In this paper we show how an upstream firm can prevent destructive
competition among downstream firms producing relatively close substitutes by
implementing a price-dependent profit-sharing rule. The rule also ensures that the
downstream firms undertake investments which benefit the industry in aggregate.
The model is consistent with observations from the market for content commodities
distributed by mobile networks

Foros, Øystein

Hagen, Kåre Petter

Kind, Hans Jarle

In this paper we show how an upstream firm can prevent destructive
competition among downstream firms producing relatively close substitutes by
implementing a price-dependent profit-sharing rule. The rule also ensures that the
downstream firms undertake investments which benefit the industry in aggregate.
The model is consistent with observations from the market for content commodities
distributed by mobile networks

HSN Open Archive

Price-dependent profit sharing as an escape from the Bertrand paradox

NHH Brage

English

NORA - Norwegian Open Research Archives

Discussion paperSAM 4   2007ISSN:  0804-6824JANUARY 2007INSTITUTT  FOR  SAMFUNNSØKONOMIDEPARTMENT OF ECONOMICSPrice-dependent profit sharing as an escape from  the Bertrand paradoxBYØYSTEIN FOROS, KÅRE P. HAGEN, AND HANS JARLE KINDThis series consists of papers with limited circulation, intended to stimulate discussion.Price-dependent profit sharing as an escape from theBertrand paradoxØystein ForosNorwegian School of Economics and Business Administrationoystein.foros@nhh.noKåre P. HagenNorwegian School of Economics and Business Administrationkare.hagen@nhh.noHans Jarle KindNorwegian School of Economics and Business Administration and CESifohans.kind@nhh.noJEL classification:Keywords: Profit-sharing, vertical restraints, investments, competition.Abstract: In this paper we show how an upstream firm can prevent destruc-tive competition among downstream firms producing relatively close substitutes byimplementing a price-dependent profit-sharing rule. The rule also ensures that thedownstream firms undertake investments which benefit the industry in aggregate.The model is consistent with observations from the market for content commoditiesdistributed by mobile networks.1 IntroductionThe Bertrand paradox may provide a plausible explanation why the majority of thecontent commodities on the Internet are oﬀered for free (marginal costs). The rivalis just “one click away”, and competing content providers have strong incentives toundercut each other as long as there are positive profit margins. In contrast, weobserve that prices for mobile phone content commodities like ring tones, footballgoal alerts and jokes are well above marginal costs (the sales value of such services inNorway in 2006 was twice as high as the total value of Internet ads). One potentialexplanation why the Bertrand paradox is not observed for such goods, is the price-dependent profit-sharing rule used by some upstream mobile access providers. Therule implies that the upstream firms charge a share of the end-user price per unitof content instead of for instance a unit wholesale price from the content providers.The crucial feature of this rule is that the share accruing to a given content provideris increasing in the end-user price. The table below shows the profit-sharing ruleused by the dominant Norwegian mobile operator Telenor; if a content provider sellshis good for 1 NOK he receives 45 % of the revenue, while he receives 80 % if hesells the good for 70 NOK.1End-user price (NOK) 1.0 1.5 3 5 10 20 70Share to the content provider 45% 54.% 62% 66% 68% 70% 80%In the formal model we consider an upstream firm selling an input (access) todownstream firms producing diﬀerentiated services. The upstream firm determinesthe access conditions, while the downstream firms decide end-user prices and in-vestments in for instance marketing. We show that by using a price-dependentprofit-sharing rule, the upstream firm induces the retailers to behave as if demandhas become less price elastic. A price-dependent profit-sharing rule is suﬃcient to1In addition to this revenue-sharing rule, the content providers are charged a fixed fee (but nounit wholesale price). Strand (2004) emphasizes that the revenue-sharing scheme creates incentivesto promote new services. The Norwegian business model is now widely taken up in Europe andAsia (Strand, 2004).1achieve the vertical integration outcome also in presence of investment spillovers. Afixed fee determines the allocation of the total industry profit.The upstream firm could alternatively use a combination of resale price main-tenance (RPM), a fixed fee and a wholesale price below marginal costs to achievethe vertical integration outcome (see Mathewson and Winter, 1984). The Bertrandparadox is then avoided by indirectly limiting the retailers’ strategy choices. Thenovelty in the above proposal is that the sharing-scheme reduces the undercuttingincentives among retailers directly by reducing the perceived price elasticity, and isless likely to raise anti-trust concerns compared to RPM .22 The ModelWe consider an upstream firm selling an input to n downstream firms. The demandcurve faced by downstream firm i = 1, ..., n is given by qi = qi(p), where p is thevector of prices charged by the n downstream firms. We assume that the demandfunctions are well behaved and downward sloping in own price (∂qi/∂pi < 0). Theconsumers perceive the goods sold by the downstream firms as imperfect substitutes(∂qi/∂pj > 0).Marginal costs both at the upstream and downstream levels are set equal to zero;however, this does not matter for the qualitative results. Hence, we can write totaloperating profit in the industry asΠ(p) =nXi=1piqi(p). (1)Below, we consider a two-stage game where the upstream firm at stage 1 deter-mines the conditions for access to the upstream good, and where the downstreamfirms subsequently compete in prices. In Section 3 we extend the model by allowingthe downstream firms to make market-expanding investments.2Moreover, with RPM the retail prices are not decided by the players with hands on marketexperience. This may obviously be detrimental to the total channel outcome.2The upstream firm uses a two-part tariﬀ, consisting of a fixed fee F and a profit-sharing rule. We specify the profit-sharing rule such that downstream firm i keeps ashare β(pi) of its operating profit, while the upstream firm gets the share (1−β(pi)).We later show that β0(pi) > 0.3Stage 2The operating profit of downstream firm i equals πi(p) = β(pi)piqi, and at thelast stage each downstream firm solves p∗i = argmaxπi. This yields the FOCs∙q∗i + p∗i∂qi∂pi¸+β0(p∗i )β(p∗i )p∗i q∗i = 0. (2)The second term in (2) would vanish if β were constant (β0 = 0), in whichcase we would get the standard result that a profit maximizing price pˆi satisfies[qˆi + pˆi∂qi∂pi] = 0. With β0 > 0 the second term on the left-hand side of equation (2)is positive, implying that the marginal profit at any given price is higher than ifβ0 = 0. This induces each of the downstream firms to behave less aggressively, andwe can state:Proposition 1: The profit-maximizing prices will be higher for β0(pi) > 0 com-pared to β0(pi) = 0.By defining εii ≡ piqi∂qi∂pias the price elasticity of demand for good i, we can rewrite(2) asε∗ii = −µ1 +β0(p∗i )β(p∗i )p∗i¶(3)Equation (3) characterizes the profit-maximizing equilibrium price for firm i. Itis well known that revenue - and thus profit for a firm facing zero marginal costs -other things equal is maximized by choosing a price for which the elasticity is equalto minus one. However, sinceβ0(p∗i )β(p∗i )p∗i > 0, we see from (3) that the profit sharingrule induces the downstream service provider to behave as if the demand has becomeless price elastic. This confirms that the profit-maximizing prices will be higher ifβ0 > 0 than if β0 = 0:3In contrast to the present paper, the literature on revenue-sharing as a vertical restraint con-ventionally assumes that the revenue share is a constant; i.e. β0 = 0 (see e.g. Lal, 1990).3Proposition 2: A profit-sharing rule β0(pi) > 0 reduces the perceived elasticityof demand for the downstream firms, making them behave less aggressively.In the sequel we assume an isoelastic sharing rule so that βi(pi) = θpλi , whereλ is the elasticity parameter determined by the upstream firm at stage 1 (for themoment we treat θ as a positive scalar). With this specification we can reformulate(2) and (3) as(1 + λ) q∗i + p∗i∂qi∂pi= 0 (4)ε∗ii = − (1 + λ) (5)Stage 1The upstream firm will use λ to induce the downstream firms to set the pricesthat maximize total industry profit. The fixed fee F is then used as a profit distri-bution parameter. Thus, we first derive the hypothetical equilibrium under verticalintegration (V I). Solving pi = argmaxΠ(p) yields the FOCs∙qi + pi∂qi∂pi¸+Xj 6=ipj∂qj∂pi= 0 (i = 1, ..., n). (6)The term in the square bracket of (6) measures the marginal profit on good i and isanalogous to the term in the square bracket of (2). The second term of (6) internal-izes the horizontal pecuniary externality when products are imperfect substitutes.Let ωpji = −∂qj∂piÁ∂qi∂pimeasure the increased demand for good j per unit reductionin the demand for good i when pi increases. The higher these ratios, the higher pishould be set in order to maximize aggregate industry profit. The challenge for theupstream firm in a vertically separated market structure is to set conditions inducingthe downstream firms to internalize this eﬀect at stage 2.Inserting for ωpji into (6) we can now characterize industry optimum asqi +"pi −Xj 6=ipjωpji#∂qi∂pi= 0. (i = 1, ..., n).By imposing symmetry this expression can be reformulated as (with subscript V Ifor vertical integration)qV I + pV I£1 − (n− 1)ωpji¤ ∂qi∂pi= 0. (7)4The optimal value of λ ensures that aggregate profit is the same in the verti-cally separated market structure as in the hypothetical equilibrium with verticalintegration. This value can be found by using equations (4) and (7) and settingq∗i /p∗i = qV I/pV I .4 We then haveλ = λ∗ ≡ −1 + 11− (n− 1)ωpji. (8)Inserting for (8) into (5) we further findε∗ii = −11− (n− 1)ωpji.If a price reduction of good i does not aﬀect demand for good j, we have∂qj∂pi=ωpji = 0. The downstream firms thus choose prices such that ε∗ii = −1, which isoptimal also from the industry’s point of view (λ∗ = 0). However, if the goods areimperfect substitutes (such that∂qj∂pi> 0), each downstream firm fully internalizesthe eﬀect its price has on the profit of the other firms when λ = λ∗ > 0. Hence,the downstream firms will not engage in destructive price competition even if theyproduce close substitutes, and the Bertrand paradox is avoided:5Proposition 3: The profit-sharing rule βi(pi) = θpλi with λ = λ∗ solves theBertrand paradox, and induces the downstream firms to maximize aggregate industryprofit.3 Market-expanding investments with spilloversWe now extend the model to allow each downstream firm to undertake market-expanding (or quality-enhancing) investments with potential spillovers. At the out-set, it is not clear how one firm’s investments aﬀect sales and profits of the other4Setting q∗i /p∗i = qV I/pV I uniquely determines the prices, since qi/pi is monotonically decreas-ing in pi when ∂qi/∂pi < 0.5As long as the horizontal pecuniary externality is the only problem to solve, we see from (8)that the scalar θ has no impact on the outcome (but in absence of the fixed fee it could be used asan instrument to allocate aggregate industry profit).5firms. The investing firm’s product will typically become relatively more attrac-tive than those of the rivals. Thereby the latter could be harmed. However, theremight also be technological or marketing spillovers from an investment such thatone firm’s investment may be to the benefit of all the downstream firms. A givenfirm’s marketing of ring tones, for instance, is also likely to benefit other firms sellingring tone services. We thus open up for both positive and negative spillovers frominvestments.We assume that the downstream profit function of firm i net of any fixed fee isgiven byπi = β(pi)piqi(p, x)− φ(xi), (9)where the new variable x denotes the vector of market-expanding investments un-dertaken by the n downstream firms, and φ(xi) is the investment cost function. Themore a firm invests, the higher is the demand it faces; ∂qi/∂xi > 0.We assume thatφ0(xi) > 0, and that it is suﬃciently convex to satisfy all second-order conditionsfor a profit maximum.Total industry profit is now given byΠ (p, x) =nXi=1[piqi(p, x)− φ(xi)] . (10)The upstream firm determines the input conditions at stage 1, with θ and λ asstrategic variables, and at stage 2 the downstream firms decide non-cooperativelyon end-user prices and investment levels.At stage 2 the first-order condition ∂πi/∂pi = 0 is given by equation (4). Simul-taneously solving ∂πi/∂xi = 0 we further findθ (p∗i )λ+1 ∂qi∂xi= φ0(x∗i ). (11)where θpλ+1i is the profit margin per unit sold.To find the optimal profit-sharing rule at stage 1, we again use vertical integrationas a benchmark. Maximizing (10) with respect to xi we find the FOCspi∂qi∂xi+Xj 6=ipj∂qj∂xi= φ0(xi) (i = 1, ..., n). (12)6If there were no investment spillovers the term ∂qj/∂xi would in general be neg-ative, and more so the closer horizontal substitutes the goods. This eﬀect, whichwill not be taken into account by independent downstream firms, tends to generateoverinvestments in a decentralized market structure. However, if one firm’s invest-ment increases demand also for its rivals, we have ∂qj/∂xi > 0. This is more likelyto be the case the poorer horizontal substitutes the goods are and the stronger theinvestment spillovers.Analogous to our procedure above, we define ωxji =∂qj∂xiÁ∂qi∂xi. The variable ωxjimeasures the increase in demand for good j per unit change in the demand for goodi resulting from a higher investment by downstream firm i. With perfect spilloversan investment by firm i benefits all firms equally (∂qi/∂xi = ∂qj/∂xi > 0), and wethen have ωxji = 1. Otherwise we have ωxji < 1 (and ωxji is negative if ∂qj/∂xi < 0 ∀i).Imposing symmetry, we can now reformulate (12) aspV I£1 + (n− 1)ωxji¤ ∂qi∂xi= φ0(xi). (13)The first-order condition ∂Π/∂pi = 0 is given by equation (7), and thus λ∗in equation (8) still applies. Clearly, aggregate profit is maximized also in thedecentralized market structure if it yields the same prices and investment levels asunder vertical integration. We can therefore use equations (11) and (13) to find thatthe upstream firm at stage 1 should setθ = θ∗ =1 + (n− 1)ωxjipλ∗V I. (14)Abstracting from the distribution of the fixed fee F, the downstream firms’ partici-pation constraint requires that θ > 0 (c.f. equation (9)). The range of permissiblevalues for θ∗ is thus in the interval (0, n/pλ∗V I ]. In the extreme case where an invest-ment by one downstream firm increases its demand by as much as the other firmsloose in sales (∂qi/∂xi = − (n− 1) ∂qj/∂xi), the investment is a waste of resourcesfrom the industry’s point of view. Then the upstream firm should set θ∗ close to zero.In the other extreme case, where we have perfect technological spillovers (ωxji = 1),we see that θ∗ = n/pλ∗V I . More generally, the upstream firm should specify a profit-sharing rule which gives each downstream firm a higher profit margin, as captured7by θ∗, the more beneficial its investments are for its rivals.We can state:Proposition 4: The profit-sharing rule βi(pi) = θpλi with λ = λ∗ and θ =θ∗ yields the downstream firms pricing and investment incentives which maximizeindustry profit.4 Concluding RemarksA major problem in many network industries is that firms may end up with de-structive competition because they produce relatively close substitutes. This mayprevent the firms from undertaking investments which could benefit the industryin aggregate. Such an outcome can be avoided by implementing a profit-shiftingrule which reduces the downstream firms’ perceived elasticity of demand. Optimalinvestment levels are ensured by giving the downstream firms an appropriate profitmargin that depends on how one firm’s investments aﬀect its rivals.Another merit of our approach is that it is easy to implement when marginal costsare low, since profit sharing then approaches revenue sharing. A general limitationof revenue sharing is the costs of monitoring the retailer’s revenue (Cachon andLariviere, 2005, and Dana and Spier, 2001). In the case at hand, this problem israrely significant, since the upstream mobile provider collects the revenue from theend users (but it is the content providers who decide end user prices).5 LiteratureCachon, G.P. and M. Lariviere. 2005. Supply Chain Coordination with Revenue-Sharing Contracts: Strenghts and Limitations, Management Science, 51(1), 30-44.Dana, J.D. and K.E. Spier. 2001. Revenue Sharing and Vertical Control in theVideo Rental Industry, The Journal of Industrial Economics, XLIX, 223-245.Lal, R. 1990. Improving Channel Coordination Through Franchising, MarketingScience, 9 (4), 299-318.8Mathewson, G.F. and R.A. Winter. 1984. An Economic Theory of VerticalRestraints, Rand Journal of Economics, 15, 27-38.Strand, J. 2004. Norwegian Operators Push Premium SMS, New Media Age, 30September 2004.9

https://openaccess.nhh.no/nhh-xmlui/bitstream/11250/163054/1/dp2007-4.pdf

Price-dependent profit sharing as an escape from the Bertrand paradox

Abstract

Similar works

Full text

Available Versions

HSN Open Archive

NHH Brage

NORA - Norwegian Open Research Archives