Minimum Quality Standards and Collusion

Ecchia, Giulio; Lambertini, Luca Working Paper Minimum Quality Standards and Collusion Quaderni - Working Paper DSE, No. 235 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Ecchia, Giulio; Lambertini, Luca (1995) : Minimum Quality Standards and Collusion, Quaderni - Working Paper DSE, No. 235, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5078 This Version is available at: https://hdl.handle.net/10419/159078 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ 1 Minimum Quality Standards and Collusion Giulio Ecchia*† and Luca Lambertini*† *Department of Economics University of Bologna and †Linacre College University of Oxford October 30, 1995 Abstract We model the introduction of a minimum quality standard in a vertically differentiated duopoly. We extend the literature in determining the standard endogenously, showing that the maximisation of social welfare entails an increase in the surplus accruing to consumers served by the low quality firm and a decrease in the surplus of the remaining consumers. Then, we consider the effects of the standard on the stability of price collusion, proving that the standard makes it more difficult for firms to collude if consumers are sufficiently rich. J.e.l. classification numbers: L13, L50 Keywords: minimum quality standard, collusion, cartel stability. Corresponding Author: Luca Lambertini Department of Economics- University of Bologna Strada Maggiore, 45- 40125 Bologna- Italy Fax.:+39-51-6402664 E-mail: [email protected] 2 1. Introduction The regulation of an imperfectly competitive market with vertically differentiated products has been long debated in the literature since the seminal contributions of Spence (1975) and Sheshinski (1976). More recently, various papers have examined the consequences of the adoption of a minimum quality standard in oligopolistic markets where each firm supplies at least one variety (see, among others, Besanko et al.,1987 and 1988; Ronnen, 1991 and Crampes and Hollander, 1995). Both Ronnen (1991) and Crampes and Hollander (1995) consider a duopolistic market with single-product firms. The introduction of the minimum quality standard is then analysed as an exogenous constraint on the low quality firm to increase its quality level. The introduction of the standard gives a strategic advantage and higher profits to the low quality firm and reduces the degree of differentiation in the market. Although the standard exerts a positive welfare effect, its consequences, as far as consumers’ surplus is concerned, are quite different depending on the cost functions of firms. Ronnen adopts a model where the provision of quality entails only a fixed cost for firms: in this case, all consumers benefit from the standard because the price-cost margin is reduced. Crampes and Hollander show that the same result holds in a model where variable costs depend on both quality and quantity, provided that the standard reduces the degree of differentiation sufficiently. Otherwise, only the consumers served by the low quality firm benefit from the standard, although social welfare always increases1. In both papers, the adoption of the standard entails a reduction of product differentiation in the market and an asymmetric change in firms’ profits. This raises the question of whether firms could find it easier to collude in presence of a standard. There are several contributions dealing with price collusion in endogenously differentiated product settings (see, for instance, Chang, 1991; Ross, 1992 and Friedman and Thisse, 1993). They mainly consider horizontal differentiation models showing that the stability of collusion increases as the degree of product differentiation increases. In this case, the symmetry of the models implies that as the degree of substitutability decreases, the gain associated with deviation from the cartel agreement decreases as well. To our knowledge, the only paper investigating the issue of cartel stability in a vertically differentiation setting is that of Haeckner (1994). He analyses the stability of 1 For an analysis of the strategic role of quality standards in an international oligopolistic setting, see Motta and Thisse, 1993, and Boom, 1995. 3 collusion in a model of endogenous vertical differentiation using a framework à la Shaked and Sutton (1982). He shows that price collusion is more easily sustained the closer the products are in the quality range. This is due to the fact that, with vertical differentiation, the punishment phase introduces an asymmetry in cartel behaviour which is absent in the horizontal differentiation models à la Hotelling (see Chang, 1991). In this paper, we extend the previous literature in two directions. First, we endogenise explicitly the choice of the minimum quality standard which maximises social welfare. Second, we consider the possibility that the reduction of product differentiation due to the standard may trigger collusive behaviours between firms in order to safeguard profits. We model a duopoly market à la Mussa-Rosen (1978) where firms produce only one variety and production involves variable costs convex in quality and an exogenous fixed cost. We show that the endogenous choice of the standard increases social welfare so that the gains for the low quality firm and low income consumers outweigh the losses suffered by the high quality firm and high income consumers. Moreover, we prove that the adoption of the standard makes it more difficult for firms to collude in prices, if consumers are sufficiently rich. This shows that the minimum quality standard, in addition to welfare gains, provides also pro-competitive benefits in the long-run. The paper is organised as follows. Section two presents the basic model and describes the duopoly and monopoly equibria without the minimum quality standard. The endogenous choice of the minimum quality and its effects are presented in section three. Section four deals with price collusion. Section five contains some final remarks. 2. The model 2.1 Assumptions and notation We consider a market for vertically differentiated products. There is a continuum of consumers whose types are identified by θ, uniformly distributed in the interval [a,b], with a=b-1 and b≥5/42. The parameter θ represents consumers’ marginal willingness to pay for quality. Each consumer is assumed to buy one unit of the vertically differentiated good in order to maximise the following indirect utility function: U = θ q - p (1) 2 This condition ensures the existence of the duopoly equilibrium (see Cremer and Thisse (1994) on this point). 4 where q indicates the quality of the product and p is the market price at which that variety is supplied. In other terms, we assume full market coverage. This assumption can be justified by envisaging a paternalistic public agency which imposes firms to guarantee universal service. The production technology involves variable costs which are convex in quality and linear in quantity and a sunk cost k, related to the development of the product. The corresponding cost function is defined as: C = t q2 x + k , t>0 (2) where x denotes the output level. We also assume that k is sufficiently small to allow for strictly positive profits for firms active in the market. We suppose that only two qualities (which we indicate as high and low) are supplied in the market, with qH> qL. Hence θi, the index of the consumer indifferent between the two varieties, is defined as θi = (pH - pL)/(qH - qL) (3) So that market demand for the two varieties are xH = bθi (4a) xL = θi - a (4b) 2.2 Duopoly equilibrium We first take into account a duopoly market where each firm supplies a single quality. Competition takes place in two stages. In the first, firms choose qualities and in the second they compete in prices. The solution concept applied is the subgame perfect equilibrium by backward induction. The profit function of firm i is defined as3 πi = (pi - t qi2) xi ; i= H,L (5) 3 For simplicity of notation, henceforth we shall consider firms’ profits gross of fixed costs k. 5 In the second stage, firms choose prices to maximise profits, given the quality levels set in the first stage. The corresponding first order conditions for a maximum are4 δπH / δpH = (pL -2pH +bqH - bqL + t qH2 )/ (qH - qL) =0 (6a) δπL / δpL = (pH -2pL +qH - bqH - qL+bqL + t qL2 )/ (qH - qL) =0 (6b) Then, the resulting equilibrium prices are pHN= (qH + bqH - qL - bqL+ 2t qH2 + tqL2)/3 (7a) pLN = ( 2qH - bqH - 2qL - bqL+ t qH2 + 2tqL2)/3 (7b) where the superscript N stands for Nash equilibrium. Substituting the equilibrium prices in the profit functions of the firms we can obtain the following equilibrium quality levels for the two firms qHN= (4b + 1)/ 8t (8a) qLN = (4b -5)/8t (8b) Since the duopoly is symmetric, demands are both equal to 1/2. The corresponding profits amount to πiN = 3/16t ; i= H,L (see Cremer and Thisse (1994)). For future reference, it is also convenient to calculate the social welfare corresponding to the duopoly equilibrium. Assuming a benevolent social planner, her utilitarian social welfare function, defined as the sum of consumer and producer surplus, can be written as ()( ) W q tq d q tq d LL a i HH i b =−+− ∫∫ θθθ θ 22 (9) 4 In this case, as in the rest of the paper, we do not present the second order conditions for optima, which, however, can be shown to hold throughout. 6 The welfare level corresponding to the duopoly equilibrium is WN = (16b2 -16b -1)/64t. In addition, we can also calculate the consumer surplus for the two segments of the market, which are, respectively, CSLN=(16b2-24b-19)/(128t) and CSHN=(16b2-8b- 27)/(128t). 2.3 Monopoly Consider now the case of a private monopolist producing two varieties. The profits of the monopolist are defined as: πM = (pH - t qH2) xH + (pL - t qL2) xL (10) The monopolist chooses prices to maximise profits, under the assumption that all the consumers must be served. Thus, monopoly prices (see Mussa and Rosen (1978); Itoh, (1983)) are: pLM = (b-1) qL (11a) pHM= (bqH - 2qL + bqL + t qH2 - tqL2)/2 (11b) The qualities result qLM=(2b-3)/4t and qHM=(2b -1)/4t; the monopolist’s profits are equal to πM= (4b2 - 8b +5)/16t.The corresponding level of social welfare is WM = (4b2 -4b - 3)/16t. It can be quickly verified that the profit maximising monopolist distorts quality levels as compared to a social optimum5. In fact, the social planner would choose the quality levels in order to maximise (9). The resulting qualities would be qLSP = (4b - 3) /8t and qH SP= (4b -1) /8t and the corresponding level of social welfare would be equal to WSP = (16b2 -16b +5)/64t. 3. The introduction of a minimum quality standard Suppose that a public authority intervenes to regulate the behaviour of firms as far as their quality choice is concerned, by introducing a minimum quality standard6. The social 5 For a seminal discussion of this point, see Spence (1975). 6 In order to induce the low quality firm to adhere to the standard, we can assume that the authority introduces a penalty which makes convenient to the firm to set a quality level equal to the standard. 7 planner sets the standard in order to maximise social welfare, taking duopolistic price competition as given. Thus, the minimum quality standard chosen by the social planner will satisfy the first order condition δW / δqL = (14b -8 -5b2 -28 tqL -20 tqL + 5t2qH2 - 10t2 qH qL -15t2 qL2 )/ 18 =0 (12) while the high quality firms will simultaneously set its quality level in order to maximise profits, so that δπH / δqH = (1 + b -tqH - tqL ) (1 + b -3tqH + tqL) /9 =0 (13) The solution of the social planner’s maximisation problem is given by qLS = (20b - 34 + 9 √6)/40t (14) which corresponds to the minimum quality standard7, whereas the quality set by the high quality firm is qHS= (20b + 2 + 3√6)/40t (15) It is easy to see that the introduction of the minimum quality standard increases the levels of quality produced in the market, that is qLS is greater than qL and qHS is greater than qH. Since the minimum quality standard is higher than the lower quality previously offered in the market in absence of regulation, the high quality firm increases its quality level since qualities are strategic complements (see Crampes and Hollander 1995, Bulow et al.1985). The effect of the standard on the degree of differentiation in the market is summarised in the following fact. Fact 1: the setting of the minimum quality standard decreases the degree of differentiation in the market. 7 Crampes and Hollander (1995, p.76) state that setting a quality standard is equivalent to granting the low quality firm the ability to commit in quality. In fact, it can be shown that the standard is slighly lower than the quality chosen by a firms acting as a Stackelberg leader in the quality stage of the game. 8 Using (8a,b) and (14-15), it is immediate to verify that (qHS - qLS) is smaller than (qHN - qLN). The quantities produced by the low and the high quality firms are, respectively, xHS= (6√6 - 21)/ (5√6 - 30) ≅ 0.355051 (16a) xLS= (9+√6)/ (30 -5√6 ) ≅ 0.644949 (16b) The effect of the introduction of the minimum quality standard on the market shares of the two firms is summarised in the following fact. Fact 2: the presence of the minimum quality standard reduces the demand for the high quality good while increasing demand for the low quality good. The corresponding profits for the firms are8 πLS= (54√6 + 261)/ 500t √6 (1- √6) ≅ 0.22153/t (17a) πHS= (756√6 - 1971)/ 500t √6 (1- √6) ≅ 0.06714/t (17b) Fact 3: the introduction of the quality standard increases the profits of the low quality firm and decreases the profits of the high quality firm due to the reduction of the degree of differentiation between products in equilibrium. It is also worth noting that total industry profits after the introduction of the minimum quality standard are smaller than total profits in the duopoly equilibrium without standard9. 8 It is easy to verify that the optimal quality choice for the low quality firm under the constraint represented by the standard coincides indeed with the latter, since both the first and the second derivatives of its profit function with respect to qL are negative in correspondence of the minimum quality standard. 15 Spence,M.,(1975),”Monopoly, Quality, and Regulation”, Bell Journal of Economics, 6, 417-29. 16 Appendix A In this appendix we will show that the level of social welfare corresponding to the introduction of the minimum quality standard, denoted by WS, cannot be increased if the social planner acts as a Stackelberg leader with respect to the high quality firm in setting the minimum quality standard. In that case, the social planner would maximise (9) with respect to qL, taking into account the reaction function of the high quality firm, which corresponds to qH= qL/3 + (1+3b)/3t (A.1) The first order condition for a maximum would be defined as δW/δqL = (130b -73 -40b2 -260 tqL +160 btqL + 160 t2qL2)/ 162 =0 (A.2) and the resulting minimum quality standard would be qLSS = (40b - 65 + 3√145)/80t (A.3) where superscript SS indicates the minimum quality standard that maximises social welfare when the social planner mimics the behaviour of a Stackelberg leader. Using (A.1), we can calculate the corresponding level of social welfare: WSS= (47780√145-451900-1972800b+100800b√145+1972800b2-100800b2√145) t(7891200-403200√145) (A.4) It is then immediate to show that WS> WSS and that the difference (WSWSS) is convex in b. 17 Appendix B In this appendix, we show the consumer surplus levels in presence of the minimum quality standard CSLS= (-22944+6309√6-15720b+1920b√6+9600b2-600b2√6) 24000t(7-2√6) (B.1) CSHS= (-127098+49803√6-26280b+10080b√6+32400b2-11400b2√6) 24000t(7-2√6) (B.2) It is then immediate to verify that CSHSCSHN<0 for b∈]2.219,∞[, while CSLSCSLN>0 for all admissible values of b. 18 Appendix C In this appendix the critical discount factors in presence of minimum quality standards are displayed: αLS*=(542042982-217073952√6+415569600b-170985600b√6-137704800b2+ 58612800b2√6-70080000b3+26880000b3√6+17520000b4-6720000b4√6)/ (66499686-20330496√6+66513600b-32169600b√6+36832300b2- 10795200b2√6-70080000b3+26880000b3√6+17520000b4-6720000b4√6) (C.1) αHS*=(-255433770+101016720√6-29044800b+13852800b√6+84602400b2- 33806400b2√6-70080000b3+26880000b3√6+17520000b4-6720000b4√6)/ (-727613226+296469936√6-378100800b+152668800b√6+259130400b2- 103214400b2√6-70080000b3+26880000b3√6+17520000b4-6720000b4√6) (C.2)