Loschian Competition under AlternativeDemand Conditions

By BRUCE L . BENSON*

Theoretical examination of the pricing be-havior of firms competing in a spatial set-ting indicates that many of the conclusionsof classical price theory are altered as thecost of distance becomes significant." Anassumption about entrepreneurial behavioris an integral component of these models ofspatial competition (see Greenhut, Hwang,and Ohta). The traditional spatial modelfollows the Loschian assumption in locationtheory—that entrepreneurs price as if theyare monopolists within their market areas.^Recently however, this Loschian assumptionhas been questioned.^ For example, Green-hut, Hwang, and Ohta noted that unex-pected conclusions derived in spatial pricetheory stem, in part, from this behavioralreaction assumption. Capozza and VanOrder further contended that the Loschianassumption does not accurately depict en-trepreneurial behavior. They concluded thatanother assumption concerning entrepre-neurial behavior " . . . is closer to the wayfirms behave in most real world situations"(p. 896).

Capozza and Van Order proposed to in-vestigate the extent to which spatial pricetheory replicates the results of spaceless pricetheory with free entry. To this end, theyoffered five "... intuitively appealing prop-erties [or characteristics] of nonspatial com-petitive theory... "(p. 897) which they felt a

'Assistant professor of economics, PennsylvaniaSlate Uoiversity. I wish to thank M. L. Greenhut andan anonymous referee for helpful comments andsuggestions.

'For example, see Melvin Greenhut, Ming JengHwang and Hiroshi Ohta; Dennis Capozza and KazemAttaran; Capozza and Robert Van Order; and myforthcoming paper.

^For example, see Edwin Mills and Michael Lav;Greenhut (1971); John Hartwick; Nicholas Stem;among others.

^For example, see Greenhut, Hwang, and Ohta;Capozza and Van Order.

reasonable model of spatial competitionshould obey:

1) As transport costs approach zero,perfect competition should be ap-proached and price should approachmarginal cost.

2) As fixed costs approach zero,concentrated production is less essen-tial; spatial monopoly power isdiminished; and again price shouldapproach marginal cost... .

3) As costs (fixed, marginal, ortransportation) rise, price should rise.

4) As demand density rises, firmsshould be able to take advantage ofeconomies of scale. Price should fall inthe long run.

5) As more firnis enter the industry,there should be increased competitionand price should fall. [p. 897]

Capozza and Van Order analyzed threemodels of spatial competition with thesecharacteristics as the criteria for acceptanceor rejection. They apparently felt thatentrepreneurial behavior was the crucial as-sumption in a spatial competition model,since, while developing their models, theyproposed; "The final assumption, upon whichour results depend, concerns the reaction of afirm to a change in a competitor's price" (p.898, emphasis added). In fact, the onlydifference between their three models wasthe entrepreneurial behavior assumptionemployed in each. Capozza and Van Orderclaimed that Loschian competition violatesall the above characteristics I** Actually,however, the characteristics can be derived

^Capozza and Van Order argued that the ap-propriate assumption is that each entrepreneur believesthat all other entrepreneurs will leave their own loca-tion and price unchanged—the Hotel! ing-Smithies as-sumption. They rejected both Loschian and Greenhut-Ohta behavior. The Greenhut-Ohta assumption is thateach entrepreneur assumes the price at the edge of hismarket area is fixed (see Greenhut, Hwang, and Ohta).

7095

VOL. 70 NO. 5 BENSON: LOSCHIAN COMPETITION 1099

for a Loschian competitor simply by alter-ing another assumption used by Capozzaand Van Order (C-V).

The purposes of the following presenta-tion are to demonstrate that: 1) the C-Vrejection of Loschian competition is notwarranted, at least for the reasons they pre-sented; and 2) the entrepreneurial behaviorassumption is not the only crucial assump-tion in a model of spatial competition. TheLoschian assumption produces results whichviolate the C-V characteristics, only whenused with the assumption they made con-cerning individual demand. Loschian com-petition does not violate the characteristicswhen the shape of individual consumer de-mand functions is altered. Note the follow-ing discussion is not designed to change theC-V assumptions. Rather, it is presented tochallenge one of their main conclusions—the rejection of Loschian competition. Fur-thermore, the amendment below does notimply agreement that the C-V properties arenecessarily the only ones a spatial modelshould obey, nor does it imply agreementthat a spatial model should obey all five ofthem.'

I. Assumptions for Two Modelsof Loschian Competition

Capozza and Van Order proposed severalassumptions which are followed here, in-cluding:

ASSUMPTION 1: A single commodity isproduced by all firms and all firms have iden-tical cost functions.

ASSUMPTION 2: The cost function is

(1) C=f+cQ

where Q is output, C is total production cost,fis fixed cost, and c is marginal cost.

five characteristics are actually intuitively una[>-pealing when put in a spatial context (see my 1979apaper). The characteristics are based upon spacelesscompetition where alteration of any one parametereffects only supply or demand. I have demonstratedelsewhere (1979a) that both costs (supply) am/demandare impacted when these parameters change in spatialcompetition.

ASSUMPTION 3: Transport cost per unit ofdistance is identical between any two pointsand equal to t dollars per unit.

ASSUMPTION 4: Firms can enter freely andwill do so until all economic profits disappear.

Two additional assumptions proposed byCapozza and Van Order will be slightlyaltered:

ASSUMPTION 5: Potential consumers oc-cupy a homogeneous unbounded linear spaceof uniform density D.

ASSUMPTION 6: All market areas consistof segments of this linear space.

Capozza and Van Order chose to workwith a plain and with circular markets. Theuse of Unear markets alters the magnitude ofthe values that are obtained (i.e., profit-maximizing price) but the general conclu-sions derived below continue to hold underthe C-V market shape assumptions. Work-ing with the linear market, on the otherhand, does substantially simplify the com-putations which follow.

Loschian behavior characterizes the spa-tial entrepreneur. That is:

ASSUMPTION 7: Each entrepreneur as-sumes his market area is fixed and prices as amonopolist within his space.

Capozza and Van Order assumed thatindividual demand was hnear, but this neednot be the case. Let us start with a generaldemand function given by

(2) P + tu^aX

where P is mill price, q is demand per con-sumer, a and b are constants, and u is unitsof distance.^ Now x can be varied to encom-

*This general demand function has been used inother spatial pricing models. Several implications ofthis general function are noted in John Greenhut,Greenhut, and William Kelly, for example, butLoschian competition was not assumed in that paper.

1100 THE AMERICAN ECONOMIC REVIEW DECEMBER 1980

pass a wide range of possible demand func-tions. For example, if x=l, the demandfunction is linear and the C-V conclusionswould follow. If x > 1, the function is con-vex upwards. With x<l, the function isconcave upwards. The function is extremelyconcave if x<0."' If - l < x < 0 and a = O,the function describes a constant elasticitydemand curve. So x is the vital parameter indetermining the shape of the individual de-mand curve. I must express individual de-mand in terms of q rather than deliveredprice (P + tu) in order to obtain the aggre-gate market demand. This yields

(3)

where u can vary from zero to U, the dis-tance to the boundary of a firm's market.

Equation (3) describes an extremely gen-eral relationship for individual demand, as xcan vary anywhere from — 1 to -h GO whilestill yielding stable equilibrium results. It ispossible, however, to depict the general con-clusions for any Loschian type spatial firmwith two values of x:

ASSUMPTION SA: x=land individual de-mand is

(4a) q=-ia-tu-P)

ASSUMPTION 8B: x = - 1 / 2 and individ-ual demand is

(4b)

We can derive the C-V results with x = 1and the derived relationships are representa-tive of all j t>0 (positive exponential de-

mands). The comparative statics using x =— 1 / 2 are the same for all — 1 < x < 0 (nega-tive exponential demand).

II. Loscfaian Reactions under AltmiativeDemand Conditions

(5)

For x=l, the aggregate spatial demand is

where Qp represents the aggregation of posi-tive exponential demands. The aggregationof demand with x = — 1/2 provides

- 2du

where Q„ represents the aggregation ofnegative exponential demands. Profits arezero in the long run, i.e.,

(7) Y=PQ-C=O

Substituting (1) and (6) into (7) yields

(8) .= ^ ( ._ ,_ i |^

while (1) and (6') into (7) provides

(8')

^It must be assumed that J O - 1 . If je< - 1 , therewill be DO prof it-maximizing equilibrium, given theassumption of constant marginal cost made earlier.Marginal revenue will be increasing if J : < - 1. and anyadjustment will move production away from tbe MR"MC level

The alternative Loschian profit-maximizingprices are obtained by maximizing (8) and(8') with respect to price. Thus

(9)

VOL. 70 NO. 5 BENSON: LOSCHIAN COMPETITION HOI

TABLE I—COMPARATIVE STATICS FOR LOSCHIAN CoMPEnnoNUNDER ALTERNATIVE DEMAND CONDITIONS

Lbscbian Reaction

Parameter Change C-V Results ( j c - 1) X-- 1/2

Marginal CostDemand DensityFixed CostsTransport Cost per Unit

of Geographic DistanceEntry of New Firms

when jc= 1, but

(9') P„ = c+[{c-a)tU+{c-a)^]'^^

Let us now examine Loschian competi-tion to see whether it is always inconsistentwith the characteristics (CI-C5) whichCapozza and Van Order set forth (the com-parative statics are summarized in Table 1):

CI) For x = l , the profit-maximizingprice rises as transportation costs {t) de-crease. This contradicts CI, as C-V noted.However, if x = — 1/2, the profit-maximizingprice falls towards marginal cost as trans-portation costs decline. Characteristic 1 issatisfied only in part, however, by Loschiancompetition and negative exponential de-mand. Price does move in the direction thatC-V argue it should as parameter t changes,but price does not approach marginal costas / approaches zero.

Examination of the C-V model revealsperfect competition need not be the struc-ture of their industry even in a spacelesssetting. Let / = 0. Then the cost function(C-V's as well as my equation (I)) results inspaceless imperfect competition or monop-

'The behavioral assumption of Loschian competi-tion implies that dU/dP — O (see C-V). Similar priceswere derived in Greenhut, Greenhut, and Kelly.

' i t should be pointed out that a in equations (4b),(5'), (6'), (8'), and (9") is not a price intercept as it is in(4a). (5). (6), (8), and (9). As f-»oo the function ap-proaches the price axis. The variable a is tbe asymptoteof the curve, or the limit the function approaches asq—trxi. Also note that in order to have a positive valueunder tbe square root (c — a)>0.

oly."* The large number of firms requiredfor perfect competition necessitates eachproducing at a relatively small scale. Thus,price must be greater than marginal cost forsurvival (unless we assume each of a largenumber of firms produce and sell an infinitequantity). This long-run cost function haseconomies of scale (actually, increasing re-turns) over the entire range so we mightexpect some firms to expand the scale oftheir output and lower price. Other firmswill therefore be forced out of the industry,and market structure approaches oligopolyor monopoly in this spaceless setting. Freeentry prevents profit taking so one firm (ora few firms) may charge a price near margi-nal cost. Price falls because of scale econo-mies, not because of perfect competition.Furthermore, it is very unlikely that pricewould equal marginal cost. Also note that aslong as / > 0, a single firm (or a few firms)cannot crowd out all the others. As trans-port costs fall, profits are available and thereis entry. TTierefore it is conceivable thatspatial competitors operate at smaller andsmaller scales as transport costs fall and

'"^Another specification of costs has been suggestedwbich results in spatial firms producing efficiently intbe long run (see Greenbut, 1971; Obta, 1976). Justicecannot be done to their interesting arguments bere.Briefly bowever, tbe spatial entrepreneur has a requiredreturn or opportunity cost because be conceives of onlya few nearby rivals as bis primary competition. As inclassical oligopoly tbeory, he must take tbe uncertainreaction of these rivals into account when makingdecisions. It follows tbat price may exceed marginalproduction costs (wbere production costs do not in-clude tbe subjective required return to tbe en-trepreneur), but production is efficient wben the en-trepreneur's opportunity costs are taken into account.

1102 THE AMERICAN ECONOMIC REVIEW DECEMBER 1980

market shares shrink. A rise in price shouldnot be surprising when there are economiesof scale over the entire range of output."However, the pricing reaction to changingtransportation costs is in the direction thatCapozza and Van Order argue it should bewhen x= ~\/l.

C2) The reactions to changes in fixedcosts are also quite different when x= — 1/2instead of unity. For x=\, price does rise asfixed costs fall. However, if jc= - 1/2, pricefalls as fixed cost decreases (3/*^/8/>0 inequation (8')). The C-V directional require-ment is fulfilled under this demand condi-tion. The limit requirement is not however,since price does not approach marginal costas /-•O.

As fixed costs fall, scale economies be-come less significant. However, as long asfixed costs are positive, one firm (or a fewfirms) can produce more efficiently than alarge number of firms. Thus, in a spacelesssetting we would expect imperfect competi-tion or monopoly as long as/>0. Of course,an attempt to reap all monopoly profitsleads to entry, but it is doubtful that pricewould approach marginal cost, given/>0.In the limit, of course, with/=O, free entryimplies price equals marginal productioncost in a spaceless setting. Thus, this criti-cism of Loschian competition based on C2appears to remain valid in part. Even thoughthe comparative statics of Loschian compe-tition with x= - 1/2 match those desired byCapozza and Van Order, price does notequal marginal cost in the/=O limit. How-ever, I have demonstrated elsewhere (1979b)that price should not equal site-specificmarginal production costs as long as />0.Price should exceed plant marginal costs inorder to cover society's costs which resultfrom the availability of multiple locations.Multiple locations are desirable, of course,because society's transport costs are reducedwith each additional location and effective(or net) demand increases. If price equals

"Actually, of course, things are not quite this sim-ple. Shares of existing demand shrink, but as transportcosts fall, effective demand rises. Whether finns oper-ate at a larger or smaller scale depends upon theelasticity of firm demand. This point Is discussed later.It is also examined in more detail in my 1979a paper.

site-specific marginal production costs, so-ciety's transport costs are relatively high be-cause a relatively small number of locationsare available.'^ In other words, an individ-ual firm's costs and revenues do not accountfor all the social costs and benefits resultingfrom spatial competition. So price shouldequal plant marginal production costs onlywhen / and t are both zero; that is, whenspace is irrelevant.

C3) As marginal costs rise, the Loschiancompetitor will raise his price just as C-Vargue. Note dPpfdc in equation (9), and9P^/3c in equation (9') are obviously bothpositive.'^

C4) The Loschian price reaction to achange in density (/)) also depends on theshape of the individual demand curve. Whendemand is linear (Jt=l) price will increasewith an increase in density. A higher densitypermits smaller market areas and the ag-gregate demand becomes more inelasticwhen the market area shrinks under an jc= 1.However, when demand is the negativeexponential (i.e., j c= - I /2 ) , price falls asdensity increases. The aggregate demandbecomes more elastic when the extent of themarket is reduced in size and .x= — 1/2 (seeGreenhut, Greenhut, and Kelly).'*

C5) As firms enter the industry, themarket area shrinks {U becomes smaller).As U becomes smaller, the mill price rises ifx= I, but falls when x is — 1/2.

How is it that the shape of the individualdemand function can cause such differentresults in a spatial world? The individualbuyer's demand certainly does not play asignificant role in spaceless economics. Yetapparently it does in spatial microeco-nomics.

'^In fact, fre« entry and Loschian behavior lead tothe socially efficient price and number of plants (seemy 1979b paper). Price equals the full marginal cost ofproduction including plant-specific costs and the in-crease in costs due to multiple locations.

'̂ Capozza and Van Order argued that dP^/^c isambiguous. This is true when there is a positive fixedcost (i.e., when we use equation (8)). The relationship isnot ambiguous, however, when the assumption of zerofixed cost in the long run is made.

'*The elasticity of spatial aggregate demand underalternative demand conditions is discussed below, inthe context of the relationships of price to density,transport costs, and entry of new firms.

VOL. 70 NO. 5 BENSON: LOSCHIAN COMPETITION 1103

FIGURE I. NET INDIVIDUAL DEMANDS AND

CHANGING ELASTICITY FOR X — 1

The underlying reason for the differentresults observed here is the difference be-tween spatial and spaceless aggregate de-mand. The price received by the firm inspaceless economics equals the price paid bythe consumer. Spatial consumers, however,incur buying costs in addition to the pricepaid to the seller.'* Producers are thereforesubject to demand elasticity that relates tothe net market price (net of transportationcosts), not the total price paid by consumers(delivered price). The price paid by con-sumers differs from the price received byproducers.

Elasticity at any given price increaseswhen transportation costs are subtractedfrom a linear gross individual demand curveto obtain net demand (the relevant demandfrom the seller's point of view). The elastic-ity of gross demand (D^) at price P inFigure 1 is less than the elasticity at price Pof the demand that is net of transportationcosts {Df^){E(j<E^). These relations applyto any gross demand with x>0. However,the elasticity at a given price decreases astransportation costs are subtracted from anygross demand of the concavity given by

"For a discussion of the difference between spatialand spaceless aggregate demand, see Greenhut (1978).

FIGURE 2. NCT INDIVIDUAL DEMANDS AND

CHANGING ELASTICITY F O R X ^ - 1 / 2

- 1 < X < 0 (as EC>EN ^t price P inFigure 2).

Elasticity of aggregate demand increasesat any price as more distant consumers areadded and x > 0. Elasticity of aggregate de-mand falls, however, as more distant con-sumers are added and - l < x < 0 , becausesuccessively more and more inelastic de-mands at a given common price are beingadded to the total.

Anything which causes a spatial firm tolose or gain distant consumers changes theelasticity of the aggregate demand faced bythe firm. For example, the impact of entryof a new competitor at a distance is to takeaway the most elastic demand for the rep-resentative seller when x > 0. The aggregatedemand left for that seller is less elastic thanwas the original demand. The resulting priceeffect is upward as Capozza and Van Ordernoted. However, when - l < x < 0 applies,distant competition removes the least elasticdemands from the representative seller'smarket space. The remaining aggregate de-mand is then more elastic than was theoriginal aggregate demand. The price effectin this case, contrary to Capozza and VanOrder, is downward.

1104 THE AMERICAN ECONOMIC REVIEW DECEMBER 1980

We can now see more clearly why changesin costs or demand density have the effectsnoted in Table 1. If transportation costs fall,elasticity of net demand for consumers ateach point in the geographic space will de-crease when x>0. The resulting aggregatedemand is less elastic and the price effect isupward. The opposite is true if - 1 <x<0.In that event, the elasticity of individual netdemands increases, the aggregate demandbecomes more elastic, and the price effect isdownward.

A similar argument applies to changes indensity. Capozza and Van Order observedthat "higher density permits smaller marketareas" and "the spatial demand curve ismore elastic for smaller market radii" (p.902). A higher density will lead to short runprofits and entry. The market areas of ex-isting firms shrink and the spatial aggregatedemand is indeed more inelastic for smallermarkets if x > 0. However, if - 1< x < 0,demand becomes more elastic as marketsbecome smaller. Therefore, price rises whenX = I, but falls when — 1< x < 0.

When fixed costs rise, each firm faceslosses. There is exit and the market areas ofremaining firms expand. Elasticity of ag-gregate demand increases as distant buyersare added and x>0. The price effect isdownward. The opposite is true, however,for - l < x < 0 . Elasticity decreases as dis-tant buyers are added in this case, and theprice effect is upward.

Alternative specifications of demand havean impact on theoretical pricing policies be-cause of the effect on the elasticity of actualaggregate demand for spatial competitors.All that the alternative behavior assump-tions really do is change the elasticity ofeach firm's ;7£Tce/t)ei/demand (see Greenhut,Hwang, and Ohta). A Loschian competitorviews his demand as more inelastic thandoes the Hotelling-Smithies entrepreneurthat Capozza and Van Order maintain is"...closer to the way most firms behave inmost real world situations" (p. 896).'* When

'*Iii fact, as Ohla (1980) has noted, the Loschiancompetitor's perceived demand is liis actual demandfollowing entry. The Hotelling-Smithies (and Greenhut-Ohta) demands are perceived demands which are be-lieved to be more elastic than the actual Loschian

we realize that alternative demand assump-tions and alternative behavioral assump-tions are just altering the demand elasticitiesto which a firm reacts, it is not surprising tofind that we can obtain similar results byvarying either assumption.

III. Conclusions

Each of the C-V "intuitively appealingproperties" apply to Loschian competitionwhen demand is negative exponential inform.'̂ It follows that one cannot rejectLoschian competition for the reasons givenby Capozza and Van Order. Perhaps oneshould reject linear (or positive exponential)demand instead? Of course not. At leastthree models predict similar results. Capozzaand Van Order prefer linear demand andHotelling-Smithies behavior on the part ofentrepreneurs. Greenhut, Hwang, and Ohtaproposed positive exponential demand andGreenhut-Ohta behavior. Here we assumenegative exponential demand and Loschiancompetition. To accept one and reject theothers requires detailed analysis of all theconsequences of each model, (as well as any

demand. Thus the Loschian assumption might be re-stated as, "...firms recognize the actual demand theyface and profit maximize accordingly" (my 1979a paper,p. 24). Hotelling-Sinithies (and Grecnhut-Ohta) compe-tition imply that entrepreneurs are ignorant of theiractual demand situation and never learn, even withcontinued entry and many observations.

'̂ Actually, of course, characteristics 1 and 2 are onlymet in part. The comparative statics are those desiredby C-V. However, the limiting requirements (that priceapproach marginal cost as t or/approach zero) are notmet. 1 maintain that these limiting requirements neednot be met in a spaceless world either, however, be-cause the cost function specified by C-V has economiesof scale over the entire range of output. Market struc-ture is more likely to be monopoly or oligopoly thanperfect competition under such cost conditions. Thus,C-V try to force a spatial model to match the results ofspaceless perfect competition when their assumptionswould prevent perfect competition in a spaceless set-ting. For more on this argument see my 1979a paper.Furthennore, I argue elsewhere that price should ex-ceed plant-specific marginal production costs as long as(>0 (see my 1979b paper). This results because ofadditional social costs and benefits that occur in spatialcompetition which are not accounted for by individualfirms. So, even though the limit requirements are notmet for characteristics 1 and 2, I maintain that theyshould not be met.

VOL. 70 NO. 5 BENSON: LOSCHIAN COMPETITION IIOS

others that might be proposed) includingempirical testing.'^ Until this is effected, onecannot accept any particular assumptionconcerning entrepreneurial behavior (or in-dividual spatial demand) as being superiorto any other assumption." One can onlystipulate that there are at least two crucialassumptions in a model of spatial competi-tion, and probably many "̂

'*The Greenhut-Ohta assumption of entrepreneurialreactions was not discussed here. This does not meanthat C-Vs questioning of the Greenhut-Obta model isaccepted. Rather, their model has many desired char-acteristics which require separate examination.

"Neither C-V's preferred model nor the model dis-cussed here will conform completely with spacelesseconomic conclusions. Incltision of costly distance stillresults in many conclusions which contradict classicalspaceless economic predictions. We have developedspatial models which conform in some ways with space-less economics then, but this does not imply that spaceis not deserving of considerable attention by micro-economists.

^For example, both this model and the C-V modelassumed constant marginal costs. However, allowingfor increasing (and/or decreasing) marginal costs willalso affect price reactions (see Greenhut, Oreenhut,and Kelly).

REFERENCES

B. Benson, (1979a) "Are the Conclusionsof Loschian Spatial Competition ReallyCounter-Intuitive?," work, paper no. 5-79-1, Pennsylvania State Univ., May 1979.

, (1979b)"Spatial Competition withFree Entry, Chamberlinian Tangencies,and Social Efficiency," work, paper no.1-79-1, Pennsylvania State Univ., Sept.1979.

., "Spatial Microeconomics: Imphca-

D. R. Capozza and K. Attanin, "Pricing andSpatial Dispersion of Firms Under FreeEntry," J. Reg. Sci., Aug. 1976, 16, 167-82.

and R. Van Order, "A GeneralizedModel of Spatial Competition," Amer.Econ. Rev., Dec. 1978, 68, 896-907.

J. Greenhut, M. L. Greenhnt, and W. H. Kelly,"A Spatial-Theoretical Perspective forBank Merger Regulations," Fed. ReserveBank Chicago Proceedings on Bank Struc-ture and Competition, 1977, 210-54.

M. L. Greenhut, A Theory of the Eirm inEconomic Space, Austin 1971.

, "Impacts of Distance on Microeco-nomic Theory," Manchester Sch. Econ.Soc. Stud., Mar. 1978, 46, 17-40.

., M. Hwang, and H. Ohta, "Observa-

tions for Market Area Delineation inAnti-Merger Cases," Antitrust Bull.,forthcoming.

tions on the Shape and Relevance of theSpatial Demand Function," Econometrica,July 1975, 43. 669-82.

J. M, Hartwick, "Losch's Theorem on Hexa-gonal Market Areas," J. Reg. Sci., Aug.1973, U, 213-22.

H. Hotelling, "Stability in Competition,"Econ. J., Mar. 1929, 39, 41-57.

A. Loscb, The Economics of Location, NewYork 1954.

E. S. MiUs and M. R. Lav, "A Model ofMarket Areas with Free Entry," / . Polit.Econ., June 1964, 72, 278-88.

H. Ohta, "On Efficiency of Production Un-der Conditions of Imperfect Compe-tition," Southern Econ. J., Oct. 1976, 43,1124-35.

, "Spatial Competition, Concentra-tion, and Welfare," Reg. Sci. Urban Econ.,Mar. 1980, W, 3-16.

A. Smithies, "Optimum Location in SpatialCompetition," J. Polit. Econ., Jan. 1941,49, 423-39.

N. H. Stem, "The Optimal Size of MarketAreas," J. Econ. Theory, Apr. 1972, 4,154-73.