Do Firms Compete When Demand is Low? A Model of Spatial Differentiation

Chirco, Alessandra; Lambertini, Luca Working Paper Do Firms Compete When Demand is Low? A Model of Spatial Differentiation Quaderni - Working Paper DSE, No. 210 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Chirco, Alessandra; Lambertini, Luca (1994) : Do Firms Compete When Demand is Low? A Model of Spatial Differentiation, Quaderni - Working Paper DSE, No. 210, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/5116 This Version is available at: https://hdl.handle.net/10419/159053 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/ DO FIRMS COMPETE WHEN DEMAND IS LOW? A Model of Spatial Differentiation Alessandra Chirco* & Luca Lambertini*# *Dipartimento di Scienze Economiche Università degli Studi di Bologna Strada Maggiore 45 40125 Bologna Italy tel 39-51-6402600 fax 39-51-6402664 e-mail [email protected] #Linacre College Oxford OX1 3JA United Kingdom e-mail [email protected] JEL classification: L 13 Keywords: market demand, competitiveness, spatial differentiation Abstract In a spatial competition model, changes in firms’ competitive behaviour may occur when the hypothesis that individual gross surplus is positive in equilibrium is relaxed. We prove that there exists a region of the relevant parameter where firms’ behaviour mimics collusion, while in another range they find it optimal to isolate from each other and behave monopolistically. Acknowledgements We would like to thank Corrado Benassi, Guido Candela and Vincenzo Denicolò for insightful comments and suggestions. The usual disclaimer applies. Please send corresponce to the second author at the Italian address. 1. Introduction The horizontal differentiation model introduced by Hotelling (1929) has produced a wide stream of literature focusing on how firms startegically exploit the possibility of choosing their respective locations in the product space in order to soften price competition. Two major points are worth stressing. First, the attention paid to product differentiation anditsbearingsonequilibriumprofitshasleftmanyquestionsonthenatureofpricecompetition insuchsettingsvirtuallyunanswered.Second,horizontaldifferentiationhasgenerallybeendealt with under the assumptions of inelastic demand and full market coverage. This paperis devoted totheinvestigationof short-run pricebehaviourwhenan exogenous shockaffecting nominalmagnitudes bringsaboutacontractioninmarketdemand, whichinturn may induce firms to noncooperatively adopt price rules of monopolistic or quasi-cooperative flavour. Thus, the present analysis gives a temptative but suggestive answer in the positive to thequestion whether the intensity ofcompetition may be inverselyrelated to the levelof market demand, as it has been informally raised by Stiglitz (1984). 2. The model Consider a duopoly in which firms 1 and 2 sell a physically homogeneous good along a segment of unit length. Unit production costs are assumed to be constant and can be normalised to zero without any loss of generality. Consumers are uniformly distributed over the segment, withtotaldensity1.Eachconsumerbuysatmostoneunitofthegood,drawingfromconsumption a gross surplus sinvariant across consumers - and paying a full price pi+d2, where piis the mill price charged by firm iand dis the distance between the consumer and the patronized firm. Total demand is equal to 1, i.e., all consumers are served, if the indirect utility function is non-negative for all consumers. In such a case, as shown by D’Aspremont et al. (1979), the Nash symmetric equilibrium prices turn out to be while the equilibrium locations are the endpoints ofthe segment, 0 and 1. However, this solution clearly requires Provided that location can be regarded as a long run choice, one might argue how firms’ pricing behaviour may be affected by an exogenous shock reducing sbelow the above threshold level. Given that U=s−pi−d2(1 pi=1, s≥5/4. 1 scan be interpreted as the reciprocal of the marginal utility of income (or money), this amounts to investigating, within the relevant range of parameters, the bearings of a shock affecting nominalincomeontheresultsyieldedbythisclassofmodels(atleastinthissimplifiedversion).1 Weshall proceed asfollows. We shall studyfirms’ optimal behaviour for Two basic settings emerge. For very low values of s, each firm’s demand is independent of the other’s, so that the market does not allow for any strategic interaction, and monopolistic pricing is necessarily observed. For higher values of s, demands overlap, so that some scope arises for strategic interaction - although this is not necessarily exploited - and firms’ behaviour must be studied by deriving their reaction functions in the price space. 2.1. Isolated markets Since for the consumer located in 1/2 transportation costs amount to 1/4, it is clear that for s<1/4 the profit accruing to each firm is independent of the rival’s behaviour. Thus, both firms behave monopolistically, maximizing a profit function defined as follows: where defines firm i’s demand. This yields: 2.2. Overlapping demands If firms may potentially compete for the consumers located in an area at the center of the linear city, which widens as sincreases. To derive the firms’ reaction functions, we have to proceed in two steps, defining (a) the profit function of firm ifor any price charged by firm j, ; (b) the optimal pricing rule for firm igiven the price charged by firm j. (a) Derivation of the profit function. First, notice that is actually independent of pjfor where for s<1, while for Thus, for any s, s∈]0,5/4[. πi M=pi√ s−pi;i=1,2,(2 √ s−pi pi*=2 3s;πi *=2 3s√s 3.(3 s∈[1/4,5/4[, π(pi|pj) π(pi|pj) ˆpj=2√s−1πi=π i M=pi√ s−pi,pj>ˆpj,ˆpj=ss≥1. 1.Providedthatlocationsareeitherexogenousorfixedintheshortrun,thepresentanalysis could be easily extended to both the case of nfirms and that of the circular city described by Salop (1979). 2 if If instead then there exists a price such that the net surplus of the consumer who is indifferent between the two firms is nil: If firm isets a price firm igains monopoly profits as defined by (2); otherwise, if she obtains the following duopolistic profits: with intersecting at from below.2Therefore, we have established the following: PROPOSITION 1: assume then, for any profits of firm iare defined as (b) Derivation of the reaction functions. The reaction functionof firm iis defined as the optimal choiceof pigiven pj. Taking into account the profit function referred to in Proposition 1, three possibilities arise. Denoting with and the prices maximizing, respectively, and whichareconcaveandsingle-peaked,thentheoptimalpriceis:(i) (ii) (iii) Consider case (i). We have if implying pj≥ˆpj. pj<ˆpj,˜pi(pj) ˜pi=pj+2√ s−pj−1. (4 pi≥˜pi(pj), pi<˜pi(pj), πi D=pi 2(pj−pi+1), (5 πi Dπi M˜pi(pj) s≥1/4; pj<ˆpj, πi(pi|pj)=min(πi M;πi D). πi(pi|pj) piMpiDπi M πi D,piMif ˜pi(pj)≤piM; piDif ˜pi(pj)>piM&piD<˜pi(pj);˜pi(pj)if piD>˜pi(pj)>piM. ˜pi(pj)≤piM pj+2√ s−pj−1≤2 3s,(6 2. Notice that for which coincide with when s<1. If as for then for all and for all firm i’s profits are given by pj<2√s−1,˜pi>0ˆpjˆpj=s, pj∈[2√s−1,s], πi M.s≥1,˜pi<0,pi>0 3 Then, for all firm i’s reaction function is For all we have to check whether (ii) or (iii) holds. Clearly, (ii) holds for all values of satisfying while for all other values of case (iii) holds. Solving (9), we get notice that For finite values of s, we may now sum up firms’ pricing behaviour in table 1. INSERT TABLE 1 HERE Notice that the strategic complementarity in prices usually observed in product differentiation models(seeBeath andKatsoulacos,1991, p.22),ariseshere only fors>9/16.However, strategic interactiondoes not necessarily yields a Nashequilibrium in prices where the latterare strategic complements. Indeed, therelation between equilibrium prices and gross surplusis described by the following: PROPOSITION 2: (i) for p*=2s/3; (ii) for p*=s-1/4; (iii) for p*=1. pj≥pj=2 3s−1+2√s 3.(7 pj≥pj, pi*(pj)=2 3s.(8 pj<pj, pj<pj piD(pj)< ˜pi(pj)⇔pj+1 2<pj+2√ s−pj−1,(9 pj<pj, pj<pj0=−5+4√ s+1; (10 pj0>0∀s>9/16. s≤3/4,s∈]3/4,5/4[, s≥5/4, 4 PROOF. In order to prove the above Proposition, it suffices to notice that if and if Q.E.D. A few remarks are now in order. For although demands overlap and thus in principle competition is possible, firms find it optimal not to compete; they behave monopolisticallyandthemarketisnotfullycovered.Furthermore,for asymmetric Nash equilibrium emerges in an area where reaction functions are downward sloping and thus strategic substitutability is observed. In such an equilibrium, the net surplus of the indifferent consumer located at the middle of the segment is nil. Finally, it is most noteworthy that in this case strategic interaction leads firms to adopt a pricing behaviour which mimics collusion. 3. Conclusions The main finding of this paper is that firms’ propension to compete is inversely related to the level of demand, here approximated by the individual gross surplus from purchase. There aredemand configurations which might support competition with positive profits, andyet firms find it optimal to set prices in a monopolistic or at least quasi-cooperative way. This is in line with that large body of the literature aimed at showing a procyclical pattern of competitiveness and thus a countercyclical pattern of the real price, e.g. Rotemberg and Woodford (1991). Two extensions,namely,theendogenisationofcostsandtheexplicitmodellingofindividualdemand price responsiveness, as suggested by Stiglitz (1984), are topics left to future research. pj≤2s/3s≤3/4; pj0≥1s≥5/4. s∈]1/4,3/4], s∈[3/4,5/4[, 5 References Beath, J. and Y. Katsoulacos, 1991, The Economic Theory of Product Differentiation, Cambridge, Cambridge University Press. D’Aspremont, C., J.J. Gabszewicz and J.-F. Thisse, 1979, On Hotelling’s ‘Stability in Competition’, Econometrica, 47, 1045-50. Hotelling, H., 1929, Stability in Competition, Economic Journal, 39, 41-57. Rotemberg, J. and M. Woodford, 1991, Markups and the Business Cycle, in O. Blanchard and S. Fischer (eds), NBER Macroeconomics Annual, Cambridge, MA, MIT Press, 63-129. Salop,S., 1979, MonopolisticCompetition with OutsideGoods, Bell Journalof Economics,10, 141-56. Stiglitz,J., 1984,Price Rigidities andMarketStructure, American Economic Review, 74(P&P), 350-5. 6 s<1/4 Table 1. Optimal pricing behaviour pi=2 3s s∈[1/4,9/16[pi=pj+2√ s−pj−1if pj<pj pi=2 3sifp j≥pj pi=pj+1 2if pj<pj0 s≥9/16 pi=˜pi(pj)=pj+2√ s−pj−1 if pj∈[pj0,pj[ pi=2 3sifp j≥pj 7