“Informative Advertisement Of Partial Compatible Products”

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TSE‐483March 2014“Informative Advertisement of Partial CompatibleProducts”Guillem Roig

Informative Advertisement of Partial Compatible ProductsGuillem Roig First draft: November 2011March 26, 2014AbstractProduct design and advertisement strategy have been theoretically studied as separatefirms decisions. In the present paper, we look at the link between advertisement and productdesign and we analyze how firms’ advertising decisions influence the market effect of product design. We consider a model of informative advertisement where two firms produce abundle of complementary products which are partially compatible. A product design withmore compatible components is associated with a larger intensity of advertisement. Highercompatibility reduces competition between firms, which incentivizes them to give factual information about their bundle. Like Matutes and Regibeau (1988), industry profit and totalwelfare is maximized with full product compatibility. However, contrary to them, we obtainthat consumer surplus is not monotone with the level of product compatibility and its maximum is attained with partial compatibility. Moreover, because consumer surplus not onlydepends on the equilibrium prices but also on the intensity of advertisement, we find that forintermediate equilibrium levels of advertising, consumers prefer fully compatible componentsrather than full incompatibility. As a result, a more compatible product design benefits allthe agents in the economy.Keywords: Informative advertisement; product design; partial compatibility; welfare.JEL classification: D21, D43, L13, L15 Toulouse School of Economics. GREMAQ. Contact information: guillemroig182@gmail.com. I am gratefulto my supervisor Jacques Crémer and Jiangli Dou for their help and support.1

1IntroductionEconomists are well aware that product characteristics beneficial to consumers might endup generating lower consumers surpluses. Even if consumers value more reliable and higherquality products, they may prefer situations where the previous features stay low. The explanation comes from how product characteristics affect market competition. Higher quality orproduct reliability, might reduce competition in the market by making the existing productsmore heterogeneous. This translates to higher equilibrium prices and hence lower consumersurpluses. A paradigmatic example is the situation of product compatibility. While makingcompatible the components of different products is welfare enhancing, because it allows fora better match between product components and consumers’ preferences, compatibility alsoreduces the competition between assembled products. In this regard, Matutes & Regibeau(1988) show that consumers with heterogeneous tastes for different components are better-offin situations where components’ compatibility is low. In their model, the equilibrium priceeffect through compatibility dominates the gains coming from a better match. Can we thenachieve a situation where product characteristics that are intrinsically beneficial to consumers,generate also larger consumer surpluses?In the present paper, we argue that an alignment of customers surplus and welfare can beachieved if we extend the firms’ strategy set. The existing marketing literature has extensivelystudied the topic of product design, and its resulting affects on the strategic interaction amongfirms and its final repercussion on consumers. However, the decision on product design hasbeen theoretically separated from the product advertisement strategy. In the present paper,we aim at filling this gap by linking product advertisement and design. We study how thestrategy to advertise influences the market effect of product design.In this regard, we consider a situation where consumers are not aware of product existence, and we explore what are the incentives to undertake informative advertisement whentwo firms produce a bundle of complementary products that are partially compatible. Firmsdo not take compatibility decisions, the level of compatibility is exogenous, but they decide onthe intensity of advertising prior to consumers’ purchasing decisions. We find that advertising increases both demand and competition, but the effect on the latter is reduced whenevercomponents are more compatible. Therefore, we find that the equilibrium intensity of adver-2

tisement increases with the level of product compatibility. Our results coincide with Matutes& Regibeau (1988) regarding industry profit and welfare, where its maximum is attained withfull product compatibility. However, we find opposite results with regards to consumer surplus. Since in our model, the level of product compatibility does not only affect prices but alsothe equilibrium intensity of advertisement, consumer surplus is not monotone with the levelof product compatibility and its value is maximized when compatibility is partial. A largerlevel of advertisement generates extra demand an the possibility of consumers to “mix andmatch” components. Moreover, for some parameters of the model we obtain that consumersprefer perfect compatible products rather than full incompatibility. Therefore, the existenceof advertisement makes firms and consumers better-off with compatibility and it works as amechanism to align players’ preferences over product compatibility.The present work is related to the existing literature on informative advertising. In thisliterature, the role of advertisement is to convey factual information to consumers about theprices and the specification of the advertised product. Therefore, consumers in the economyhave a passive role as they only learn about the existence of a product when they receive anadvertisement from the firm. Most of the existing literature builds from the work of Butters(1977), where the advertisement process is specified by assuming that firms send independentadvertisement messages and have no ability to target advertisement to consumers. Grossman& Shapiro (1984) employ this framework to compare private and social advertisement ina model of highly differentiated products. They find that equilibrium prices are decreasingwith the level of advertisement and its equilibrium level is socially excessive. Soberman (2004)extends the previous model and shows that whenever the level of product differentiations ishigh, prices increase with the level of advertisement. By assuming an exogenous small levelof product differentiation, we obtain that the equilibrium prices decrease with the intensity ofadvertisement. Finally, because consumers do not incur to shopping cost, the advertisementof prices does not have any effect in our model. In this way, we do not have the possibilitythat firms incur to a “loss leader” advertisement strategy as studied in Chen & Rey (2012)and Ellison (2005).The remaining of the paper is organized as follows. In section (2), we present the set-up ofthe model, and in section (3) we derive consumers’ demand and firms’ profits. Later, in section(4) we obtain the general formulation of the equilibrium prices and intensity of advertisement.3

We proceed with welfare analysis in section (5). Finally, we conclude in section (6). All proofsare in the appendix.2ModelWe consider a model of informative advertisement and partial product compatibility. Following Matutes and Regibeau (1988), we have a duopolistic market where each firm i A, Bproduces and sells two maximally differentiated components xi and yi with zero cost of production. Firms are located in the extremes of a Hotelling square, firm A is at the origin (0, 0)and firm B is located at (1, 1). Consumers are uniformly distributed with mass normalizedto 1. A consumer (x, y) has a preferred first component that is x away from firm’s A firstcomponent, and a preferred second component that is y away from firm’s A second component. Similarly, her preferred point and firm’s B components are (1 x) and (1 y) awayfrom each component respectively. Therefore, the distances between consumer (x,y) and thespecification of the x and y component sold by either firm A or B are:Definition 1. dxA 0 x ,dyA 0 y ,dxB 1 y , and dyB 1 y .We have a static game that consists of two stages. At stage 1, each firm decides on itsintensity of advertisement φi [0, 1] and sets prices pxi and pyi for each component. The firmsdo this simultaneously and non-cooperatively. The intensity of advertisement φi representsthe fraction of the target population that is exposed to the message of firm i. In this regard,consumers in our model are passive because they only learn about the existence of a productwhen they see an advertisement. We denote by E(φ, α) the total expenditure of advertisement,and the parameter α represents its effectiveness. Therefore, the higher the effectiveness, the00 (·) lower is the cost to reach a given fraction of consumers. We assume Eφ0 (·) 0; Eφφ00 (·) 0, and that the Inada conditions are satisfied.10; Eα0 (·) 0; and EφαAt stage 2, consumers make their purchasing decisions. They need a unit of each component to form a system which gives them a gross utility of V. If consumers do not receiveany advertisement they are uninformed and cannot effectuate any purchase. If they receiveone advertisement, they are captive consumers and can only effectuate a purchase from the1Decreasing returns to scale on advertisement might be due to media saturation or heterogeneity of consumers on viewing ads.4

firm of which they are aware. Finally, consumers that receive two advertisements are selectiveand can “mix and match” components produced by different firms. Their ability to “mix andmatch” depends on the degree of incompatibility between components, represented by theparameter z 0, which represents the loss in utility for consumers who consume a bundle ofcomponents produced by different firms.2Therefore, the utility of uninformed consumers is 0 and they are in the economy with aproportion of (1 φA ) (1 φB ). The utility of a captive consumer (x, y) isV (dxi dyi ) pxi pyifor i A, B,and their proportion in the economy is φA φB 2φA φB . Finally, the utility of selectiveconsumers, in proportion φA φB , is the same as captive consumers unless when they buyform different firms. In this latter case, their utility is V dxA dyB pxA pyB z,if they buy component x from firm A and component y from B.Finally, since firms cannot differentiate consumers that have seen both advertisements oronly their own; price discrimination is not feasible.3Demand and PayoffsWith the utility functions of consumers, we can derive the form of the demand function. Thisis obtained by identifying the consumer that is indifferent between two alternatives. In orderto simplify the calculations, we make the following assumption regarding the gross utility ofconsumption.Assumption 1. The gross utility of consumption V is large enough to ensure full marketcoverage.3The assumption implies that consumes will always effectuate a purchase. Hence, the indif2Consider for instance the installation of some extra plug-ins for a certain software to work with a specifichardware.3Later in the paper, we show that for an equilibrium intensity of advertisement φ̂ and prices, this is sufficient5

ferent selective consumers are represented in the figure below. Because the characteristics ofthe consumers are uniformly distributed, the demand is just the area of the regions representedin the figure.12 xpxA pB z2FB (1, 1)DAB12 DBBpyB pyA z212DAA pyA pyB z2DBAFA (0, 0)12 xpxB pA z2Figure 1: Demand for selective customers, where dashed lines represent the set of indifferent consumers.They are analytically given by: 1 (1 pxB pxA z)(1 pyB pyA z) 2z 2 ,4 1 (1 pxB pxA z)(1 pyB pyA z) ,4DA,A DA,Bwhere the first and the second under-script stands for the identity of the firm for the first andsecond component respectively. We refer to the appendix in page 22 for the calculation of thedemand functions.Finally, the profit of each firm is the revenue obtained from each group of consumers minusthe total expenditure of advertising. Hence, firm A obtains πA (φA φB ) (pxA pyA ) DA,A pxA DA,B pyA DB,A φA (1 φB ) pxA pyA E(φA , α).to have 2 2 φ̂zφ̂ (1 z) 4 φ̂ φ̂ 2z 1 φ̂ V (1 φ̂) φ̂ (1 z)6.(3.1)

The first line are the revenues obtained by consumers who are aware of both bundles, and thesecond are the ones obtained by the consumers that are only know the bundle produced byfirm A. Having defined the profit function, we proceed to obtain the equilibrium price andintensity of advertisement.4EquilibriumFirms make simultaneous decisions to choose the intensity of advertisement and the price foreach component.4 From the previous demand functions, and the assumption of full coverage,we observe a conflict between the pricing strategies for each group of consumers: because ofthe different competitive pressure, firms want to set low prices to selective consumers andhigh prices to those consumers that are captive. Due to the fact that firms cannot pricediscriminate, the market equilibrium price is a compromise between those two conflictinginterests, and a pure strategy equilibrium may fail to exist.The following proposition characterizes the equilibrium in prices and the intensity of advertisement for a symmetric Nash equilibrium in pure strategies and full coverage of consumers.Proposition 1. In a symmetric equilibrium with full coverage, the price of each componentand the intensity of advertisement is implicitly defined by the following system of equations(p̂, φ̂)p̂ Eφ (φ̂, α) (2 φ̂)φ̂(1 z)(2 φ̂)2φ̂(1 z), p̂ (2 φ̂),(4.1)(4.2)whenever the equilibrium intensity of advertisement is φ̂ 1. The equilibrium price is p̂(z) 11 z ,when the equilibrium intensity of advertisement is φ̂ 1.The previous expressions are easily obtained from the first order condition of prices andintensity of advertisement. We relegate the calculation in the appendix page 18, where weverify that no firm wants to deviates from the proposed equilibrium.4In our model consumers do not have shopping costs and they are passive. Only firms’ advertisement givesthem information about the existence of the product and market prices. If consumers had shopping costs, itmight not be optimal for the firms to disclose information about prices. At this regard, an optimal strategy isto advertise one of the components at a loss in order to attract consumers.7

The expressions in the proposition give an implicit solution of the problem and later, weprovide an explicit solution by making use of a specific advertising cost function. Condition(4.1) tells us that the equilibrium price of each component decreases in both the level ofincompatibility and the intensity of advertisement. When incompatibility rises, selectiveconsumers tend to buy both components from the same firm. Because in equilibrium, firmscompete for the bundle rather than for separate components, competition is intensified andthe equilibrium price drops. Similarly, when the intensity of advertisement increases, therelative importance of selective consumers with respect to captive consumers rises, and sincecompetition occurs only with respect to selective consumers, prices fall.Condition (4.2) states that the marginal cost of advertising equals its marginal benefit.5The intensity of advertisement increases with the effectiveness parameter α and decreaseswith the level of incompatibility. Because with a larger effectiveness, advertisement becomescheaper, a higher proportion of consumers are informed about the product. Since incompatibility increases competition for selective consumers, and advertisement intensifies competitioneven further, firms decide to reach a lower proportion of consumers in order to keep competition milder.Therefore, the level of incompatibility has two opposite effects on the equilibrium price.The direct effect is that a higher degree of incompatibility makes the selective market morecompetitive and the equilibrium price of each component falls. The indirect effect comes fromadvertisement. Since advertisement works as mechanism to increase competition, the equilibrium intensity of advertisement decreases with the level of incompatibility. Consequently, therelative importance of captive consumers is larger and prices increase. The following lemmashows that the direct effect is of first order and an increase of incompatibility unambiguouslyreduces the equilibrium market price. The formal proof is in page 20 in the appendix.Lemma 1. An increase of incompatibility decreases the market equilibrium prices:dp̂ 0.dzWe proceed with welfare analysis, and we see that consumers obtain a larger surplus with5A unique equilibrium exists. Since the cost function is convex and the right hand side is decreasing withthe intensity of advertisement, there exists only one value of φ such that both expressions cross.8

higher levels of product compatibility. The reason is that whenever firms need to inform aboutthe existence of the product, consumers’ surplus is not only affected by equilibrium prices butalso by the intensity of advertisement.5WelfareIn order to make the analysis tractable, we assume that the reservation price of consumers issuch that the equilibrium stated in proposition 1 exists.Total welfare in our model is the sum of both industry profit and consumer surplus.W Π CS.Firms’ profits is the sum of the revenues obtained from selective and captive consumersminus the expenditure on advertisement: 2 2 φ̂(·)1π(z, α) φ̂2 (·) 2p̂(·) φ̂(·) 1 φ̂(·) 2p̂(·) E(φ, α) E(φ, α), (5.1)21 z and due to symmetry, the total industry profit is 2π(z, α). Consumer surplus is equal to CS(z, α) φ̂(·) 2 φ̂(·) (V 2p̂(·)) IC(z, α) P C(z, α),(5.2)where IC(z, α) and P C(z, α) are now defined.Incompatibility costs, IC(z, α), are incurred only by selective consumers who “mix andmatch” components from different firms and equalIC(z, α) φ̂2 (·) (DAB DBA ) z z(1 z)2 φ̂2 (·),2(5.3)and preference costs, P C(z, α), come from the fact that consumers cannot perfectly matchtheir preferences with the actual systems in the market. On average, selective consumers havea lower preference cost than captive consumers, and this is because the former are able to“mix and match” components from both firms. This equals to9

2P C(z, α) φ̂ (·) 1 z22 2φ̂(·) (1 φ̂(·)),(5.4)where the first part corresponds to selective consumers and the second to captive consumers.The derivation of the preference costs is not straightforward and the interested reader isreferred to the appendix page 22. By introducing expressions (5.3) and (5.4) into consumersurplus we obtain h z1 z22CS(z, α) φ̂(·) 2 φ̂(·) (V 2p̂(·)) φ̂(·) (1 z) 22 i 2 1 φ̂(·) ,(5.5)and adding this with the industry profit, yields total welfare 2W (z, α) φ̂(·) 2 φ̂(·) V φ̂ (·) (2 2φ̂)1 z2 z (1 z)2 22φ̂(·)! 2φ̂2 (·). (5.6)αWe proceed to obtain a closed solution of the model by assuming a specific form of theexpenditure of advertisement.5.1Explicit SolutionWe assume a particular form for the cost of advertisementE(φ, α) φ2.α(5.7)Differentiating this expression with respect to φ and substituting into (4.2), the equilibriumintensity of advertisement is equal to(φ̂(z, α) min2αα )p,1 .2α(1 z)(5.8)Hence, for α 2(2 z) ᾱ(z) there is partial information. By substituting the equilibriuminvestment to all previous expressions, we obtain the following results.In the figure below, we see how the equilibrium intensity of advertisement and prices evolvewith the level of incompatibility.10

θp̂(z, α)Π(z, α)IC(z, α)P C(z, α)CS(z, α)SW Π CSα ᾱ(z)1α 2(1 z) ᾱ(z) 2αα 2α(1 z) 2α(1 z)8α 11 z21 z 2α 2α(1 z)2α2 (1 z)2 z 2 α 2α(1 z)2αz(1 z)22h i2α 2 2α(1 z) α(z 2 1) 2 α 2α(1 z)hi 2α 8 2(V 1) 2α(1 z) α(1 z)(1 z 2 ) 2 α 2α(1 z)hi 2α 4 2(V 1) 2α(1 z) α(1 z)(1 z 2 ) 2 α 2α(1 z) 1 z 222V (z 1) (5 2z) z 42(1 z)2V (z 1) (1 2z) z 42(1 z) 2αFigure 2: Intensity of advertisement, equilibrium price and welfare indicators. The column on theleft stands for a situation where the effectiveness of advertisement is such that there is partial advertisement, i.e φ̂ 1 and the one on the right stands for full informed market i.e. φ̂ 1.The dashed lines represent a situation with full advertisement for low values of incompatibility. In this case, the decrease in price is more severe than when we have an equilibrium withpartial advertisement. When the market is not fully informed, an increase of incompatibilitycreates a decrease on the intensity of advertisement, and this boosts the proportion of captiveconsumers in the economy. This last effect, smooths the reduction in prices.6If we turn to the analysis of welfare, we see that with full advertising, the results aresimilar to Matutes and Regibeau (1988). In such a case, both industry profit and welfareattains its maximum with full product compatibility and consumer surplus is maximized withfull incompatibility CS [z 0 α ᾱ(z)] CS [z 1 α ᾱ(z)]. Here, all results are drivenby equilibrium prices, because all consumers are informed, the intensity of advertisement doesnot have any effect on the total demand.Our contribution stands for analyzing the case where there is an equilibrium with partial6We have already mentioned that the competition effect is of first order with comparison to the changein the consumers’ composition. We compare how prices changes with the level of incompatibility in bothequilibria. We observe that the change of prices with the level of incompatibility is larger with a full informedequilibrium. p z, α φ̂ 1 pz,α φ̂ 121 α 2. zα(1 z)2(1 z)2 zand this is always the case for most of the parameters of efficiency considered.11

φ̂ , p̂10.930.830.710.5801zFigure 3: Equilibrium price and intensity of advertisement as a function of the level of incompatibilityz. The thick line represents a situation where the parameter of advertisement efficiency is equal toα 2, and the dashed line stands for α 3.advertisement and not all consumers are aware of the existence of both systems. In order toperform this analysis, for the rest of the paper we consider that V 6α(z) 1/2(1 z) α(z)(5.9)In the appendix, page 23, we show that this is necessary to obtain a pure strategy equilibriumin prices.Industry profit does not depend on the reservation price of consumers, and this decreaseswith the level of incompatibility. Here, we distinguish two effects. When incompatibilityincreases so does the equilibrium price, and because the intensity of advertisement also decreases, demand is consequently reduced. However, the fall of profits is more accentuatedin a situation when all consumers are informed about the products, since the reduction ofprices in this case is larger. In general, higher efficiency of the advertising technology entailslower industry profits. With a higher effectiveness of advertisement, firms advertise more inequilibrium, and the proportion of selective consumers increases, brining about a more intensecompetition. This competition effect dominates the increase in demand and the reduction ofthe cost of advertisement coming from a more efficient advertising technology.With regards to consumer surplus, the following proposition states the main result ofthe paper. This reveals that consumers are better-off with an intermediate level of productcompatibility.12

Proposition 2. With V 6 and (5.9) holding, consumer surplus attains its maximum foran intermediate level of product compatibility whenever the advertising efficiency belongs to(α̂, ᾱ), and α̂ is defined by 10 α̂ α̂ 2 α̂3 8. ˆ ᾱ , where α̂ˆ is defined byMoreover, whenever α α̂, p pˆ4 2 5α̂ˆˆ 8 10 2α̂ α̂ p 2p 2 ,ˆˆ2 α̂2 α̂consumers are better with complete product compatibility than full incompatibility.7The formal proof is in the appendix, page 21. Whenever the market is partially informed,the level of incompatibility affects not only the equilibrium price, but also the total demand aswell as the demand composition in the market. While with a low level of incompatibility, thedecrease of the equilibrium price dominates the decrease in the intensity of advertisement, forhigh levels of incompatibility, the reduction in the level of advertisement, and the subsequentincrease in the preference costs, dominates the effect of the price decrease. Consequently, theaverage consumer is worse-off. In general, consumers prefer to be better informed about theproducts offered in the market at the expense of higher equilibrium prices. In the figure 4below, we illustrate how consumers’ surplus varies with z, the level of incompatibility and asshown in the proposition, its maximum is attained with an intermediate level of compatibility.The level of incompatibility also plays a role the evolution of both incompatibility andpreference costs. For low values of incompatibility, the total incompatibility cost increases withincompatibility as selective consumers who “mix and match” pay this cost. For large valuesof incompatibility, the total incompatibility costs decrease, because there are less selectiveconsumers in the market, due to lower advertisement, and also a smaller proportion of theseconsumers decide to “mix and match”. Preference costs are an increasing function on the levelof incompatibility. A lower intensity of advertisement, due to an increase of incompatibility,reduces the proportion of selective consumers while the proportion of captive consumers isincreased. Because captive consumers have on average a larger preference costs, the total7ˆ that solve the previous equations. See page 21 in the appendix for the formalThere is only one α̂ and α̂proof.13

preference cost in the economy is increased. A lower proportion of selective consumers will“mix and match” components, and the preference costs are accordingly increased.Furthermore, we observe that increases in the advertising effectiveness unambiguouslyincreases consumers surplus. Not only prices are lower in equilibrium, but also consumers arebetter informed about the existing products, because the intensity of advertisement is bigger.SW , CS , Π4.84.54.44.173.833.53.481.3410.680.5801zFigure 4: Industry profit, consumer surplus and social welfare as a function of the level of incompatibility z. The thick line represents a situation where the parameter of advertisement efficiency is equalto α 2, and the dashed line stands for higher efficiency α 3.Finally, welfare always decreases with the level of incompatibility. Because prices are justa transfer from consumers to firms an increase in incompatibility unambiguously decreaseswelfare.8 Incompatibility does not only create a higher preference cost as the proportionof consumers who “mix and match” is decreased, but in our model it also decreases theequilibrium intensity of advertisement. Less advertisement, creates a reduction of demand asthe proportion of uninformed consumers increases.Therefore, whenever firms have to undertake advertisement to inform about the existence8This is the case as we are working with full coverage. Otherwise, the equilibrium price would have an effecton social welfare as it has an effect on total demand.14

of the products, all agents have similar preferences regarding the level of product compatibility.We have shown that for some parameters firms and consumers are better-off with full productcompatibility. Moreover, efficient advertising in our model works as a way to increase potentialdemand but it also fosters competition. This second effect is reduced by making productscompatible. Hence, we have found two instruments that firms might use to increase profits.One is to increase compatibility, the other is to agree on having an inefficient advertisingtechnology.6ConclusionIn this paper, we have shown that intrinsic beneficial characteristics, such as product compatibility, create larger consumer surpluses if we extend the firms’ strategy set. In our model,higher product compatibility generates lower competition which translate to higher prices.However, lower competition incentivizes firms to convey information about their product,which generates a demand increase and a better match between products and consumers’ preferences. In contrast with previous literature, we find that consumer surplus is non-monotonicon the level of incompatibility and it has an inverse U-shaped form. A competition authorityshould then be careful about possible actions aiming at increasing the consumer surplus suchas increasing market competition. Larger competition between firms might be achieved bymaking existing products less compatible. However, we have seen that increasing the levelof competition might have detrimental effects to consumers, because they are less informedabout the purchasing possibilities in the market.Following the existing literature on informative advertising, we have considered that consumers are passive as they do not engage into any active search to find about the productsoffered in the market. Furthermore, consumers do not incur to any shopping costs whenthey commute to the firm to effectuate a purchase. This assumption simplifies the analysistremendously, because the decision of firms referring to what component and what price toadvertise is not relevant. However, if consumers experienced some positive shopping costs itmight be in the interest of the firms to practice some sort of “loss leader” advertising strategyas studied in Chen & Rey (2012). We believe that as long as a “loss leader” strategy softenscompetition, the intensity of advertisement will be larger in equilibrium. However, the results15

on consumer surplus are not clear. Prices will be larger in equilibrium but consumers willalso be mo

TSE‐483 “Informative Advertisement of Partial Compatible . & Regibeau (1988) regarding industry pro t and welfare, where its maximum is attained with full product compatibility. However, we nd opposite results with regards to consumer sur-plus. Since in our model, the level of

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