American Economic Association A Generalized Model of Spatial Competition Author(s): Dennis R. Capozza and Robert Van Order Source: The American Economic Review, Vol. 68, No. 5 (Dec., 1978), pp. 896-908 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/1811321 . Accessed: 25/06/2014 00:17 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review. http://www.jstor.org This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions A GeneralizedModel of Spatial Competition By DENNIS R. CAPOZZA AND Recentresearchon the theoryof the firm in economic space (for example, see M. L. Greenhut,M. Hwang, and H. Ohta, 1975; Capozza and Kazem Attaran) has argued that in a spatial contextmany of the conclusions of classical price theory are reversed.Since mostfirmsoperate in a spatial ratherthan a nonspatial environment,the resulthas farreachingimplications.For example, Greenhut, Hwang, and Ohta have shown that some formsof (spatial) competitionwill resultin higherpricesthan those realizedunder(spatial) monopoly. Capozza and Attaranhave shown that spatial comparativestatics can be different fromnonspatial. In particular,cost increases (fixed, marginal, or transport) may cause lower marketequilibriumprices. Many of the spatial resultsappear to be sensitiveto the assumptionmade concerningcompetitionamong firms(see Greenhut, Hwang, and Ohta, 1975; the authors, 1977a,b). The purposeof thispaper is to investigatethe extentto which spatial price theory replicates the results of classical pricetheorywhenthereis freeentry.To this end we analyze in detailtwo models of location under freeentrywhichdifferprimarily in their assumption about price competition. An importantaspect of the analysis is the use of the second model as a basis for analyzing monopolistic competition. Indeed, we believe that monopolistic competitioncan bestbe viewedas a resultof the spatial dimension and that the results of monopolisticcompetitionshould be derived accordingly. The traditionalmodel of spatial competition follows the assumption in location theory(see, for example, Edwin Mills and Michael Lav; Martin Beckmann; Greenhut * University of SouthernCaliforniaand Department of Housing and Urban Development,respectively.We are indebtedto anonymous readers for helpfulcomments. ROBERT VAN ORDER* and Ohta; John Hartwick; Nicholas Stern among many others)that firmsset price as iftheyare monopolistswithintheirmarket area. In this paper a second model is developed froman assumption about pricing that is similar to that assumed by Harold Hotelling and followed by Arthur N. Smithiesin the analysisof spatial duopoly. Under the Hotelling assumptioneach firm conjecturesthat all other firmswill leave theirpriceunchanged.' We believe that this assumption is closer to the way firmsbehave in most real world situations.A third assumption developed by Greenhut and Ohta is also analyzed. To previewthe conclusions,we findthat manyof the perverseresultsfoundin spatial price theoryusing the traditional assumption are also found in the second model using the Hotelling monopolistic competition assumption,but usually as special extremecases. Thus while we are not able to rule out all perversespatial results,we are able to showthattheconditionsunderwhich theyare likelyto be realized are much more restrictive than earlierbelieved. The second model also leads to a generalized class of models encompassinga wide range of competitivereactions. In the next section we consider the relationshipbetweenspace and the theoryof the firm.In addition we list the characteristics that we think reasonable spatial models should possess; these are subsequently used in discussing alternative models. The second section outlines the standard assumptions of location models and discusses the possible price reactions. The thirdand fourthsections analyze the two models. Finally, the last two sections presentand evaluate a generalizationof the earlierresults. I The term"zero conjecturalvariation" is sometimes used to describe this assumption, see, for example, B. CurtisEaton and Richard Lipsey. 896 This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions VOL. 68 NO. 5 CAPOZZA AND VAN ORDER: SPATIAL COMPETITION 1. Spatial Microeconomics Beforedevelopingany spatial model, it is worthwhileto inquire into the relationship between spatial and nonspatial microeconomics. There are two essential distinguishing featuresof spatial competition. The firstis transportationcost. If transportationwere costless, firmswould have no protection from spatially separated rivals. A firm1,000 miles away would be just as formidablea competitoras one next door. As a result,space would be of no consequence; firmsmightas well all be located on the head of a pin, perfectcompetition would prevail. Transport cost gives the spatial firmitsmonopolypower over customersclose to it. The second essentialfeatureis that average cost curvesmust be downward sloping over some range. The curve may have the negativeslope, for instance,eitherbecause thereare some fixedcosts or because there are economies of scale in production. If averagecostcurvesare nonnegatively sloped throughoutthe range, there will be no advantage to concentratingproduction at specific locations. Each consumer could produce his required consumptionjust as cheaplyas any concentratedfirm,and again spatial competitionwould disappear. If either of the above features fails to hold in practice, there is no need for a spatialmodel. In fact,both are characteristic of mostreal worldindustries;but thatis not the interestingpoint for model building. The issue that must be raised here is that nonspatial perfectcompetition would appear to be essentiallya special case of imperfectspatial competition that arises if eitherof the above featuresis absent.Therefore,we should expect a reasonable spatial model to behave like the perfectcompetition model in the limit. In particular,we should expectthe followingcharacteristics: 1) As transportcosts approach zero, perfectcompetitionshould be approached and priceshould approach marginalcost. 2) As fixedcosts approach zero, concentratedproductionis less essential;spatial monopoly power is diminished;and again 897 priceshould approach marginalcost. The above would have to hold fora model to be valid over a wide range of values. In additionto 1) and 2), we mightalso expect that,at least forsome values of the parameters(i.e., low transportand low fixedcosts), the model should obey the intuitivelyappealingpropertiesof nonspatialcompetitive theorythat: 3) As costs (fixed,marginal,or transportation)rise,priceshould rise. 4) As demand density rises, firms should be able to take advantage of economies of scale. Price should fall in the long run. 5) As more firmsenter the industry, thereshould be increased competitionand priceshould fall. The most widelyused Loschian model of location violates all of the above. The inconsistenciesof the Loschian model can be attributedto the factthat it is a special extremecase of a more general class of location models. In the next section we begin to develop themodels. ofLocationModels II. TheAssumptions underFreeEntry Following the writingsof August Losch a somewhat standard set of assumptions has evolved for partial equilibriummodels of location withfreeentry.We accept most of these,including: ASSUMPTION 1: Thereis a singlecommoditythat can be produced in two-dimensionalspace withthesame costfunction(i.e., ubiquitousresourcesand technology). ASSUMPTION 2: The cost functionhas constantmarginalandfixedcosts C =f + cX (1) where X = output; C = productioncost; f = fixedcost; c = marginalcost. 3: Transport cost per ASSUMPTION mileis identicalbetweenany two points and equal to t unitsper mile. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 898 THE AMERICAN ECONOMIC ASSUMPTION 4: Potential consumers occupy a homogeneousunboundedplain at unijormdensityD. ASSUMPTION 5: All consumers are identicaland have a demand curve that is linearindelivered price(p + tu). REVIEW DECEMBER 1978 (G-O) ASSUMPTION 8c: Greenhut-Ohta competition Each firm assumes the price at theedge of themarketarea (borderprice) to befixed. All threeassumptionscan be interpretedin termsof how the firmexpectsrivalsto react to a change in its price. The differencesare (2) x = a - b(p + tu) crucial to the results that follow and rewhere p = the mill price; x = demand per quire some elaboration. consumer; u = distance to the firm; and Note that the buyerhas two decisions to a, b > 0. There is no price discrimination. make: whereto buy and how much to buy. Firmsset the mili price, and transportation The firstchoice is made by buying where the delivered price (including transport costsare paid by theconsumer. costs) is smallest.If we considertwo neighASSUMPTION 6: Firms continueto enboringfirms,the boundaryof theirmarket ter untilprofitsfor all firmsare drivento areas will be at a point where both firms' zero. productssell for the same delivered price. Hence, ifwe have, as in Figure 1, two firms To theabove we add: at a distance U apart, and draw each firm's deliveredprice(mill price plus transport)as ASSUMPTION 7: All marketareas are a functionof distance,then the boundary circular. betweenfirmswill occur at the intersection of the two deliveredprice lines. At this inAs is well known,circles will not cover a tersection,consumers will be indifferent plane while hexagons will; however, the betweenthetwo sellers. analysis of hexagonal market areas is not This intersectionoccurs at that radius R, materiallydifferent. Extensionto hexagonal satisfying areas is straightforward (see fn.4). Assumpp + tR = p + t(U - R) tion 7 narrowsand simplifiesthe investigation. Since we are not concernedwith the where p = themillpriceof thefirstfirm p = themillpriceof the second firm shape of marketareas, thereis no loss of = the distancebetweenfirms U generalityby includingthis assumption as = the radius of the first firm's R long as we restrictthe discussion to twomarketarea dimensionalmarketareas. The finalassumption,upon whichour re- SolvingforR we have sultsdepend,concernsthereactionof a firm I R = to a change in a competitor's price. The + tU) (3) (P-p usual assumptionwhich has been analyzed p+Ap +t(U-U) extensivelyin thiscontextis: ASSUMPTION 8A: Loschian competition-Eachfirm assumesits marketarea to befixed, and sets prices like a monopolist withinitsmarketarea. However, other assumptions are possible including: ASSUMPTION 8B: Hotelling-Smithies (H-S) competition Each firmassumes the pricesof competitors to befixed. p+Ap+tU p+t (U-U B ~-Ap+t(o-U) ptU +Ap p+Ap p __ _ __ _ 0 FIGURE _I RD 1. THREE RC PRICE - . 71U RA REACTION This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions ASSUMPTIONS VOL. 68 NO. 5 CAPOZZA AND VAN ORDER: SPATIAL COMPETITION That is, theradiusof the firm'smarketarea dependson transportcosts,distanceto other firms,and itspricerelativeto others'prices. Since we assume that all firmsare identical, thisformulaapplies in the same way to all itscompetitors,so thatR is the "radius" of themarketin all directions. ConsiderAssumption8. The representative firmlocated at U = 0 in Figure 1 is chargingprice p; its competitorat U = U chargesprice p = p initially.The market will be split betweenthem, each having a marketof radiusRA = U/2. If the firstfirm should raise itsprice to p + Ap, then under the Lbschian assumptionthe marketradius must remain unchanged at RA. However, theonly way this resultcould be obtained would be if the rival raises its price at the same timeto p + /p. Thus, one implication of the Lbschian assumptionis that the price reactionof a competitoris unitary,that is, dp/dp= 1. From (3) we have I dp 1 2t dp dR dp (4) and since dp/dp = 1, we have dR/dp = 0, whichis the Lbschian case. Under Assumption8B of Hotelling and Smithies,the competingfirmsare assumed not to changetheirprice(dp/dp = 0). If so, the intersectionof the price lines would move to pointC in Figure 1 and the market radiusof thefirstfirmwould fallto Rc= 2t (p - - p - Ap + tU) < RA In this case, from(4) above, since dp/dp = 0, we have dR/dp = - 1/2t. Under Greenhut-Ohtacompetition (8c) the borderprice is fixed.2In the contextof our analysisthe border price could remain constantwhen the firstfirmraises its price if rivals lower theirprice to p - Ap (i.e., dp/dp = - 1). The intersection would then be at point D in Figure 1. Market radius 2It should be noted that G-O competitionwas developed in thecontextof zonal pricingby competitors ratherthan by mill pricing.Thus the application in this context is somewhat strained. We retain the stylizedversionhere because it helps to highlightthe generalizationin Section V. TABLE 1-EXPECTED 899 RADIUS Loschian competition H-S competition G-O competition Spatial Monopoly AND PRICE RESPONSE Radius Change dR/dp Conjectural VariationforPrice dp/dp 0 - 1/2t - l/t - l/t 1 0 -1 would fallto RD D= - 2t (P - p 2Ap + tV) < Rc and since dp/dp = -1, we have from (4), dR/dp = - I /tforthe G-O case. For completenesswe consider also the spatial monopolist. The maximum price a consumerwill pay is given by the price interceptof the demand curve, which is a/b. The monopolist'smarketwill extendto the point where delivered price equals this maximumprice or p + tR = a/b, whence R = (1/t)((a/b) - p), and dR/dp = - 1/t. Thus each assumptioncan be converted to an equivalent assumption concerning eitherthe change in market radius or the pricereactionof competitor'sprice. Table 1 summarizesthesecases. The classificationaccording to expected pricereactionis particularlyuseful,because it offerscomparison with the nonspatial market structureliterature.In this light, Lbschian competition is seen to be the spatial equivalent of noncompetitiveoligopoly. Firms raise and lower price in unison eitherbecause of collusion or price leadership. H-S competition is comparable to monopolistic competition or competitive oligopoly.3Firms assume no reaction from 3The terminologyemployed in this paper is somewhat differentfrom standard usage in the spatial economics literature.The ULschian model described hereas spatial noncompetitiveor collusive oligopoly is oftenreferredto elsewhereas spatial monopolistic competition.We preferto reservethe term "monopolisticcompetition"forthe zero price reactioncase to retain comparabilitywith the nonspatial microeconomics literature.The term "spatial monopolistic competition" is used advisedly since many authors argue thatmonopolisticcompetitionis not possible in a spatial contextwhere there are only six immediate competitors. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 900 THE AMERICAN ECONOMIC competitors.G-O competitiondoes not appear to have a nonspatialanalog. Whileit is clearlypossible forany of these marketstructuresto be realized in practice, our suspicion is that the most common marketstructureis somethingclose to H-S competition. We have two reasons for favoringthe H-S assumption: first,it incorporatesthecontestformarketarea which Lbschian competitionignores by assuming marketareas to be fixed;and secondly, it allows us to analyze spatial monopolistic competition,whichmany believe to be the most common industry structure. This analysiscan be done in a relativelyrigorous mannerwhich gives a precise way of distinguishingindividual consumer demand fromdemand facedby thefirm.In addition, the model degeneratesto perfectcompetition under intuitively appealing conditions. We turnnow to the actual models. X = DjR (a - b(p + tu))2ll udu = HDR2(a - bp - (2/3)btR) Profitsare Y = pX- C Substituting (1) and (5) into(6) we have (7) Y = HDR2(a - bp - (2/3)btR) *(p- 1978 P P= b -tR b L L ~~~~Z PI p2 R2 Rmax D R FIGURE 2. THE L6SCHIAN MODEL Making use of Assumptions 1-8, the equilibriumpriceand radiuscan be derived. Total demand in a marketof radius R will be4 (6) DECEMBER N RI III. TheLoschianModel (5) REVIEW c) -f From Assumption6, profitswill be zero in long-runequilibrium.This impliesthat (8) HDR2(a - bp - 2/3btR) (p - c) -f = 0 Equation (8) definesa curve in the (R,p) 4Notice that equation (5) can be writtenX = DAxT whereA is marketarea and x is demand of a representativeconsumer.Writtenin this form,the equation is applicable to all marketareas that are regularpolygons. For example, if market areas are hexagonal A = 3.45R2 and x = a - bp - .76tR. plane for each level of fixed costs f. This curveis shownin Figure2 forthreelevelsof fixedcost and is designatedthe zero profit locus (ZPL). Two items are worthnoting: first,this curve is invariantto the type of competitionprevailingin the industry,so thatit will be repeatedin our discussion of H-S and G-O competition;second,the curve collapses around the point M as fixedcosts rise.This is significantforthreereasons: 1) M is theZPL forf = frax; forf> fnax nonnegativeprofitsare not possible, the ZPL is not defined,and no firmswill survive. 2) The point M gives the price-radiuscombination a regional monopolist would choose' (i.e., the mill price at each production point and the optimal spacing of production points). 3) M also defines the 5Since the monopolistis unconstrainedby competition,profitsneed not be zero. The monopolist's price is found by substitutingdR/dp = - I/t into equation (9) and solvingforp which gives p = 1/4 (a/b + 3c). The monopolist'spriceis independentof density,fixed costs,and transportcosts. It should be noted that this resultholds only in a marketof uniformdensity.See Dennis Heffleyfor a discussion of the spatial monopolist in a marketwithvariabledensity. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions VOL. 68 NO. 5 CAPOZZA AND VAN ORDER: SPATIAL COMPETITION maximum possible extent of the market given by Rmax= (3/4)t((a/b) - c). Radii beyondRmaxare ruled out because beyond thatpointpriceis greaterthan ((a/b) - tR) and some customerswould have negative demands. A second condition is derived from the profit-maximizing behavior of firms.Maximizing(7) withrespectto pricegives (9) ddY= dp = 2(p- c)(a- bp- btR) dR _ R(a - 2bp - (2/3)btR + bc) dp Under Lbschian competitiondR/dp = 0 so that(9) becomes 91) tR a + c p and givestheprofit-maximizing price that a Loschian firmwill charge in the short run givena marketof radius R. It is shown as LL' in Figure2. Notice that the profit-maximization condition (9') and the zero profitsconditions are satisfiedtwice fora given level of fixed costs. However, for demand to be nonnegativeat theedge of the marketarea, it is requiredthat(a/b) - p > tR > 0. Therefore all pointsto therightof thelinep = (a/b) tR whichis shown as ND (nonnegativedemand) in Figure 2, are not admissible. If fixedcosts are fl, long-runequilibriumoccurs at (R,, p1) wherethe profit-maximization line (LL') cuts the ZPL in the admissibleregion. TABLE 2-EFFECT ON PRICE 901 For marketswitha radius largerthan R1, positiveprofitswill be earned and firmswill enterthe industry.Entrywill cause market areas to shrinkand the industrywill move towards long-run equilibrium. For radii smaller than R, profitsare negative, firms will leave the industry,marketradii will increase, and again the industry moves towardslong-runequilibrium.6 It is straightforward to show that this modelis inconsistent withthe characteristics 1)-5) outlinedabove. We note the following behaviorof the Loschian model. IA) As transportcosts approach zero, the profit-maximization line LL' becomes flat (note equation (9')), and price approaches the nonspatial monopoly price ((a/2b) + (c/2)) rather than the perfectly competitivemarginalcost price. 2A) As fixedcosts approach zero, the ZPL expands, the equilibrium price rises and again approaches p = (a/2b) + (c/2), the nonspatial monopoly price,ratherthan themarginalcost price. 3A) As fixedcosts rise fromf1 to f2 in Figure2, theZPL shrinks,and price fallsin thelong run to P2. An increasein marginal costs shiftsboth the ZPL and the profitmaximizationline (LL') so that the price 6Readers may be more familiarwith the tangency solutionthatoccurs in the(X,p) plane. The problem is actually three-dimensionalin (X,p, R) space. The tangencysolution arises graphicallyin the projection onto the(X,p) plane. Our graphicalsolution(Figure 2) is the projection onto the (R,p) plane. (See the authors, 1976.) The point (RI, P1) corresponds to the pointof tangency. IN THE LOSCHIAN AND SMC MODELS SMC Loschian Noncompetitive Normal Parameter Spatial Change p <a/b - 4/3 tR Oligopoly a + D f t + c i - - 4 + - + + Perverse p > a/b - 4/3tR Spatial Monopolist 4 + + 0 0 i - - This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions + 0 902 THE AMERICAN ECONOMIC REVIEW DECEMBER 1978 K change is ambiguous. The comparative statics are developed more fully in ApatR b 4~ 3 pendix A. The signs are summarized in Table 2. 4A) Appendix A shows that price inZPL,~ creaseswhenthereis an increasein density. Intuitivelythis occurs because the higher densitypermitssmaller market areas. The spatial demand curve is more inelastic for smallermarketradii;7 that is, the firmhas M moremonopolypowerand raisesprice. fi m imizotion (SMC) 5A) As firms enter,marketarea shrinks; p13 the firmmoves back along the profit-maximizationline,and pricesincrease. The above might be acceptable for a model of organized (price-leader) spatial Cl oligopolybut is counterto what one might expectto hold in spatial monopolisticcomI ~K _ ~ R petitionor unorganized oligopolistic comR3 Rmox petition.It is particularlyimportantto note FIGURE 3. A MODEL OF SPATIAL MONOPOLISTIC thatin the limitas transportcosts and fixed COMPETITION: NORMAL CASE costs approach zero, the Loschian model approaches the nonspatial monopoly price ratherthan the perfectlycompetitiveprice. (9") p = c + tR(a - bp - 2/3btR) The model of the next section does ap(a - bp) proach perfectcompetitionin thelimit. This can also be writtenas IV. A ModelofSpatialMonopolistic p = t+R (9"') Competition (SMC) xo The model to be developed is one with Hotelling-Smithiescompetition (Assumption 8B) instead of Loschian competition. With this zero conjectural variation assumptionwe have shown above that dR/ dp = - 1/2t. Substitutingin (9) gives the profit-maximization line forthe SMC firm.8 7The elasticityof the spatial demand curve in the Loschian model is 17= -bp a - bp - 2/3 btR and becomes moreinelasticas R decreases. 8To the individualfirmR is endogenous. The firm's price equation is obtained from substituting(3) into (9"). In equilibrium if firmsare identical they will move untiltheyare equidistant.We can thereforeinterpret(9") as an equilibriumcondition with R set equal to one-halfthedistancebetweenfirms(assuming firmsare not regional monopolists). For a briefdiscussionof short-rundynamics,see AppendixB. where x = averagedemand per person xo = demand perpersonat R = 0 That is, profit-maximizing price is an additive markup over marginalcost where the markupis given by the second termon the rightin (9"'). Equation (9") has an invertedU-shape in the admissible region and is shown in Figure3. This profit-maximizing pricecurve startsat marginalcost (p = c, whereR = 0) and rises to a maximum at p = (a/b) (4/3)tR, thendeclines.This conditionfora maximumis satisfiedif average demand is unitaryelastic with respect to transportation cost.9 The SMC curve is downward 9This elasticityis givenby 775FI = dx-T -- and is unitaryifp = 3- btR 3 =2 a - bp - - btR a/b - 4/3 tR. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 3 VOL. 68 NO. 5 CAPOZZA AND VAN ORDER. SPATIAL COMPETITION sloping,however,ifaveragedemand is elasticwithrespectto transportcost. The zero profitlocus (ZPL) is as before and is superimposedon Figure 3. Long-run equilibriumoccurs at (R3, p3) if fixedcosts are fl. For market radii larger (smaller) thanR3 firmsearnpositiveprofits;new firms enter(exit) and move the industrytowards equilibrium. In thismodel two forcesdetermineprices. Firstis the demand effect:the more elastic is averagedemand,the lower the price. The second is the competitiveeffect:the closer competingfirmsare, the greater is competitionand theloweris price.When market radius is small (i.e., when firmsare very close together)price is close to marginal cost. Competition forces price down to marginalcost in this model whereas in the Loschian model price approaches (a/2b) + (c/2) > c as radiusapproacheszero. There is, however,a point (given by p = (a/b) - (4/3)tR, thelineKK' in Figure3) wherefurtherincreasesin marketarea correspondto lower prices. For small market areas with densely packed firms,competitionis important.However,as marketareas approach the maximal size Rmax, competition over the size of the market becomes less and less importantsince the consumer at theedge of the marketis buyingless and less (note that at Rmax he is buyingnothing and is not worthfighting over). Indeed, the SMC profit-maximization line approaches the ULschianline and intersectsit at Rmax. Hence, the Loschian-type effects noted above wheremonopoly power decreases as marketarea increases may tend to be importante.venin the SMC model. Thus, we cannotruleout a priorithe "perverse" case in which enteringfirmsraise rather than lower the equilibriumprice. This perverse case occurs when firmsare so spread out that individual firms are almost regional monopolists. Mathematicallywe solve the model by substitutingthe profit-maximization equation (9") into the zero profitsequation (8) to obtain (10) (a - bp)2(p- C)3= ( f) x [(a-bp-(? 3 ()( ( HID 903 I -1)2 (a -bp)(p -C)2] Explicitsolution forprice is difficult, but we can outlinesome propertiesof the longrunequilibrium.Note the following: 1) The zero profitslocus is the same as thatin theLbschian model (equation (8)). 2) Changes in f or D affectthe zeroprofitslocus but not the profit-maximization curve(equations (8) and (9")). 3) Since t,f, and D enterequation (10) as t2f/D, solvingforthe impact of changes in one impliestheeffectsof theothers. There are threepossiblecases: CASE 1: Degenerate case nonnegative profitsare notpossible in the admissibleregionand no firmsexist. This mightoccur if fixedcostswereverylarge or populationdensityverysmall. CASE 2: Normalcase (p < (a/b) - (4/3)tR, i.e., to the leftof KK' in Figure3)-zeroprofits curve cuts the profit-maximization curveon the upward-sloping portion as in Figure3. If additionalfirms enter,pricefalls. CASE 3: Perverse case (p > (a/b) (4/3)tR, i.e., to the rightof KK' in Figure 3) zero profitscurve cuts the profit-maximizationcurveon thedownward-sloping portionas in Figure4. If additionalfirmsenter, prices rise. This could occurfor large fixed costs or small populationdensity.However theimplicationis that transportcosts are a largeproportionof deliveredprice (p + tR) at theedge of themarketarea. Greenhutand Greenhut,Hwang, and Ohta (1976) present empiricalevidenceindicatingthat in general thisis unusual.It wouldcertainlybe unusual in urbanareas wherethepacking of firmsis relatively dense. Only the lattertwo cases are of interest. In Table 2 the comparative statics have been summarized. To derive the results consider firstthe effectsof increasing fixed costs f Rising fixedcosts affectonly the zero profitslocus, shrinkingit. Price rises if the profit-maximization line is upward sloping (normal This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 904 P THE AMERICAN ECONOMIC K a -4 P= b --4 tR ZPLf.f M c > ~ DECEMBER 1978 approaches marginalcost. 3B, 4B) The comparative statics are normaliftheintersection takes place on the rising portion of the profit-maximization line as discussed above. That is, increases in costs will raise price; increasesin density will lowerprice. SB) Entryforcespricedown if the profit-maximization line is upward sloping,i.e., in the normalcase. t P4 REVIEW V. SpatialCompetition-A Generalization profit maximization (SMC) We have not developed a model with G-O competition,but it would be straightforward to do so. Since dR/dp = - 1/t underG-O competition,(9) would become K (9"") I c + tR(a p 2(a R FIGURE 4. THE PERVERSE CASE OF THE SMC case) and fallsif the line is downwardsloping. From property3) the same holds fora risein transportcosts t, or a fall in density D. Increasing marginal cost c raises the profit-maximization line and shrinks the ZPL, raisingprice in the normal case, but withambiguous effectin the perversecase. Similarly,a rise in the demand intercepta expandsthe zero profitlocus and raises the profit-maximization line, leading to an ambiguous effecton price in the normal case and an increasein theperversecase. The effectson price are quite different in thetwo cases. In theabsence of a restriction on the sign of (a - bp - 4/3btR) nothing can be said about the sign of the price changes. If we compare the signs for the SMC model with those for the Loschian model,we findthat the Loschian signs are identicalto the perversecase of the SMC model. Note that characteristics1)-5) hold for theSMC model. I B) From (9 ") as t approacheszero the profit-maximizationline becomes horizontal at p = c, themarginalcost price. 2B) As fixed costs approach zero the intersectionof the two curves moves back line and price along theprofit-maximization - bp - 2/3btR) bp - 1/2btR) This equation definesa curve in the (R,p) plane that starts at p = c and rises to a maximumat M, the monopoly price, and falls afterward.The comparativestatics in this model will be unambiguous (since the profit-maximization curve is monotonically increasing in the admissible region) and similarto the normalcase of theSMC. The threeformsof competitionsuggesta generalization.Equation (9) definesa family of curvesin the(R,p) plane which is shown in Figure 5. The two extremecases in the p a c Loschion b2 \dp dp M dp dp Rmax FIGURE 5. THE PROFIT-MAXIMIZATION CURVE DIFFERENT FORMS OF COMPETITION This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions UNDER VOL. 68 NO. 5 CAPOZZA AND VAN ORDER: SPATIAL COMPETITION figureare the Loschian profit-maximization curvewheretheprice reactiondpj/dpis plus one and the G-O curve where the reaction is minus one. Hotelling-Smithiescompetition is one intermediatecase; but we can conceive of any number of other intermediate cases as the price reaction varies from1 to -1. Notice that as the price reaction increases,themaximumpoint on the curvemovesto the left. profit-maximization This impliesthat the perverseresultsnoted fortheLoschian model and forsome values oftheSMC model are more likelythe larger the price reaction,that is, the greater the tendencyforfirmsto collude. A relevantquestion to raise at this point is which price reaction is most likely. We feelthat in practice the value probably is fairlyclose to zero. In two-dimensional space, thereare six competitorssurrounding each firm.Each surroundingfirmis a small proportionof the total competition faced by the firm.Therefore,it seems reasonable to argue that unless thereis collusion,firmswould not react dollar fordollar to price changes of just one of their competitors,and would probably have a comuch closer to zero than unity.In efficient any case, the qualitativeresultsof theS MC model would also hold for any value of pricereactionbetween 1 and - 1. We view the Loschian model as extreme,largelybecause it fails to meet the requirementsset out earlier in the paper. In particular: it does not allow price to approach marginal cost as fixed or transportationcosts approachzero. It unambiguouslyrulesout the plausible cases where fixed and marginal cost increasesraise price. It rules out the possibility'that a densityincrease will decrease price.The argumentthatfirmsmatch price increases of rivals dollar for dollar seemsimplausible.As a model of collusive oligopoly,however,it is moreacceptable. The Greenhut-Ohtamodel has the advantagethat it does give plausible and unambiguouscomparativestatics results,and handles the extremecases properly. However, in the contextof spatial competition withmillpricing,the impliednegativeprice reactionseems unlikely.In another context or zonal (forexample,underdiscriminatory 905 pricingwhere a negative price reaction is not necessarilyimplied) it may be more appropriate.10 competitiondoes The Hotelling-Smithies satisfythecriteriawe have set up. While we should not expect price reactions to be exactlyzero, we should (for reasons mentionedabove) not expectit to be especially large, so that in the absence of collusion the zero conjectural variation assumption would appear to be a suitable approximation. VI. Conclusion We have shown that it is indeed the case that many perverse results are possible whenthefirmis placed in a spatial context. We have developed an imperfectcompetition model that shows that the perverseresultsappear under relatively"unusual" circumstancesand that provides a convenient way of interpretingimperfectcompetition models as an outgrowthof spatial considerations. In particular we have shown that the widely applied Loschian model, which exhibits the perverse results, can be interpreted as an extreme case of a class of models defined by the price reaction of firms.The perversecomparativestatics results appear when the profit-maximization curve is negatively sloped in the (R,p) plane. The negativeslope is more likelyto be obtained if the price reaction(dp/dp) is close to one; if transportcosts are a large proportionof deliveredprice or, more specifically,if average demand is elastic with respect to transportationcost; or if fixed costs are so large that firms are barely viable." All of these are clearlypossible; but we have arguedthatin urban areas each would be an exception. Thus, the results of the theoryof the nonspatial firmare usually ?0Seefn.2 above. I I There is a fourthfactorthatinfluencesthe slope of curve that has not been anathe profit-maximization lyzedin thispaper. Ohta has shown that the convexity of the individual demand curve also influencesthe slope. The negativeslope is less likelythe more convex individualdemand is. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 906 THE AMERICAN ECONOMIC qualitativelysimilarto those forthe spatial firm;however, there are conditions under whichthetraditionaltheoryis violated. We have not addressed the welfareissue raised by the lack of perfectcompetition. We have shown in our 1977a paper that total costs (production and transport)are not minimized.There are too manyfirmsso that not enough advantage of scale economies is taken. This is a standardmonopolistic competition result which we conjecturewould carryover to the model here. We do not however,offera proof. Also, we have deliberatelykept withina partialequilibriumframework.This framework is in the spiritof most work in monopolisticcompetition,but it is not entirely satisfactory.In particular,the distribution of population is assumed to be given; but one mightexpectthe distributionto depend upon the price structure and transport costs; that is, if transportcosts are high, consumerswould concentrateat production sites and producers would cluster. We do not attempt to incorporate this aspect, viewingour analysis as a kind of market equilibriumcurve,whichshould eventually be imbeddedin a moregeneralsystem. APPENDIX A-THE COMPARATIVE STATICS OF LONG-RUN EQUILIBRIUM PRICES IN THE LOSCHIAN MODEL In theshortrun,withfixedmarketareas, priceis given by equation (9') and is an increasingfunctionof demand densitya and marginalcost c but a decreasing function of transportcost t. To determinethe long-runprice we combine equations (8) and (9') which gives the quarticinp (A1) a (p-b2b 2 )2(p - c)2 = ft2 9171Db The solutionis a (A2) p = 4 + 3c - 2 I a [ 4b~ ( 4t( 3 - C)2 c)1/2]1/2 HDb The relevantroot, after taking economic REVIEW DECEMBER 1978 admissibilityintoconsideration,is the positiveone, so that (A3) p =! (' + 3c + 2Z) 4 b whereZ = _ c)2 -4t 3 -4(a fb)1/2]1/2 Differentiating with respect to the parameterswe have (A4) aP aa I 8 ( - c + 6bZ) > 0 8~b_Z b (A5) ap =-8b1[2a(q 8bZ ab b - c + Z) ) 1/2] t< -4 t ( 3 11Db (A6) a1 Oc p (A7) a= aD (A8) ap Of (A9) aP At 0 O (a -c-6Z) 8Z b ( 6DZ t 6DZ I 3Z D )1/2 11Db (f)1/2 11Db 11)/2 H1Db > 0 < 0 < 0 The sign of ap/la follows directlyfrom the requirementthata/b > p > c for firms not to shut down. The others follow from the requirementthat Z be real forthe solutionto exist. Intuitivelythe results come from the workingof two effects.In Figure 2, changes in some parameters shift the short-run profit-maximization lineLL', butat thesame time the zero profitslocus may expand or contract.An increase in a, which can be as an increase in income, shifts interpreted the profit-maximizationline up and expands the zero profitslocus. Thus we have theunambiguouspositivesign. Increases in fixedcosts shrinkthe ZPL and raise prices. Increasesin transportcost lower the profitmaximizationline and shrinkthe ZPL so thatpricesfall. The sign of marginal costs is ambiguous because increasesin c shiftthe profit-maximizationline up, but shrinkthe ZPL. The resultingchange in price depends on the slope of the profit-maximization line. If This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions CAPOZZA AND VAN ORDER: SPA TIAL COMPETITION VOL. 68 NO. 5 transportcosts are low, prices will rise; but iftransportcosts are sufficiently high,prices can fall. Some of the resultsare somewhat counterintuitive. One would not expect pricesto fall if fixed or marginal costs rose. Nor would one expect population density to have a positiveeffectand transportcosts a negativeeffect.In general,prices are lower in urban areas than in low density rural areas because of highervolume and greater competitionamong sellers. Similarly,high transportcosts are oftencited as a cause of high prices not low prices, because of the lessenedcompetitionbetweenfirms. APPENDIX B-SHORT-RUN DYNAMICS The "true" short-runprice equation for thefirmis (9") with R = U- -t (p- p-) substitutedforR in theequation. This gives thefirm'spriceas a functionof the distance betweenfirms(a proxyforthe "number" of firms) and other firms' price. Short-run equilibriumis givenin Figure6. The firm'sprice line is AA'. For pricesto theleftof Pe firmsthinktheyhave charged more than theirrivals price p but discover theyhave not and increase their estimates ofp, increasingprices.Similarlyto the right p X ~~~A' of Pe prices must fall, and Pe is a stable equilibrium.It is possible to show that 1) if p-is zero firmsshould set a positive price and 2) a unitincreasein T leads to less than a unitincreaseinp so thatAA' looks as it is drawnin thefigure. Then, in short-runequilibrium,p must equal F and R mustequal 1/2U, so that(9") can be intepretedas givingboth profitmaximization and (with R = 1/2U) short-run equilibrium. We assume that short-run equilibriumis obtained "instantaneously" and suppressthe dynamicsof short-runadjustments.For a more completediscussion, see our 1976paper. REFERENCES M. Beckmann, "Equilibrium vs. Optimum Market Areas," disc. paper no. 16, BrownUniv. 1970. D. R. Capozza and K. Attaran, "Pricing and Spatial Dispersion of Firms Under Free Entry," J. Reg. Sci., Aug. 1976, 16, 167-82. and R. Van Order,"An Equilibrium Model of Location and Pricing with Spatial Competition," work. paper no. 676, Univ. SouthernCalifornia1976. , (1977a) "A Simple and Model of Spatial Pricing Under Free Entry,"SouthernEcon. J., Oct. 1977, 44, 361-67. , (1977b) "Pricing Under and Spatial Competitionand Spatial Monopoly," Econometrica, Sept. 1977, 43, 1329-338. Edward H. Chamberlain,The Theoryof MonopolisticCompetition, Cambridge,Mass. 1938. B. C. Eaton and R. Lipsey, "The Principleof Minimum Differentiation Revisited," Rev. Econ. Stud.,Jan. 1975,42, 27-49. MelvinL. Greenhut, A Theoryof the Firm in EconomicSpace, Austin 1971. andH. Ohta,"SpatialConfigurations and Competitive Equilibrium," Weltwirtsch.Achiv.,Mar. 1973,109, 87-104. M. Hwang, and H. Ohta, Pe FIGURE 6. PRICE ADJUSTMENT IN THE SHORT RUN 907 "Observa- tionson the Shape and Relevance of the Spatial Demand Function," Econometrica,July1975,43, 669-82. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions 908 THE AMERICAN ECONOMIC , and , "An Empirical Evaluation of the Equilibrium Size and Shape of Market Areas," Int. Econ. Rev., Feb. 1976,17, 172-90. J. M. Hartwick, "Losch's Theorem on Hexagonal Market Areas," J. Reg. Sci., Aug. 1973,13, 213-22. D. R. Heffley, "Transport Rate Changes and the PricingBehavior of an Urban Spatial Monopolist," mimeo.,Univ. Connecticut 1976. E. M. Hoover,"Transport Costs and the Spacing of Central Places," 1970 Papers Reg. Sci. Assn.,Vol. 25, 255-74. H. Hotelling,"Stability in Competition," REVIEW DECEMBER 1978 Econ. J., Mar. 1929,39, 41-57. AugustLosch, The Economics of Location, New York 1954. E. S. Mills and M. R. Lav, "A Model of MarketAreas with Free Entry,"J. Polit. Econ., June1964, 72, 278-88. H. Ohta, "Spatial Competition,Concentration, and Welfare," memo., Aoy-ma Gakuin Univ. 1977. A. W. Smithies, "Optimum Location in Spatial Competition," J. Polt. Econ., Jan. 1941,49,423-39. N. H. Stern, "The Optimal Size of Market Areas," J. Econ. Theory,Apr. 1972, 4, 154-73. This content downloaded from 195.34.79.101 on Wed, 25 Jun 2014 00:17:45 AM All use subject to JSTOR Terms and Conditions
