Quality Competition and Social Welfare in Markets with Partial Coverage: New Results

Transcription

1 Feruary, 00 Quality Cometition and Social Welfare in Market with Partial Coverage: New Reult Gregory S. Amacher 1 Erkki Kokela Markku Ollikainen 1 Deartment of Foretry, Virginia Commonwealth Univerity, and College of Natural Reource, Virginia Tech, Blackurg, VA 061 USA, Deartment of Economic, P.O. Box 5, FIN-0001 Univerity of elinki, Finland Deartment of Economic and Management, P.O. Box 7, FIN-0001 Univerity of elinki, Finland. gamacher@vt.edu, erkki.kokela@helinki.fi, markku.ollikainen@helinki.fi. We would like to thank Profeor uca amertini for ueful dicuion, and Ph.D. Chiara omardini-riiinen for comutational hel and comment. Kokela thank Dea rtment of Foretry of Virginia Tech and Reearch Deartment of the Bank of Finland for their hoitality. Thi aer i a art of the roject Studie in Environment and Reource Economic financed y the Academy of Finland.

2 Quality Cometition and Social Welfare in Market with Partial Coverage: New Reult Atract We ue a vertical roduct differentiation model under artial market coverage to tudy the ocial welfare otimum and duooly equilirium when convex cot of quality roviion are either fixed or variale in term of roduction. We how that, under fixed cot, at the ocial welfare otimum only one quality variant of the good i rovided, while oth variant are otimal under variale cot. In the duooly equilirium the quality read i too wide under variale cot, ut too narrow under fixed cot, relative to the ocial otimum. Finally, in oth the fixed and variale cot cae, average quality rovided y the duooly equilirium i too low from the erective of a ocial welfare maximizer. Keyword: Product Differentiation, Partial Market Coverage, Social Welfare JE Claification: 1, D60

3 1. Introduction A well-etalihed reult in vertical roduct differentiation model i that a duooly coniting of high and low quality firm lead to roduct quality dierion that i too high and average level of quality that are too low, relative to the ocially otimal outcome ee Crame and ollander Thi reult ha een etalihed under the aumtion that market are fully covered, i.e., that all conumer urchae oitive quantitie of the good in quetion. A conequence of full market coverage i that, even though duooly qualitie differ from the ocially otimal one, the quantitie roduced y the firm are alway equivalent. The cae of a artially covered duooly i more aealing, in that it allow for ome conumer who do not urchae from either firm ut could otentially enter the market. In thi cae, if the duooly and ocially otimal outcome differ, then not only the qualitie ut alo the quantitie differ. In the voluminou literature on artial market coverage, the ocial welfare outcome ha mainly remained an oen iue. Our work fill an imortant ga in thi literature. We characterize the roertie of ocially otimal qualitie and olve for the divergence etween duooly and ocial outcome when a market i artially covered. Unlike other work, we analyze and comare oth cae of fixed and variale cot of roduction. 1 For variale cot we characterize the ocial otimum in the ame way that Crame and ollander 1995 do for the fully covered market cae. Ecchia et al. 00 have argued that, under fixed cot, it i otimal to rovide jut one quality level. owever, they do not tudy the rolem of ocially otimal rice etting, nor do they how how the fixed cot cae may differ from the variale cot cae. Motta 199 allow for oth variale and fixed cot with artial market coverage ut doe not exlicitly olve for the ocially otimal outcome. Intead he ue numerical illutration to comare conumer urlue in the different equiliria. ike mot exiting roduct differentiation model, we aume conumer derive utility from oerved quality of roduct. Indeed, we retain thi aumtion ecaue we

4 eek to rovide cloure on the modeling of one cla of roduct differentiation model under the aumtion of artial market coverage. Alo like other work, our duooly outcome i olved a the ugame erfect equilirium of a two-tage game. Firm maximize rofit y firt cometing in qualitie and then cometing in rice. Our characterization of the ocially otimal outcome rovide everal new finding. Unlike in the duooly equilirium, it i ocially otimal in the fixed cot cae to rovide only the high quality variant of a good, while the rofit maximizing duooly rovide two variant. Moreover, roviion of the ocially otimal quality level i higher than the high quality variant rovided y the market and, therefore, average quality rovided y the market i lower than the ocially otimal average quality. The ocial lanner in thi cae i free to charge either a zero or oitive rice, and to ervice either ome or all of the conumer in the market. Under the aumtion of variale cot the high quality firm in a duooly will have higher rofit ut lower market hare than the low quality firm. A a new reult we how that the read of roduct quality oerved under a rofit maximizing duooly i too high relative to the ocially otimal outcome. At the ocially otimal outcome, oth firm roduce the ame amount, and total outut i greater than with the duooly outcome. At the ocial otimum the major difference etween fixed and variale cot cae i that, under fixed cot, only one roduct variant i rovided, while oth variant are otimal under variale cot. In duooly equilirium the quality read i too wide under variale cot, while it i too narrow under fixed cot. Average roduct quality in oth cae i too low comared to the ocially otimal equilirium. We roceed a follow. Section reent a aic duooly model and the rofit maximizing olution with artial market coverage for oth fixed and variale cot of roduction. In Section we comare the ocially otimal and rofit-maximizing qualitie. Finally, we rovide a rief concluion. 1 Under the aumtion of fixed cot, Ronnen 1991 conider minimum quality tandard without analyzing the ocially otimal quality roviion. amertini 1996, in turn, conider the variale cot cae ut doe not examine the ocially otimal outcome.

5 5. A Duooly Model of Vertical Product Differentiation with Partial Market Coverage Under an aumtion of artial market coverage, each conumer i tyically aumed to urchae either one unit of the good or nothing. et a conumer have a utility function u ee Tirole 1988, 96-97, 96-98, u k k, 1 where k and k are the quality and rice of the kth good. In 1, rereent the conumer tate arameter, o that the conumer derive a urlu equal to k k from a good of quality k and rice k. Aume there are two oile qualitie of good roduced y two tye of firm, k high quality and k low quality. A tandard aumtion i that the conumer tate arameter are uniformally ditriuted over qualitie on a definite interval, [, ] ee e.g. Motta 199, or Cremer and Thie We aume that the high and low quality firm have quadratic and convex cot function for roviding quality, 1 ck k k for k,. Becaue conumer can urchae either one unit or nothing, the conumer who i indifferent etween high and low quality good ha a threhold tate arameter defined y ˆ. Under artial market coverage, ome conumer do not enter the market. More ecifically, the lowet marginal willingne to ay value can e defined for the conumer who i indifferent etween uying and not uying the good, i.e., Throughout the aer, derivative of function with one argument will e denoted y rime, while artial derivative will e denoted y ucrit of function with many argument.

6 6 c. Recalling the uniform ditriution of conumer tye, the demand for high and low quality roduct then ecome q ˆ and q c ˆ, where q and q are the numer of conumer urchaing from the low and high quality firm, reectively. Baed on the aove aumtion, we will focu on cae where the cot of roviding quality are either fixed or variale with reect to outut. The aumtion of fixed cot ha een widely alied in the literature. Kuhn 000 recently argued that the variale cot cae might e more aealing than the fixed cot cae, ecaue it avoid an imlauile feature of fixed cot. Thi i that the high quality firm ha oth higher rofit and a larger market hare in equilirium. In conformity with oervation from ractice, our variale cot cae reult in an equilirium where the rofit of the high quality firm are higher that thoe of the low quality firm. owever, the market hare of the high quality firm i lower than the low quality firm..1 Price and Quality Game: Fixed cot The analyi of duooly cometition under fixed cot of roduction wa originally rovided y Ronnen In what follow we develo the feature of hi model very riefly. When the cot of quality roviion i fixed in term of qua ntity roduced, then given the demand quality firm are: qk and the cot function in, the rofit function of the high and low π q c, for k,. k k k k k There are then two tage of the duooly game: quality roviion tage 1, and rice cometition conditional on quality rovided tage. Firm move imultaneouly in each tage. We can olve for the ugame erfect equilirium of thi game. Thi equilirium relie, a uual, on commitment y firm in term of quality. In the econd Thi reult wa originally dicovered y ehmann-grue e alo howed that it hold irreective of whether the firm chooe their qualitie imultaneouly or equentially.

7 7 tage, firm chooe rice given the cot of quality roduction. From the firt -order condition, 0 π and 0 π, we can olve for the otimal rice and their difference a follow, ; ; * * Thu, duooly rice deend on the quality difference and the uer ound of the conumer tate ditriution. The lower ound of the tate ditriution doe not matter here, ecaue in artially covered market the lowet critical value of marginal willingne to ay i endogenou. Inerting the aove rice into the reective rofit function yield the indirect rofit function for each firm choice of quality, 1 π ; 1 π. 5 Differentiating equation in 5 with reect to qualitie give, 0 8 π 6a 0 π 6 Solving thee firt -order condition for high and low quality and their difference with Mathematica yield, ; ; In fact, amertini 1996 ha hown that the imultaneou move game i the only ure trategy equilirium oile for a artial market coverage model with variale cot of roducing quality.

8 8 Thu, the equilirium duooly qualitie and the quality difference etween firm deend oitively on the quare of the uer ound of tate ditriution, the marginal cot arameter of quality roviion., and negatively on Uing thee otimal qualitie, we can now olve for the rice and demand of oth quality variant a a function of exogenou arameter: ; q ; q The overall demand, which indicate the reulting coverage in the market, i therefore given y q q If we now normalize 1 and 0, then we can conclude that aout 79% of conumer enter the market and uy one of the two quality variant. Becaue the high quality firm charge a higher rice and face a larger demand, it ha higher rofit and greater market hare than the low quality firm. Thi can alo e een ; from the rofit olution for high and low quality firm, π and π. A we hall ee, thi reult mut e modified for the cae of variale cot of roduction.. Price and Quality Game: Variale cot Next we aume the cot of roviding quality are variale in term of outut. Under thi aumtion, and given the demand for each firm are written, q k and the cot function in, the rofit function k [ k c k k ] q k π, for k,. 8 A efore, in the econd tage firm chooe rice given the cot of quality roduction. From the firt-order condition, π 0 and π 0, we can olve for otimal rice,

9 9 [ ] 1/ 9a [ ] 1/. 9 1/ * * 9c Again, duooly rice and their difference deend on quality difference and on the uer ound of the conumer tate ditriution. Sutituting thee otimal rice into the rofit function, we can exre indirect rofit in term of quality a, [ ] π 10a [ ] * π. 10 Otimal econd tage qualitie then follow from the firt-order condition, [ ] Ω π 11a [ ] Λ π, 11 where [ ] Ω and [ ] Λ. Given the comlexity of the firt-order condition, olving for the actual equilirium qualitie i a it laoriou. Without lo of generality we define d for ome 1 > d, where d indicate the degree of roduct differentiation etween firm

10 10 exreed in term of the quality read etween high and low quality firm. Note that thi aumtion doe not redetermine the reult reented later concerning difference etween ocially otimal and duooly outcome. It imly imlie that the high quality firm roduce higher quality than the low quality firm, which i alway the cae in thee model. Uing d and olving 11a - 11 with Mathematica, we otain the following equilirium qualitie and their difference, ; 0.987, Equilirium duooly qualitie and the degree of quality differentiation are oitive function of the uer ound of the quare of the tate ditriution,, and a negative function of the marginal cot arameter of quality roviion,. Thi reult i qualitatively imilar to thoe found in full market coverage model. Note alo that the quality difference i higher with variale cot comared to the fixed cot cae. Thi lat finding can e interreted a follow. Under fixed cot, the cot of roducing oth quality variant of the good in the econd tage are zero even though the cot of roviding quality differ, ut they are trictly oitive under variale cot. Thu under variale cot of roduction, quality cometition etween the firm i tighter ecaue the firm otain greater rent from differentiating comared to the fixed cot cae. Finally, uing the otimal qualitie aove, we can olve the reviou firt-order condition for equilirium rice and demand: 0.51 ; q ; q Interetingly, for our cae of variale cot of roduction, we find that the high quality firm ha higher rofit ut lower market hare than the low quality firm. The overall demand i.e., coverage in the market i given y q q Thu, under variale cot, overall market coverage i maller than in the cae of fixed cot. Thi i a natural reult ince roduction cot are now oitive and the quality read i wider, which erve to relax rice cometition etween firm ;

11 11 and allow the firm to charge higher rice. The firm indirect rofit function can now e olved to otain ~ π ; ~ π.. Socially Otimal veru Profit-Maximizing Quality Deciion Now we turn to the main art of our aer, i.e., the determination of the ocially otimal qualitie and their relationhi with the equilirium duooly qualitie under oth aumtion of fixed and variale cot of roduction. The ocially otimal level of quality are thoe that maximize a ocial welfare function, which i the um of urlu to conumer net of cot to roduce high and low quality good, SW ˆ 1 1 d d. 1 ˆ c.1 Fixed Cot and the Socially Otimal Qualitie We tart y analyzing the roertie of the firt-et olution under fixed cot. The ocial lanner imultaneouly chooe rice and qualitie to maximize 1. The lanner account for the critical tate arameter earating conumer of high and low quality variant, while keeing it oen whether it i ocially otimal to erve the whole market or not. Thu, the lanner ue the following critical value of the tate arameter ˆ ; c. 1

12 1 Differentiating firt the ocial welfare function 1 with reect to high and low quality rice give. 5 Uing thi in the ocial welfare function and differentiating it with reect to high and low qualitie yield, 1 1 SW 0, SW Solving for otimal high quality yield w. Note however that SW < 0, imlying that roduction of low quality variant i zero. Thu, it i ocially otimal to rovide jut one quality variant high quality, w, a ointed out y Ecchia et al. 00. Conider now the relationhi etween ocially otimal qualitie and duooly qualitie which ha een characterized in equation 7, * w < 0 16a * w > 0 16 a w a < 0, 16c where the ucrit a refer to average quality. Clearly, a duooly rovide too little high quality and too much low quality good. Thi imlie that rofit maximization reult in a quality dierion that i ocially u-otimal. Moreover, the average quality rovided y the market i too low from the ocial lanner erective. We ummarize thee finding in: 5 The firt-order condition for the rice of the high and low variant are SW 0, SW 0.

13 1 Prooition 1: Under fixed cot of roduction, the ocially otimal outcome involve roduction of only the high quality variant. Comared to the ocially otimal outcome, the rofit maximizing duooly rovide too little high quality and too much low quality. Uing the ocially otimal quality, we can alo olve for the ocially otimal rice. Inerting otimal qualitie into the firt-order condition for rice would imly that the otimal rice i zero, ecaue, when quality i given, the cot of commodity roduction i zero. There are everal oile way to olve for the otimal rice. Firt, the ocial lanner could offer the high quality commodity to conumer at a zero rice, given that invetment in quality i indeendent of the commodity rice level, and invetment a uch rereent a unk cot. The econd way i to aume that ociety charge a oitive rice uch that either ome uet or all of conumer urchae the commodity. The latter rice can e determined from the indifference relation etween uying and not uying for the conumer having the lowet reference for quality. By inerting the ocially otimal quality into thi indifference relationhi, we have w 0. Recall that we olved for the ocially otimal quality uch that. w Uing thi yield the following ocially otimal rice, w. Under thi rice, demand for the commodity i imly given y wf q, o that relative to the duooly equilirium, the ocially otimal olution with thi ricing trategy yield 1% higher demand for the good ee our earlier analyi of duooly in Section.1.. Variale Cot and the Socially Otimal Qualitie Next we comare the equilirium duooly olution with the ocially otimal one in the cae of variale cot of roduction. Unlike with fixed cot of roduction, the aumtion of variale roduction cot allow the ocial welfare maximizer to offer roduct at a nonzero marginal cot. Therefore, relacing duooly rice y the marginal cot of quality roviion in the critical tate arameter ˆ and c, we can define new threhold critical tate arameter for the uer and lower ound of the tate ditriution,

14 1 ˆ 1 C 1 ;. 0 Uing equation 0 and differentiating the ocial welfare function 1 with reect to the qualitie and then give the following firt-order condition, SW SW ˆ 0 ˆ, 1a C ˆ C 0 ˆ. 1 The ocially otimal qualitie can then e olved from 1a and 1 to otain, w, 5 w and 5 w w. 5 ike the quality difference in the rofit maximizing duooly cae, the ocially otimal quality difference deend oitively on the quare uer ound of the tate ditriution, and negatively on the marginal cot arameter of quality roviion. A for the relationhi etween ocially otimal qualitie in the duooly and ocial welfare maximization cae, we otain uing equation 1, * w > 0, a * w < 0. The magnitude of thee exreion deend on the ize of the quared uer ound of the tate ditriution, which indicate how many conumer can otentially e catured y differentiating roduct qualitie. Unlike in the cae of fixed cot, the rofitmaximizing duooly roduce too much high quality and too little low quality than

15 15 would the ocial lanner. Further, if we comare quality difference acro the outcome, we ee that rofit maximization give a quality dierion that i too wide, i.e., w w * * 0.10 < 0 5. c Thi imlie that, in order to relax rice cometition, firm will ehave in a manner that increae the read of quality dierion too much y maximizing rofit. Such ehavior decreae ocial welfare. We can ummarize thee finding in: Prooition : Under variale cot of roduction, the ocially otimal outcome involve roviion of oth high and low quality variant. Comared to the ocially otimal outcome, the rofit maximizing duooly rovide too much high quality and too little low quality. Next, we olve the demand for qualitie in the ocially otimal outcome. Uing 1 in 9a and 9 and accounting for the definition of demand yield q w w q 0.. ence, the high and low quality firm will have demand of equal ize at the ocial welfare otimum. The difference etween total demand in the ocially w w otimal and duooly outcome are given y: q q q q > 0, imlying that roduction of each variety and market coverage are oth too mall under the duooly. Intuitively, the duooly retrict roduction in order to charge rice higher than marginal roduction cot. Thi i trikingly different from the well-known reult derived in fully covered market, which tate that the ize of the economy roduction of quality i equal under duooly and ocially otimal outcome. Our new finding may have imortant olicy imlication for achieving efficient level of quality in market. Finally, it i intereting to comare the feature of ocially otimal quality roviion under fixed and variale cot. The major difference etween thee two cae i a follow: under fixed cot we howed that only one quality variant of the good i rovided. Both variant are otimal under variale cot, and we can how that roviion of the high quality variant i greater than in the cae of fixed roduction cot. In the duooly equilirium, we alo howed that the quality read i too wide under variale

16 16 cot, ut it i too narrow under fixed cot of roduction. In oth the fixed and variale cot cae, average quality i too low from the erective of the ocial welfare maximizer.. Concluion We have ued a vertical roduct differentiation model under the aumtion of artial market coverage to characterize the ocial welfare and rofit maximizing duooly outcome, in term of quality roviion and quantitie roduced. We conider thee under oth variale and fixed convex cot of roduction. Under an aumtion of fixed cot, the high quality firm ha higher rofit and greater market hare than the low quality firm. Unlike in the duooly equilirium, however, we demontrated, a ha een ointed out in Ecchia et al. 00, that it i ocially otimal to rovide the high quality commodity and et low quality roduction to zero at the ocial otimum. We alo demontrated that roduction of high quality at the ocial otimum i higher than that rovided y the market, and therefore the average quality rovided y the market i lower than at the ocial otimum. A roduction i cotle and invetment in quality i lum um, the lanner may e free to et a zero or oitive rice, and to erve either all or art of the market. Under an aumtion of variale cot, we alo find a new reult that the read of roduct quality in the rofit maximization duooly outcome i too high relative to the ocial welfare maximizing outcome. At the ocial welfare otimum, overall outut i greater than the outut roduced under the duooly. At the ocial otimum there are two major difference etween fixed cot and variale cot aumtion. Firt, under fixed cot only one variant of quality i rovided, while oth variant are otimal under variale cot. Second, roviion of high quality i greater under variale cot of roduction. Comaring ocial welfare maximizing and duooly outcome, we alo how that the roduct differentiation read i too wide under a variale cot aumtion, ut it i too narrow under fixed cot. Average quality in oth variale and fixed cot duooly cae i too low from a ocial lanner erective.

17 17 Reference Crame C. and A. ollander Duooly and Quality Standard. Euroean Economic Review 9, Cremer,. and J. Thie On the Taxation of Polluting Product in a Differentiated Indutry. Euroean Economic Review,, , Ecchia, G.,. amertini and C. Scara 00. On the Regulation of Vertically Differentiated Market thr ough Minimum Quality Standard. Rivita di Politica Economica 91, Kuhn, M ow Quality eaderhi in Vertically Differentiated Duooly. Univerity of York, Deartment of Economic, Working Paer Serie. amertini, Chooing Role in a Duooly for Endogenouly Differentiated Product. Autralian Economic Paer, 05-. ehmann-grue, U Strategic Choice of Quality When Quality i Cotly: the Peritence of the igh-quality Advantage, Rand Journal of Economic, 8, 7-8. Motta, M Endogenou Quality Choice: Price v. Quantity Cometition. Journal of Indutrial Economic, 1:11- Ronnen, U Minimum Quality Standard, Fixed Cot and Cometition. Rand Journal of Economic,, Tirole, J The Theory of Indutrial Organization. MIT Pre, Camridge, Ma.