Product Specification with Differentiation by Attributes: Is There Too Much Innovation? Reynald-Alexandre LAURENT∗

Abstract This paper considers a probabilistic duopoly in which consumers follow a random decision rule based on products’ attributes. We study a two-stage game in which firms choose, first, the specific attributes of their product and, then, compete in prices. The existence of a perfect Nash equilibrium is proven under different cost assumptions. When costs are exogenous, firms classically choose the highest attributes indices, which are also socially optimal. When unit or fixed costs are attributesdependent, firms select heterogenous attributes indices and product differentiation is both horizontal and vertical. However, the equilibrium attributes gap is excessive compared to the social optimum, achieved for a pure horizontal differentiation. The firm selling the less appreciated product makes the highest profit, its margin being preserved by the horizontal differentiation : this result contrasts with the high quality advantage highlighted in pure vertical differentiation models.

jel classification : D11, D43, D60, L13. Keywords : Product differentiation, quality choices, welfare analysis.

∗ PSE,

Paris School of Economics / Paris-Jourdan Sciences Economiques (CNRS, EHESS, ENS, ENPC). Contact : PSE-

ENPC, 28 rue des Saints-Pères, 75007 PARIS or [email protected]. I thank Bernard Caillaud, Gabrielle Demange, Dominique Henriet, Philippe Jehiel, Laurent Linnemer, Jacques-François Thisse and other seminar participants at PSE and CREST-LEI for their helpful comments.

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Introduction From 1930 to 1982, the American company Caterpillar Tractor, an heavy earth-moving equipment

manufacturer, kept improving its product with new attributes and services, putting aside cost considerations (Miller, 1990, p. 22). Caterpillar’s product was perceived as a benchmark by its competitors, always imitated but never equaled. As underlined by Peters and Waterman in their book “In Search of Excellence” (1982, p 171), the firm was also reputed for its reliability, making the high prices acceptable for its customers. This case gives rise to several questions. How to justify such a strategic positioning? What is the effect of this systematic product improvement policy on social welfare? Some answers can be given by studying firms’ choices of innovation when products are described and differentiated by their attributes. Standard economic models suppose that products are differentiated either along a vertical dimension, the level of quality, or along a horizontal dimension, the type of variety. However, these two dimensions generally co-exist and this distinction seems somewhat arbitrary to describe consumers’ and firms’ behaviors. Specifically, consumers select a product according to the attributes it possesses or not. For example, the presence of a particular safety accessory can be a decisive factor for the purchase of a car. Within firms, product managers have perceived long ago the necessity to develop specific attributes for their products in order to build a competitive advantage (for instance, see Nash, 1937, p 256). The decision problem of a marketing service may be better described by a choice made in a set of attributes than by a choice of “varieties” or “qualities”. In particular, a firm may either add to its product an attribute already existing on the market in order to make up for the competitive advantage of a rival (imitation), or develop a new attribute in order to gain a competitive advantage (innovation). However, such a policy usually leads to higher fixed R&D costs and/or unit production costs. The choice of qualities with endogenous (quality-dependent) costs has been largely studied in the economic literature, both in models with vertical differentiation (Ronnen, 1991 ; Motta, 1993 ; LehmannGrube, 1997) and in the logit oligopoly (Anderson, de Palma and Thisse, 1992). The duopoly with “differentiation by attributes” (Laurent, 2007a), also called “DBA”, provides an alternative and relevant framework to describe firms’ choices of product specification. In this model, consumers are not heterogenous according to an individual characteristic, as the taste for quality (Mussa and Rosen, 1978), the income (Gabszewicz and Thisse, 1979) or more generally an individual preference (as in random utility models): they follow a probabilistic reasoning and the heterogeneity of their behavior stems from the random decision rule they use. At the price equilibrium, the model provides a general framework which embodies existing types of differentiation. Thus, when the specific attributes of the two goods provide the same utility index to consumers, differentiation is horizontal. When a single product possesses all the specific attributes available on the market, and the other none, differentiation is vertical. Finally, when each good has some specific attributes but that provide different utility levels to the consumers, the above two dimensions of differentiation are simultaneously taken into account. Such a two-dimensional differentiation has been analyzed by Neven and Thisse (1990), Economides (1993) and Irmen and Thisse (1998). However, in these models, the number of characteristics of differentiation is given a priori, whereas the nature of product differentiation is a consequence of firms’ strategic choices in the model used here.

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This paper extends the analysis of the price equilibrium by studying a two-stage game in which firms choose the new specific attributes of their product and, then, compete in prices. We consider the cases of exogenous costs, attributes-dependent unit costs and attributes-dependent fixed costs. An analysis of firms’ decisions in terms of social welfare is also realized. From a descriptive point of view, we characterize the subgame perfect Nash equilibria in the sequential game. When costs are not linked with the attributes, firms choose the highest attributes indices. When unit costs are convex and individual utilities concave in the attributes index, a subgame perfect Nash equilibrium always exists. One firm chooses the highest possible attributes index, while its rival selects a level that equates the marginal utility and the marginal cost of its attributes. This outcome is linked with the reference status of the firm selling the most appreciated product and provides a possible explanation of Caterpillar’s behavior in the 1980s. When fixed costs depend on attributes, we use the classical linear utility and quadratic fixed costs setting and show that the differentiation is both horizontal and vertical at the equilibrium. The firm selling the less appreciated product always makes the highest profit when costs are endogenous : this result contrasts with the high quality advantage highlighted in models with pure vertical differentiation. In order to carry out a normative analysis, the consumer surplus is defined as the expected utility of consumption, which includes both the specific attributes of the goods and the attributes they share. When costs are exogenous, the equilibrium attributes index is also socially optimal but this result no longer holds when costs are attributes-dependent. When unit costs are endogenous, there is an excess of vertical differentiation in that the social optimum is achieved under pure horizontal differentiation. When fixed costs are endogenous, this excessive differentiation effect is strengthened: the degrees of vertical and horizontal differentiation are too high. The paper is organized as follows. The properties of the Nash price equilibrium in the DBA duopoly are resumed in section 2. Section 3 presents the concept of attributes index and the sequential game considered. Section 4 studies firms’ attributes choices at the perfect Nash equilibrium. A normative analysis of these choices is carried out in section 5. The last section concludes and proofs of some propositions are presented in the Appendix.

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Price equilibrium in the DBA duopoly In an other paper (Laurent, 2007a), we propose a discrete choice duopoly in which products are

described by their specific attributes, an approach which recalls that of Lancaster (1966). For instance, the relative advantage of a car over its rival may come from a specific accessory, such as a sun roof. The specific attributes of a product i provide the consumers the utility ui , with i = {1, 2}. Each firm sells its product at a price pi , these prices being also perceived as attributes by the consumers, assumed relatively “affluent” : buying the least expensive good allows a save of a certain amount of money in the perspective of a future expense. In other words, the price gap between the products is perceived by the consumers as a specific attribute of the least expensive good. Thus, when pi > pj , product j possesses an additional attribute providing the utility θ(pi − pj ). The parameter θ measures the relative importance of the price attribute compared to non-price attributes. In the DBA duopoly, consumers follow a probabilistic reasoning : the probability Pi of buying a

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particular good i increases with the specific non-price and (eventually) price attributes of this product, represented at the numerator of the probability. We suppose that N consumers in the market purchase exactly one unit of one product and the demand Xi = N Pi for a product i depends on the price hierarchy : - if pi ≥ pj , Xi =

N ui ui + uj + θ(pi − pj )

(2.1)

Xi =

N (ui + θ(pj − pi )) ui + uj + θ(pj − pi )

(2.2)

- if pj ≥ pi ,

The choice probabilities of the DBA duopoly are consistent with the model of Restle (1961) and with the binomial “Elimination By Aspects” model (Tversky, 1972a,b). As shown by Rotondo (1986), considering the price gap as an attribute is the most relevant method to integrate prices in these structures1 . Suppose that each firm i bears a unit cost ci and a fixed cost Fi and chooses the price which maximizes its profit. Despite the kink in the demand curve, the existence of a Nash equilibrium in pure strategies has been proven : Proposition 1 (Laurent, 2007a, p 9) There exists a Nash price equilibrium verifying pi ≥ pj , with i, j ∈ {1, 2} and i 6= j, if and only if : ui ≥ uj

(2.3)

and √ ci − cj ≥

ui uj − ui θ

(2.4)

Moreover, this equilibrium is unique. Equilibrium prices when pi > pj are : p∗i

√ ui + ∆ = + ci 2θ

p∗j =

ui + uj + ci θ

(2.5)

(2.6)

with ∆ = u2i + 4ui (ui + uj + θ(ci − cj )). Thus, firm i, whose product is the most appreciated by consumers selects a higher price than its competitor. When ci 6= cj , it could seem surprising that p∗j increases with ci and not with cj . But the analysis of strategic interactions between firms shows that firm i is locally insensible to a little price variation of pj from the equilibrium : this observation reminds practices of pricing imitation like those described by Lazer (1957 p. 130-131), and particulary the case in which the firm selling the “best quality” good sets a reference price on the market (or “focal price” in the sense of Schelling, 1960). In this framework, the other firm chooses the reference price minus a certain amount, 1 Note

however that the DBA choice probabilities are never equivalent to that of the binomial logit of product differen-

tiation.

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which depends on the quality gap with the benchmark firm. The type of differentiation in the model depends on the utility parameters : a) When ui = uj = u > 0, products are differentiated and the specific attributes of each good are appreciated in the same way within the population of consumers. When ci = cj , pi = pj . Such a configuration, in which each consumer purchases its preferred variety when products are sold at a same price, refers to a pure horizontal differentiation, as in the Hotelling (1929) model. b) When ui > 0 and uj = 0, product i possesses all the specific attributes of the market. Thus, at equal prices, all consumers prefer purchasing good i rather than good j: this observation is the sign of a pure vertical differentiation in which a preference hierarchy between products exists, as in the models of Gabszewicz and Thisse (1979) and Shaked and Sutton (1982). c) Thus, the general case in which ui > uj > 0 corresponds to a double differentiation: differentiation is horizontal up to the level uj , the specific attributes of the two products providing the same utility, and vertical for a level ui − uj , product i also proposing additional attributes. d) Finally, when ui = uj = 0, any form of differentiation vanishes. For ci = cj = c, the market outcome is similar to a price competition à la Bertrand with homogeneous goods : pi = pj = c. At the outcome, profits are given by : Π∗i =

Π∗j

N ui − Fi θ

√ N (ui + uj + θ(ci − cj ))( ∆ − ui ) √ − Fj = θ( ∆ + ui )

(2.7)

(2.8)

Moreover, the firm selling the “most appreciated” product (such that ui > uj ) makes the highest profit when the gap of differentiation is sufficiently high compared to the gap of costs : Π∗i > Π∗j ⇔ θ(ci − cj ) < ui − uj

(2.9)

This finding is consistent with models of vertical differentiation, in which the high quality firm always makes the highest profit when exogenous unit costs are equal (Shaked and Sutton, 1982 ; Tirole, 1988, Choi and Shin, 1992 ; Wauthy, 1996). However, a comparative statics analysis of profits shows that differentiations by attributes and by qualities are not equivalent. For instance, when the low quality raises in a vertically differentiated duopoly, goods become closer substitutes and competition is increased. Conversely, when the low utility index of attributes raises in the DBA duopoly, product differentiation is strengthened and price competition relaxed.

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A game with attributes choices After having reminded the properties of the price equilibrium, this section presents our assumptions

on the attributes choices of the firms and the stages of a sequential game with price competition.

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3.1

The concept of “attributes index”

During the choice of product specification, a good i is endowed with some specific attributes ki chosen by its producer among a set of available attributes noted Ki (ki ⊂ Ki ). Each firm is perfectly informed if a given attribute is specific to its product or not2 : consequently, the attributes sets available for the two firms are disjoint : Ki ∩ Kj = ∅. Moreover, it is supposed that any set of specific attributes can be represented by a synthetic attributes index (or “index”), a continuous positive variable qi : ki → [0; Qi [ with Qi = q(Ki ). A given index q can be affected to several sets of specific attributes. This index is viewed as a measure of the level of “accessories” of the considered product. As the shared attributes are not included, note that q is not equivalent to the qualities in vertically differentiated models. Comparison between firms is easier when attributes are represented by indices rather than by vectors : if a firm i realizes a product innovation, its attributes index qi is simply assumed to increase. A given index q provides the consumers the utility u : q → R+ . Thus, a consumer purchasing a good i with a set of attributes ki obtains the utility ui = u(qi ) (we normalize u(0) = 0). If firms choose the sets of specific attributes k1 and k2 verifying q1 = q2 , their products provide the same utility to the consumers. The function u also verifies u0 (qi ) ≥ 0 : any increase in the attributes index implies an additional utility for the consumer. Finally, we make the classical assumption that the marginal utility of the attributes decreases with the index : u00 (qi ) ≤ 0. The following Inada conditions are supposed to hold lim u0 (qi ) = 0 and lim u0 (qi ) → +∞. The effect of attributes index on costs is detailed in section 4.

qi →+∞

3.2

qi →0

Stages of the game and equilibrium solution

Competition between firms is modelled by a two-stage game in pure strategies. In the first stage, firms choose simultaneously3 an attributes index for their products. In the second stage, the indices previously chosen are of a common knowledge and firms compete in prices. Equilibrium prices are given by equations (2.5) and (2.6). The game is solved by backward induction. For a first stage vector of equilibrium indices (q1∗ ; q2∗ ), the price subgame at the second stage is solved by a couple of prices (p∗1 (q1∗ ; q2∗ ); p∗2 (q1∗ ; q2∗ )) such that the following inequality holds for each firm i (i = {1; 2}) : Πi (p∗i , p∗j ; q1∗ , q2∗ ) ≥ Πi (pi , p∗j ; q1∗ , q2∗ ) ∀pi ∈ [0; +∞[, ∀i, j ∈ {1, 2}, i 6= j At the first stage, the equilibrium of the “attributes” subgame is solved by a vector of attributes indices (q1∗ ; q2∗ )

such that the index chosen by i satisfies : ci (qi∗ , qj∗ ) ≥ Π ci (qi , qj∗ ) ∀qi ∈ [0; Qi [, ∀i, j ∈ {1, 2}, i 6= j Π

ci (q ∗ , q ∗ ) = Πi (p∗ (q ∗ , q ∗ ), p∗ (q ∗ , q ∗ ); q ∗ , q ∗ ) corresponding to the profit evaluated at the second with Π i j i i j j i j i j stage. 2 We

investigate only the “stationary” choices of new attributes, eventually achieved after a preliminary learning : in

other words, we suppose that any initial wrong belief in the specific nature of an attribute has been corrected. 3 A simultaneous choice of product specification is more appropriate than a sequential one if the new generations of products are developed by multiple firms, as in the automobile industry (Aoki and Prusa, 1997, p 104). Such a perspective is followed here.

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A subgame perfect Nash equilibrium is defined by a vector of equilibrium indices (q1∗ ; q2∗ ) and a vector of equilibrium prices (p∗1 (q1 , q2 ), p∗2 (q1 , q2 )) for all indices (q1 , q2 ).

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Product innovation and equilibrium attributes indices This section solves the two-stage game for various costs assumptions : exogenous costs, attributes-

dependent unit costs and attributes-dependent fixed costs.

4.1

Attributes choices with exogenous costs

In this section, unit and fixed costs are not linked with the indices and are assumed to verify c1 = c2 and F1 = F2 . At the end of the first stage, suppose that the attributes choices verify the conditions (2.3) and (2.4). In this case, equilibrium prices are given by (2.5) and (2.6). Each firm’s profit strictly increases with its attributes index and the highest indices are chosen when costs are exogenous. The existence of a perfect Nash equilibrium is guaranteed only if the equilibrium indices effectively verify the condition (2.3), which imposes the inequality Q1 ≥ Q2 . These conclusions are resumed here : Proposition 2 When costs are independent of the attributes, there exists two subgame perfect Nash equilibria differing only by the identity of the firms and verifying pi ≥ pj with i, j ∈ {1, 2} and i 6= j if and only if Qi ≥ Qj . The highest possible indices are chosen. This finding is similar to that obtained in the logit oligopoly (Anderson, de Palma and Thisse 1992, p 237) : the DBA model also respects the “principle of maximum differentiation”, each firm having an interest in pushing as far as possible its advantage. Differentiation is horizontal for a level Qj and vertical for a level Qi − Qj . However, this result depends crucially of the assumption of costs independence.

4.2

Attributes choices with endogenous unit costs

This section supposes that unit costs of production increase with attributes indices whereas exogenous fixed costs are equal. Thus, each firm possesses a set of available specific new attributes, discovered previously by its applied research service : the objective is to choose the attributes which will be finally added to the product. Such a decision is generally taken at the end of the innovation phase after a cooperation between the development and marketing services. Each firm’s unit cost is modelled by the same function c : q → R+ with c(0) = 0. For firm i, this unit cost is constant with the quantity produced but increasing and convex in the index : c0 (qi ) > 0 and c00 (qi ) ≥ 0. The attributes realizing the best balance between the cost and the utility provided are selected in first and the less “efficient” attributes are chosen afterwards. The maximum available index is identical for the two firms: Q1 = Q2 → +∞. Thus, we can set lim u(qi ) = uα where uα is a “reservation qi →Qi

utility” threshold reached by the consumers when the attributes index is maximum. We study the existence of a perfect equilibrium in which firms choose indices and then prices. At the first stage, the attributes choices are resumed in this proposition :

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Proposition 3 When unit costs are endogenous, there exists two perfect Nash equilibria verifying pi ≥ pj and which differ only by the identity of the firms. Firm i chooses the highest possible index such that u0 (qi ) = 0. Firm j chooses the index equating the marginal utility of the consumer and the marginal cost of production : u0 (qj )/θ = c0 (qj ). Proof : this proof is presented in Appendix 8.1. For firm i, whose product is the most appreciated, the introduction of a unit cost depending on the attributes does not modify the result obtained in the previous section. The choice of the highest index is a consequence of the weak nature of the price equilibrium in the model. Indeed, i being a “reference” firm, any increase in its unit costs also induces a raise of its rival’s price and the position of i in the market is not weakened. As its profit is invariant with ci , this reference firm selects the highest index, whatever the cost beard. Obviously, the relevance of this result does not lie in the exact values of attributes index computed4 but rather in the underlying strategic interactions we highlight. The choice of firm j is modified when unit costs become endogenous : a lower attributes index is chosen. The equilibrium index decreases with θ and if θ → +∞, firm j does not develop any specific attributes, leading to a market outcome with pure vertical differentiation. In the general case, differentiation is both horizontal for a level qj∗ (firms offering similar attributes indices) and vertical for a level qi∗ − qj∗ (firm i providing additional attributes). As mentioned in section 2, when exogenous unit costs are identical, the firm with the highest index always makes the highest profit : is this property still verified when unit costs are attributes-dependent ? In this framework, condition (2.9) becomes : Π∗1 ≤ Π∗2 ⇔ θc(q1∗ ) − u(q1∗ ) ≥ θc(q2∗ ) − u(q2∗ )

(4.1)

As the utility is concave and the unit costs convex, the following property is verified : d(θc(q) − u(q)) > 0 ∀q ≥ q with q > 0 such that θc0 (q) = u0 (q) dq The index q2∗ verifying q2∗ = q < q1∗ , the inequality (4.1) is always true : the firm with the lowest attributes index always realizes the highest profit. Indeed, the choice of a low attributes index reduces the unit cost while allowing to fix a relatively high price because of the horizontal dimension of differentiation on the market. In a pure vertical differentiation model with quality-dependent unit cost, the converse property holds : when cost function is quadratic, as in Motta (1993, p 124), the high-quality firm makes the highest profit at the perfect Nash equilibrium. The absence of horizontal differentiation increases competition and thus the disadvantage of proposing a low quality. This analysis can be linked with firms’ practices and in particular with the strategy of “quality supremacy”. Miller (1990) shows that such a strategy has been followed by the Caterpillar company until the beginning of the 1980’s. This search of quality turns to obsession : “Caterpillar had devoted itself long and single-mindedly to building a better, more efficient crawler tractor than anybody else in the world” (op. cit. p 22). Caterpillar developed new attributes increasing vertical dimension of differentiation, like an exclusive after-sales service : “Caterpillar offered its customers forty-eight-hour guaranteed 4 In

particular, as there is no outside option in the model, the attributes index is not bounded upside.

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parts-delivery service anywhere in the world, from a construction site in Nebraska to a village in Zaire. If it couldn’t fulfill that promise, the customer got the part free” (op. cit. p 23). Consequently, Caterpillar acquires a status of reference, allowing it to set very high prices5 . This strategy has also been partially responsible of Caterpillar’s difficulties at the beginning of the 1980’s, the company being confronted to a profit erosion and to the growing success of its main competitor, Japanese company Komatsu: “Cat’s obsession with quality had boosted expenses to the point where it could no longer compete. Its production methods had become too inefficient to enable it to match Komatsu’s prices” (op. cit. p 226). This observation is also consistent with the finding that the low-index firm makes the highest profit. To conclude, when unit costs are attributes-dependent, index choices in the DBA model provide a good description of a maximum quality strategy. However, note that firm i is not incited to choose the maximum index when a threat of imitation exists on the market : in this case, the market outcome gets closer to the traditional quality setting in pure vertical differentiation models 6 .

4.3

Attributes choices with endogenous fixed cost

Whereas the assumption of endogenous unit costs follows the perspective of a product development service, the assumption of an endogenous fixed costs rather corresponds to the upstream point of view of an applied research service. The quantity of new attributes proposed by this service depends on the investment realized in research, which generates a fixed cost. This section keeps the assumption Q1 = Q2 = +∞ and the properties of the function ui = u(qi ) presented previously. But we now suppose exogenous unit costs c1 = c2 = c whereas each firm bears a fixed cost F : q → R+ , with F (0) = 0. This function is increasing and convex in the attributes index of i : F 0 (qi ) > 0 and F 00 (qi ) ≥ 0. The raise in the number of specific attributes makes them more and more costly to discover. However, the proof of equilibrium existence can not be realized with functions of a general form. That is why we use the following setting : S1 : The utility has a linear form ui = qi and the fixed costs are quadratic Fi = qi2 . These functions are both meaningful and quite standard in the literature : for instance, a linear utility is used by Anderson, de Palma and Thisse (1992) and a quadratic fixed cost function is assumed by Ronnen (1991), Motta (1993) or Pepall and Richards (1994). As previously, we study the existence of a perfect equilibrium in which firms choose attributes indices and then prices. When pi ≥ pj at the second stage, equilibrium prices are given by the equations (2.5) and (2.6) and profits become : √ N N (u(qi ) + u(qj ))( ∆ − u(qi )) √ Πi (qi ) = u(qi ) − F (qi ) ; Πj (qj ) = − F (qj ) θ θ(u(qi ) + ∆) with ∆ = u(qi )2 + 4u(qi )(u(qi ) + u(qj )) 5 Peters

and Waterman (1982, p 171) write of their experiences in ordering equipment for the Navy: “We would go to

almost any ends, stretching the procurement regulations to the limit, to specify the always more expensive Cat equipment. We had to, for we knew our field commanders would string us up if we didn’t find a way to get them Cat.” 6 see Laurent, 2007b

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The study of the index choices at the first stage leads to the following proposition : Proposition 4 When qi ≥ qj for i, j ∈ {1, 2} and i 6= j, Πi is quasi-concave and the maximum verifies u0 (qi )/θ = F 0 (qi )/N . The first derivative of Πj is zero if : u0 (qj ) 4ui (ui + u(qj )) F 0 (qj ) √ √ = θ N ∆( ∆ + ui ) The setting S1 guarantees the quasi-concavity of Πj and the existence of a global perfect Nash equilibrium. Proof : presented in Appendix 8.2. Thus, concavity of utility and convexity of fixed costs are necessary but not sufficient conditions to prove the existence of the equilibrium. The attributes index chosen by firm i equates the marginal utility weighted by θ and the marginal fixed cost. The index selected by firm j is lower : differentiation is both horizontal and vertical. The two indices decrease with θ and increase with N , the number of consumers, as in the logit model. Indeed, the marginal investment in quality is more profitable the more consumers they are (Anderson, de Palma et Thisse 1992, p 245). Under S1 and when θ = N = 1, equilibrium indices are qi = 0.5 and qj = 0.317954. Moreover, firm j realizes a higher profit than its rival7 . Here again, such a property is not verified in pure vertical differentiation, the high-quality firm making the highest profit whatever the endogenous fixed cost function used (Lehman-Grube, 1997).

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Equilibrium and socially optimal differentiation The previous section proved the existence of a perfect Nash equilibrium in which firms choose their

attributes indexes and then compete in price. In this section, we study if these choices are beneficial for the consumers and optimal in term of social welfare.

5.1

Product differentiation and consumer surplus

The determination of consumer surplus in probabilistic discrete choice models is not obvious because consumers do not exhibit a perfect rationality. In the multinomial logit model, Anderson et al. (1992, p 202) propose to define the consumer surplus as the sum of the individual expected utilities of consumption. We adopt here a similar logic : consumer surplus is defined by the sum of option’s utilities weighted by these option’s choice probabilities. But such a formula requires to be able to define a global utility for each option, which is less “natural” in the DBA duopoly than in the logit oligopoly. Indeed, the utility provided by shared product attributes is not taken into account in the DBA model but should be integrated in a global utility function. Thus, we suppose a form of disconnection between consumers’ choices and their evaluation of these choices, between the descriptive part of the model and its normative part. When people choose among options, the cognitive cost of the decision plays a role. But in the evaluation of 7 This

property seems true for many settings although a general conclusion can not be drawn with attributes-dependent

fixed costs

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their decisions, consumers only consider the consistence between the decision and their preferences (the “quality” of the decision). The global utility of a product depends on its attributes, which are specific or shared with some other goods, and of its price. As previously defined, the specific attributes ki of a product i provide the consumers the utility ui = u(qi ). Similarly, we assume that the attributes belonging to all the products in the choice set, noted k0 , provide the consumers the utility u0 = u(q0 ). These utilities are weighted by the coefficient θ measuring the relative importance of these non-price attributes in comparaison with the price of the product (the higher θ is, the less non-price attributes are essential). The net utility of a product i is given by : Ui =

u0 + ui − pi θ

To simplify the problem, we suppose that the N consumers following the random decision rule have P exactly the same preferences on the attributes : their surplus can be expressed by SC = i∈A N Pi Ui . At the price equilibrium verifying p1 ≥ p2 (product 1 is the most appreciated), consumer surplus becomes :

SC

! √ √   u0 u1 − ∆ N ( ∆ − u1 ) u0 − u1 √ + − c1 + − c1 θ 2θ θ u1 + ∆ √ ! 2u1 (u1 − ∆) √ u0 − θc1 + u1 + ∆

=

2N u1 √ u1 + ∆

=

N θ

(5.1)

with again ∆ = u21 + 4u1 x and x = u1 + u2 + θ(c1 − c2 ). First, note that consumer surplus strictly increases with the utility of the common attributes u0 and decreases with the importance of the price attributes θ : the higher the disutility of a monetary amount loss is, the lower the surplus obtained by the consumption is. Second, when consumers’ preferences are inobservable, the effect of product differentiation on consumer surplus is generally ambiguous. On the one hand, differentiation allows to realize a better match with consumers’ tastes. On the other hand, differentiation increases firms’ market powers, which relaxes price competition at the detrimental of the consumers. When u1 and u2 vary, the equilibrium consumer surplus in the DBA duopoly is affected in the following way : −4N (x − u1 ) ∂SC −8N u31 ∂SC √ √ = √ ; = √ 0 = ∂u1 θ ∆( ∆ + u1 )2 √ ∂W 4N u1 [u1 (u2 − u1 ) + (u1 + u2 ) ∆] √ √ = >0 ∂u2 θ ∆(u1 + ∆)2 The highest possible values of u1 and u2 are socially optimal. As shown in section 4.1, these values are precisely chosen by firms 1 and 2 at the equilibrium, which leads to the following proposition : Proposition 5 When costs are symmetric and exogenous with the attributes, equilibrium attributes indices express a maximum differentiation are socially optimal. This result is equivalent to that obtained by Anderson et al. (1992) in the logit oligopoly : when utilities increase, firms’ profit gain overcomes consumer surplus loss.

5.3

Social optimum with attributes-dependent unit costs

We use here the same utility and costs functions as in section 4.2. The index q1W and q2W maximizing the social welfare can be determined by the following equations :

N ∂W = ∂q1 θ

! √ 4u1 [x ∆(u0 (q1 ) − θc0 (q1 )) + (x − 2u1 )(u1 (u0 (q1 ) + θc0 (q1 )) − u0 (q1 )x)] √ √ u (q1 ) − θc (q1 ) − ∆(u1 + ∆)2 0

0

√ ∂W 4N u1 (u0 (q2 ) − θc0 (q2 ))(u1 (x − 2u1 ) + x ∆) √ √ = ∂q2 θ ∆(u1 + ∆)2 The derivative with q2 is zero when u0 (q2 ) = θc0 (q2 ) and welfare function is quasi-concave : √ ∂ 2 W 4N u1 (u00 (q2 ) − θc00 (q2 ))(u1 (x − 2u1 ) + x ∆) √ √ = ≤0 ∂q22 ∂W =0 θ ∆(u1 + ∆)2 ∂q2

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The socially optimal index q2W verifies the equality between the weighted marginal utility of the attributes and the marginal unit cost of firm 2. As previously seen, this level is also identical to the equilibrium index q2∗ . This result also implies that the derivative with q1 is zero for u0 (q1 ) = θc0 (q1 ). Indeed, when q1 = q2 , we have : ∂W N = ∂q1 θ



−4u1 u0 (q1 )(x − 2u1 )2 √ √ ∆(u1 + ∆)2



and thus x − 2u1 = u2 − u1 + c1 − c2 = 0, utility and costs functions being identical between firms. We verify that the welfare function is quasi-concave : ∂ 2 W N 2u1 (u00 (q1 ) − c00 (q1 )) √ = ≤0 2 ∂q1 ∂W =0 θ u1 + ∆ ∂q1

Thus, the equilibrium index chosen by firm 1 is too high compared to the socially optimal threshold. These conclusions are resumed here : Proposition 6 When unit costs depend on attributes, the socially optimal attributes indices of firms i and j are given by qiW = qjW = q and u0 (q) = θc0 (q). At the perfect equilibrium verifying pi ≥ pj , the index chosen by j also maximizes the welfare whereas the index selected by i is too high compared to the social optimum. These optimal indices are equivalent to the optimal qualities computed in the logit oligopoly by Anderson et al. (1992, p 242). However, equilibrium qualities are socially optimal in the logit whereas it is not true in the DBA model, the index chosen by firm 1 being too high. At the social optimum, differentiation is purely horizontal whereas the equilibrium exhibits an additional vertical differentiation : in other words, there is an excess of vertical differentiation at the equilibrium9 . Indeed, vertical differentiation is the main origin of the high price level : the high-index firm does not carry attention to the additional cost implied by an increase in its index, as prices are positively correlated due to the “reference price” effect. This global increase of prices reduces drastically consumer surplus. For instance, the quality obsession of Caterpillar induced a general price increase on the earth-moving equipment market, diminishing therefore consumer surplus.

5.4

Social optimum with attributes-dependent fixed costs

The setting S1 (linear utility and quadratic fixed cost) presented in section 4.3 is used here to carry out a welfare analysis of the perfect equilibrium (with again c1 = c2 = c). The first and second derivatives of the welfare function with q1 and q2 are : N ∂W = ∂q1 θ ∂2W N = ∂q12 θ 9 If

! √ 2q12 ( ∆ + 9q2 ) √ √ − 2q1 ∆( ∆ + q1 )2

! √ √ √ 2q12 q2 [4( ∆ + 9q2 ) ∆ − 9(5q1 + 2q2 )( ∆ + q1 )] √ √ −2 ∆( ∆ + q1 )3

a partial heterogeneity in consumers’ preferences is introduced, the socially optimal differentiation can be horizontal

and slightly vertical : a little degree of vertical differentiation provides a better match with heterogenous tastes for attributes. However, the result of excessive vertical differentiation is not affected.

13

∂W N = ∂q2 θ N ∂2W = ∂q22 θ

! √ 4q1 ( ∆(q1 + q2 ) − q1 (q1 − q2 ) √ √ − 2q2 ∆( ∆ + q1 )2

! √ √ 8q13 [( ∆ + q1 )(5q1 + 4q2 ) + (q1 − q2 )(3 ∆ + u1 )] √ √ −2 ∆( ∆ + q1 )3

with ∆ = q12 + 4q1 (q1 + q2 ). Under S1, there exists a vector of indices (q1W , q2W ) such that the first order conditions are verified. For these levels, the welfare function is quasi-concave in q2 inside the interval [0, q1 ] and in q1 inside the interval [q2 , +∞]. The optimum is defined by the following proposition : Proposition 7 When utility is linear and fixed costs quadratic with the attributes, the socially optimal attributes indices of firms i and j verify qiW = qjW = 1/4. At the perfect Nash equilibrium, attributes indices chosen by i and j are too high compared to the social optimum. Indeed, under S1, equilibrium indices are qi = 0.5 > 0.25 and qj = 0.317954 > 0.25. Here again, the social optimum exhibits a purely horizontal differentiation. Introducing a vertical product differentiation in a horizontally differentiated market increases firms’ power and diminishes welfare. But, contrary to the case of endogenous unit costs, the attributes index chosen by firm j is also too high compared to the socially optimal level : there is an excess of vertical and horizontal differentiation in this framework.

6

Conclusion When consumers follow a random decision rule focused on products’ attributes, we study a two-

stage game in which firms choose their attributes index and then compete in prices. The existence of a perfect Nash equilibrium is established with exogenous costs and with attributes-dependent unit and fixed costs. When costs are exogenous with the attributes, firms choose the highest indices, as in the logit oligopoly. When unit or fixed costs are endogenous, firms select different attributes indices and product differentiation is both horizontal and vertical at the equilibrium. Moreover, the low-index firm realizes the highest profit at the equilibrium : a weak attributes index keeps costs at a low level while allowing the choice of a relatively high market price because of the horizontal dimension of product differentiation. This result contrasts with the high quality advantage of vertical differentiation models. In term of social welfare, the highest indices chosen under exogenous costs are also optimal, as in the logit oligopoly. When costs are attributes-dependent, a pure horizontal differentiation is socially optimal. At the equilibrium, the vertical differentiation is excessive as it strengthens firm’s market power : prices increase and consumer surplus is reduced. In the particular case of endogenous fixed costs, horizontal differentiation is also excessive at the equilibrium. These results are obtained under the (rather classical) assumption that firms can not imitate the competing products. If this assumption is not expected to affect the low-index advantage, the excess of differentiation may be reduced if firms behave strategically when a threat of imitation exists. Consequently, introducing a third stage of product imitation in the game is a natural extension of the current work, realized in Laurent (2007b).

14

7

References

Anderson S.P., de Palma A. and Thisse J-F. (1992), Discrete choice theory of product differentiation, Cambridge, MIT Press. Aoki R. and Prusa T.J. (1997), “Sequential versus simultaneous choice with endogenous quality”, International Journal of Industrial Organization, 15, 103-121. Beath J., Katsoulacos Y. and Ulph D. (1987), “Sequential Product Innovation and Industry Evolution”, Economic Journal, 97, 32-43. Choi C.J. and Shin H.S. (1992), “A Comment on a Model of Vertical Product Differentiation”, Journal of Industrial Economics, 40, 229-231. Economides N. (1993), “Quality variations in the circular model of variety-differentiated products”, Regional Science and Urban Economics, 23, 235-257. Gabszewicz J.J. and Thisse J-F. (1979), “Price Competition, Quality and Income Disparities”, Journal of Economic Theory, 20, 340-59. Hotelling H. (1929), “Stability in Competition”, Economic Journal, 39, 41-57. Irmen A. and Thisse J-F. (1998), “Competition in multi-characteristics. spaces : Hotelling was almost right”, Journal of Economic Theory, 78, 76-102. Klemperer P. (1987), “Markets with Consumer Switching costs”, The Quarterly Journal of Economics, 102, 375-394. Laurent R.-A. (2007a), “A Duopoly with Differentiation By Attributes”, revised version of the working paper PSE 2006-17, “Differentiated duopoly with ’elimination by aspects” ’. Laurent R.-A. (2007b), “Innovation and imitation in the Duopoly with Differentiation by Attributes”, mimeo. Lazer W. (1957), “Price Determination in the Western Canadian Garment Industry”, The Journal of Industrial Economics, 5, 124-136. Lehmann-Grube U. (1997), "Strategic Choice of Quality When Quality is Costly: The Persistence of the High-Quality Advantage", RAND Journal of Economics, 28-2, p 372-384. Miller D. (1990), The Icarus Paradox, How exceptional Compagnies Bring About Their Own Downfall, Harper Business. Motta M. (1993), “Endogenous quality choice : Price vs. Quantity Competition”, The Journal of Industrial Economics, 41, 113-131. Mussa M. and Rosen S. (1978), “Monopoly and Product Quality”, Journal of Economic Theory, 18, 301-17. Nash B. (1937), “Product Development”, Journal of Marketing, 3, 254-262. Neven D., Thisse J.F. (1990), “On Quality and Variety in Competition”, in Gabszewicz J.J.,Richard J.F., Wosley L.A. eds., Decision Making: Games, Econometrics and Optimization, North Holland, Amsterdam, 175-199. Pepall L. M. and Richards D. J. (1994), “Innovation, Imitation, and Social Welfare”, Southern Economic Journal, 60, 673–684. Peters T. J. and Waterman R. H. (1982), In search of excellence: Lessons from America’s best-run companies, Harper & Row (New York).

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Ronnen U. (1991), “Minimum Quality Standards, Fixed Costs, and Competition”, The RAND Journal of economics, 22, 490-504. Rosenkrantz S. (1997), “Quality Improvements and the Incentive to Leapfrog”, International Journal of Industrial Organization, 15, 243-61. Rotondo J. (1986), “Price as an Aspect of Choice in EBA”, Marketing Science, 5, special issue on Consumer Choice Models, 391-402. Schelling T.D. (1960), The strategy of conflict, Harvard University Press, Cambridge Mass. Shaked A. and Sutton J. (1982), “Relaxing Price Competition through Product Differentiation”, Review of Economic Studies, 69, 3-13. Tirole J. (1988), The Theory of Industrial Organization, Cambridge, MIT Press. Tversky A. (1972a), “Elimination by aspects : A Theory of Choice.”, Psychological Review, 79, p. 281-299. Tversky A. (1972b), “Choice by Elimination”, Journal of mathematical psychology, 9, p. 341-367. Wauthy X. (1996), "Quality Choices in Models of Vertical Differentiation", The Journal of Industrial Economics, 44, 345-353.

8

Appendix : proofs on the attributes choice

8.1

Equilibrium with endogenous unit costs

The proof of proposition 3 is presented here. Suppose that firm 1 sells the most appreciated product (q1 ≥ q2 ). We identify the local equilibrium and show afterwards that the equilibrium is also global. Local maximum. When q1 ≥ q2 , equilibrium prices at the end of the second stage have the following expressions : √ u(q1 ) + ∆ u(q1 ) + u(q2 ) = c(q1 ) + ; p∗2 = c(q1 ) + 2θ θ 2 with ∆ = u(q1 ) + 4u(q1 )x and x = u(q1 ) + u(q2 ) + θ(c(q1 ) − c(q2 )). p∗1

For firm 1, first and second order conditions are given by : ∂Π1 N ∂ 2 Π1 N = u0 (q1 ) = 0 ; = u00 (q1 ) ≤ 0 2 ∂q1 θ ∂q1 ∂Π1 =0 θ ∂q1

As utility is concave, we have a maximum and firm 1 chooses the highest index q1c , verifying u0 (q1c ) = 0. For firm 2, the following derivative is obtained after simplifications: ∂Π2 4N u1 x(u0 (q2 ) − θc0 (q2 )) √ √ = ∂q2 θ ∆( ∆ + u1 )

(8.1)

It equates zero for a value q2c verifying u0 (q2c ) = θc0 (q2c ). At the equilibrium, the second order condition is : ∂ 2 Π2 4N u1 x(u00 (q2 ) − θc00 (q2 )) √ √ = ≤0 ∂q22 ∂Π2 =0 θ ∆( ∆ + u1 ) ∂q

(8.2)

2

Thus, Π2 is quasi-concave if u00 (q2 ) ≤ θc00 (q2 ) : this condition always holds under the assumptions of concave utility and convex unit cost. 16

Finally, attributes indices verify q1c > q2c which implies u(q1 ) ≥ u(q2 ) and c(q1 ) ≥ c(q2 ). Consequently, conditions (2.3) and (2.4) are respected and a price equilibrium verifying p1 ≥ p2 exists at the second stage. Global maximum. We prove here that firms have no interest in deviating from the local maximum previously computed, which constitutes a perfect subgame Nash equilibrium. Consider first the case of firm 1 : there is no attributes index q1cc such that q1cc ≤ q2c and verifying Π1 (q1c ) < Π1 (q1cc ). As u0 (q1c ) = 0, the reference profit is Πc1 (q1c ) = N uα /θ (see assumptions in section 3.1). √ √ c cc cc c c If q1cc exists, Πcc ∆)] with ∆ = (uc2 )2 + 4uc2 x and 1 is similar to Π2 : Π1 (q1 ) = [N x( ∆ − u2 )]/[θ(u2 + x = u(q1cc ) + uc2 + θ(cc2 − c(q1cc )). The first derivative looks like the equation (8.1) and, consequently, the optimal index is given by u0 (q1cc ) = θc0 (q1cc ) (symmetrically, the profit is quasi-concave, as in equation c (8.2)). As utility and cost functions are identical between firms, we have q1cc = q2c and thus ccc 1 = c2 c and ucc 1 = u2 . This utility level is now defined by uβ . When firm 1 deviates from the local maximum, cc c c it chooses the same attributes index as firm 2 and, for this value, Πcc 1 (q1 ) = N uβ /θ. But as q2 ≤ q1 ,

we have uβ ≤ uα and thus Πc1 ≥ Πcc 1 . Firm 1 can never improve its profit by deviating from the local maximum. Consider now the case of firm 2. We study if there exists an index q2cc belonging to the interval √ q2cc ≥ q1c and verifying Π2 (q2c ) < Π2 (q2cc ). The reference profit of firm 2 is given by Πc2 (q2c ) = [N x( ∆ − √ u(q1c ))]/[θ(u(q1c ) + ∆)]. If firm 2 deviates from the local maximum, the attributes index chosen can however not be strictly higher than that of firm 1 since q1c = Q1 = Q2 : firm 2 necessarily selects q2cc = q1c . cc c In this case, firm 2 realizes a profit Πcc 2 (q2 ) = N u(q1 )/θ. Comparison of profits leads to the condition :

c c c c c Πcc 2 ≤ Π2 ⇔ θc(q1 ) − u(q1 ) ≥ θc(q2 ) − u(q2 )

(8.3)

As utility is concave and unit cost convex, this inequality is always true. Indeed, the following property is verified : d(θc(q) − u(q)) > 0 ∀q ≥ q with q > 0 such that θc0 (q) = u0 (q) dq As q2c verifies u0 (q2c ) = θc0 (q2c ), we have q2 = q and (8.3) is always true when q1c > q2c . Thus, firm 2 has no interest in deviating from the local maximum. Finally, potential deviations are unprofitable and a price equilibrium exists at the second stage for the attributes indices chosen at the first stage : we have a perfect Nash equilibrium.

8.2

Equilibrium with endogenous fixed cost

We suppose again that firm 1 chooses the highest index (q1 ≥ q2 ) and prove the Proposition 4 in this Appendix. First, the local equilibrium in attributes is identified. Second, we show that a perfect Nash equilibrium exists when the setting S1 is used. At the last stage of the game, profits are given by equations (2.7) and (2.8). We study the attributes choices of firm 1 : ∂Π1 N = u0 (q1 ) − F 0 (q1 ) ∂q1 θ 17

At the equilibrium, the index chosen verifies u0 (q1 )/θ = F 0 (q1 )/N and the second order condition is : N ∂ 2 Π1 = u00 (q1 ) − F 00 (q1 ) ≤ 0 2 ∂q1 ∂Π1 =0 θ ∂q1

As utility is concave and fixed cost convex, the profit is quasi-concave. Consider now the attributes choice of firm 2. After simplifications, the first derivative is given by : 4N u1 xu0 (q2 ) ∂Π2 = √ √ − F 0 (q2 ) ∂q2 θ ∆( ∆ + u1 )

(8.4)

At this extreme point, the index chosen by 2 verifies : 4u x F 0 (q2 ) u0 (q2 ) √ √1 = θ N ∆( ∆ + u1 ) The second order condition is : √ √ 4N u1 [u1 u02 (q2 )( ∆ + u1 + 2x) + xu00 (q2 )(u1 + ∆)(u1 + 4x)] ∂ 2 Π2 √ √ = − F 00 (q2 ) ≤ 0 ∂q22 ∂Π2 =0 θ ∆(u1 + 4x)(u1 + ∆)2 ∂q2

The assumptions of convex fixed costs and concave utilities are not sufficient to ensure that profit is quasi-concave in the general case : the fixed cost function must also be sufficiently convex (or the utility strongly concave). When profits are quasi-concave, the equilibrium indices verify q1 ≥ q2 . Indeed, cost and utility functions are identical between firms and the following inequality is respected : √

4u1 x 4u x √1 √