute between BT and eac of T-Mobile, Vodafone, O and Orange about BT s termination carges for 8 calls, Final determination, January, available via Ofcom s website. Ofcom, 3, Determination to resolve disputes concerning BT s tiered termination carges in NCCNs, 7 and 46, Final Determination, 4 April, 3, available via Ofcom s website. Pan Kewen, Leung Steven C. and Xiao Di,, Revenue saring versus wolesale price mecanisms under different cannel power structures, European Journal of Operational Researc, 3:

26 Raut Sumit, Swami Sanjit, Lee Eunkyu, and Weinberg Carles B., 8, How complex do movie cannel contracts need to be?, Marketing Science, 7: Rocet Jean-Carles and Stole Lars, 3, Te Economics of Multidimensional Screening, in Advances in Economics and Econometrics: Teory and Applications - Eigt World Congress, series Econometric Society Monograps, no. 36, Cambridge University Press. Supreme Court, 4, Judgment: Britis Telecommunications Plc (Appellant) v Telefónica O UK Ltd and Oters (Respondents), 9 t July 4. Judgement available at ttps:// 4_Judgment.pdf Spengler, Josep J., 95, Vertical integration and antitrust policy, Journal of Political Economy, 58: Wilson Robert B., 997, Nonlinear Pricing, Oxford University Press, Oxford. Endnotes A supply cain is said to be coordinated if contract arrangements lead to joint profits being maximised. Te case first came before te Competition Appeals Tribunal, wo concluded tat tere may well be incentives to reduce average retail price, notwitstanding te fact tat complex forms of retail price discrimination are practiced in tese markets (network operators offering a range of tariff menus ). Competition Appeals Tribunal Case Numbers 68/3/3/,69/3/3/,5/3/3/. Transcripts, including te final judgement, are available from te CAT website: ttp:// Economic analysis of te incentive to reduce retail price, and on associated welfare consequences, can be found in reports presented as annexes to te Ofcom (, 3) publications. 3 Notation: q( p, ) q( p, ) / p; q( p, ) q( p, ) / etc. 4 For (motivational) reasons tat lie outside te model, tere will no doubt be trade off, and it will rarely be optimal to pus too ig. 5 Te slope coefficient can be normalised to - by suitable coice of units. 6 Tis equivalence can be illustrated numerically if one makes more specific assumptions about te demand function and te distribution for retailer marginal cost (for example, wit linear demand and a uniform distribution for retailer marginal cost a derivation for tis case is omitted, given tat it is a limiting result for te bivariate distribution case examined in section 3 below). 7 A full derivation of te solution for tis case is available at te autor s website ttps:// 8 For example small retailers, wo purcase less, will be paying iger prices for te wolesale product. 9 Tat is, any practical step function wolesale pricing scedule can be viewed as similar to a form of revenue saring contract. 6