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Monopolistic competitionwith outside goodsSteven C. SalopBureau of EconomicsFederal Trade Commission

The Cham berlinian monopolisticaUy competitive equilibrium has been exploredand extended in a number of recent papers. These analyses have paid onlycursory attention to the existence of an industry outside the Chamberliniangroup. In this article I analyze a mod el of spatial competition in w hich a secondcommodity is explicitly treated. In this two-industry econom y, a zero-profitequilibrium w ith symm etrically located firms may exhibit rather strange prop-erties. Eirst, demand curves are kinked, although firms m ake "Nash" conjec-tures. If equilibrium lies at the kink, the effects of parameter changes areperverse. In the short run, prices are rigid in the face of small cost changes.In the long run, increases in costs lower equilibrium prices. Increases in m arketsize raise prices. The welfare properties are also perverse at a kinked equilibrium.

1 . Introduction The Chamberiinian (1931) zero-profit monopolisticaUy com petitive equilib-rium has been explored and extended in a number of recent papers. Theseanalyses have focused on the monopolisticaUy competitive industry and havepaid only cursory attention to the existence of an industry outside the Chamber-linian g roup. In this paper, a model of spatial com petition is analyzed in whicha second commodity is explicitly treated.

In this two-industry econo my , a zero-profit equilibrium with symm etricallylocated firms may exhibit rather strange properties. First, demand curves arekinked, even though firms make "Nash" conjectures. If equilibrium is a tangencysolution awayfrom he kink, the short- and long-run responses to parameter changesare conven tional. Ho we ver, if equilibrium lies at the kink, the effects of param-eter changes are perverse. In the short run, prices are rigid in the face of smallcost changes. In the long run, increases in costs lower equilibrium prices.Interpreting the cost increase as an excise tax, this result states that the incidenceofthe tax is negative. Increases in market size raise prices. The welfare prop-

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142 / THE BELL JOURNAL OF ECONOMICSerties are also perv erse at a kinked equilibrium. D ecreases in cost and increasesin market size lower both consumer and aggregate welfare.In the next section the formal model is presented and the symmetric zero-profit Nash equilibrium (SZPE) is defined. Conditions for existence of theSZPE are derived in Section 3. Comparative statics and welfare properties areexplored in Sections 4 and 5, respectively. The paper concludes with a shortdiscussion ofthe detenence equilibrium concept.2. The basic model In this section we analyze a variant ofth e traditional Hotelling (1929) modelof spatial competition v/hich is derived from Lerner and Singer (1937). In thisvariant the economy that is envisioned consists of two industries. The oneupon which we focus is monopolisticaUy com petitive with differentiated brandsand decreasing average costs; the other is a competitive industry producinga homogeneous commod ity. Each of Z. consum ers p urchases either one unit ornone' ofthe differentiated commodity according to preferences, prices, and thedistribution of brands in product space. Remaining income is spent on thehomogeneous commodity.

Each consumer has a most-preferred brand specification/*. A brand/differ-ent from the most preferred specification is valued lower according to prefer-ences in product space U(IJ*). The product space ofth e industry is taken to bean infinite line or the un it-circumference of a circle. While neither assump tionis realistic, both allow the "corner" difficulties ofthe original Hotelling modelto be ignored and an industry equilibrium with identical prices by equally-spacedfirms to obtain. Eliminaiing the technical difficulties makes it simpler to analyzethe qualitative equilibrium properties ofthe model. Thu s, the model is a bench-mark for subsequent analyses with nonuniform preferences across empiricallyvalidated product spaces. By eliminating technical problems, this model allowsa focus on the essential interactions of firms in an industry.

If there are n brands of the differentiated comm odity available at prices Piand locations Ij, a consumer whose most preferred specification is /* will purchaseone unit from some brand if the maximum surplus of utility less price acrossbrands outweighs the surplus from the homogeneous other good. Denoting thatsurplus by i , we have the decision rule: Purchase one unit ofthe brand satisfyingmax [U(U*) -Pi] ^s. (1)

i

The t radi t ional model of constant t ransport costs is captured with preferencesgiven byU(U*) = u - c | / , - / * | , (2 )

whe re the "d is ta nc e" | / j - /* | refers to the shortest arc length be tw een / , and/ *.In this case, equation (1) may be rewritten as follows:max [t; - c\li - l*\ - p j ^ 0 , (3)

i

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SALOP / 143FIGURE 1TYPI CAL DEM AND CURVE

PRICE,

MONOPOLY

COMPETITIVE

SUPERCOMPETITIVE

- ^ O U A N T I T Y

We now explore the existence and properties of a symmetric zero-profitNash equilibrium (SZPE). By symmetric we mean an equilibrium in which thebrand s are equally spaced around the circular produc t space and charge identicalprices. By zero profits, we mean an equilibrium in which free entry leads to asituation of each brand earning zero profits. The equilibrium is Nash in that eachbrand chooses a best price, given a perception that all other brands hold theirprices constant. The conclusion discusses the requirement that the number ofbrands be integer-valued.The methodology here consists of deriving the perceived demand curve for

a single representative brand as a function of other brands' prices and locationsand then finding a tangency between that demand curve and the average costcurve. Three regions ofthe representative brand's demand curve may be dis-tinguished: the "mono poly ," "c om petitive ," and "supercom petitive" regions. The"m on op oly " region consists of those prices in which the bran d's entire m arketconsists of consumers for whom the surplus of no other brand exceeds the surplusofthe homogeneous outside good. The "competitive" region is composed ofthose prices in which customers are attracted who would otherwise purchasesome other differentiated brand. The "supercompetitive" region consists ofthose prices in which all the customers of the closest neighboring brand arecaptured. These three regions of a typical demand curve are illustrated inFigure 1.

Suppose the representative brand charges price p and its nearest competitorslocated at distance l/n charge p, as shown in Figure 2. We derive the regionsof the demand curve as follows. In the absence of competition from otherdifferentiated brands, the representative brand captures all consumers livingwithin a distance where the net surplus given in (3) in nonnegative. Denotingthe maximum distance by i and substituting into (3) we have ,i = ^ i^ . (5)

If there are L consum ers around the circle, since the brand captures customers

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144 / THE BELL JOURNAL OF ECONOMICSFIGURE 2THE CI RCUL AR M ARKE T

P

-1/n a4 1/n-II

Those consumers residing in the potential monopoly market of two brands pur-chase from the one offering higher net surplus. If the brands are located adistance apart of \/n and the neighboring brand on one side charges a price p,then from (3) the representative brand captures all those consumers within adistance x given by

v~cx-p^v-ci x\ - p. (7)Denoting by x, the value for which (7) holds with equality, we have

X = - - (p + c/n - p) (8)2c

and hence a firm's competitive demand q'^ = ILx is given byL

q" = (p + c/n - p). (9)cDifferentiating (6) and (9), the slopes sl(D) of the demand curve in these

two regions are given byt = -c/2L (10)sl(D')= --cIL. (II)

Thus, we have the unusual result that demand is more elastic in the monopolyregion than in the competitive region. Moreover, as illustrated in Figure 1,the monopoly region comes at higher prices.

The two regions fit together as follows. Suppose the right-side neighbor hasa potential monopoly market illustrated in Figure 3. At prices above v, therepresentative brand obtains no customers. As it begins lowering prices below v,it captures demand from the homogeneous good according to the monopoly

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SALOP / 145FIGURE 4MA R K E T OV E R LA P

slope c/2L. Eventually its price becomes low enough that its monopoly marketoverlaps the monopoly market of its neighbor as illustrated in Figure 4. Now as itlowers price further, it begins to capture customers from its neighbor accord-ing to the steeper competitive slope c/L. At the kink in Figure 2, the m onopolyregions just touch . Note that the kink arises here from the existence o fthe otherindustry, not from the non-Nash perceptions discussed by Sweezy (1939). Itgeneralizes to higher dimensional spaces.^At some lower price, even those customers residing at the neighbor are in-different between the representative firm at p and the neighbor at p (and addi-tional surplus c(l/n)). This price p^ is given by

p,= p - c/n. (12)At prices below p^, the representative firm captures the entire market of itsneighbor, for not only are those consumers residing at the neighbor willing toincur the surplus loss c/n for the price differential p - p, but so are all thecustomers o fth e neighbor. Thus the representative brand 's demand has a dis-continuity at P; from this "predatory" pricing.

The demand curve in Figure 2 displays the typical shape of these threeregions. It shifts according to the prices and locations of the neighboring brand s.Since demand can never exceed the monopoly demand, the kink always ties onthat monopoly curve, as illustrated in Figure 5 (with supercompetitive regionsdeleted). Note that the market may be so competitive as to make the kinknonexistent. This occurs when the neighbors' potential monopoly market in-cludes the location of the representative brand.3. Existence of a symmetric zero profit equilibrium (SZPE) A SZPE is defined as a price p and a number of brands n such that everyequally spaced^ Nash price setter's maximum profit price choice earns zeroprofits. We ignore the additional requirement that the num ber of brand s must bean integer and discuss it later. In addition , the potential nonexistence of equilib-rium arising from the discontinuity in demand is also postponed . If an equilibriumexists, the representative brand's demand curve and average cost curve will betangent, for then the zero-profit point is surely also one of maximum profits.Three equilibrium configurations are possible, as illustrated in Figure 6. wherethe monopoly, kinked, and competitive equilibrium prices are denoted by sub-scripts (m,k,c), respectively.

At the monopoly equilibrium, some consu mers lying between tw o neighbor-ing brands may not purchase the differentiated commodity. Thus, the markets

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146 / THE BELL JOURNAL OF ECONOMICSFIGURE 5FAMILY OF DEMAND CURVES

P

of neighbors may not overlap and each can act as a monopolist, constrainedonly by the outside commodity. Monopoly equilibria with and without overlapare pictured in Figure 6. At a kinked equilibrium, markets just touch. As illus-trated in Figure 6, since lhe extension ofthe monopoly demand curve lies abovethe average cost curve, the monopoly pricep^ lies below the kinked equilibriumpricepfc. At the competitive equilibrium configuration, monopoly markets com-pletely overlap. However, Pc may be above or below p ^ , depending on demandand technologies.

It is easy to show graphically which equilibrium configuration obtains forany set of technology and demand parameters {E,m,v,c,L}. Simply drawingthe average cost curve and the entire family of demand curves, existence of anequilibrium configuration requires maximum profits equal to zero-point E inFigure 7. A zero-profit point like G does not satisfy maximum profits becauseit is dominated by a point like E.

FIGURE 6EQUILIBRIUM CONFIGURATIONS

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SALOP / 147FIGURE 7EXISTENCE OF EQUILIBRIUM

L/n-

The S ZPE satisfies two conditions : marginal revenu e (less than or) equal tomarginal cost and price equal to average c ost . For cons tant marginal cost mand fixed cost E, the SZPE is given byp + q -~^mdqp = m + F/q,

and from symmetry, if the equilibrium has no gaps,q = L/n.

(13)(14)

(15)At the monopoly equilibrium dp/dq is given by sl(D'") in (10); at the competitiveequilibrium by sl( >*) in (11); and at the kinked equihbrium by a slope betw eensKD*") and j /(DO- Substituting (15) and (10) into (13) and (14), the monopolyprice and number of brands* are given by= m -\- clln,

1 ^-^.

Using (11) instead of (10), the competitive equilibrium is givenPr ^ m + c/n^

(16)(17)

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148 / THE BELL JOURNAL OF ECONOMICSThe values (Pk,f^k) for a kinked equilibrium lie between the values givenin (16)-(19). Since there is no tangency at a kinked equilibrium, (13) holds asan inequality. Instead of being given by the equality in (13), price is given by

the monopolistic demand function, orPk = V - (c/L)q = V - c/n. (20)Solving (20) and the price equal to average cost condition given by equa-tion (14) for equilibrium variety n^, we have

F nk + c/nk = V - m. (21)The m onopoly equilibrium configuration requires the very restrictive condi-tion that the exogenously given average cost curve be tangent to the exogenously

given demand curve or i' - m = V2cF/L.^ We ignore this limiting case for theremainder ofthe analysis. The competitive equilibrium configuration occurs forall parameter values such that v - m ^ ^/iVcFjL.'^ The kinked equilibriumconfiguration occurs for values of ^ - m in the interval [\/2cF/L,^z'VcF/L],which is small relative ;:o the range of values v - m can assume.**The demand discontinuity can imply the nonexistence of any SZPE."Recalling from (12) that the represen tative firm can ca pture its neighb or's entiremarket at prices below p - c/n, an additional condition for existence of a SZPEis that such pricing behavior is unprofitable. A sufficient condition for this isthat the predatory price p - c/n does not exceed marginal cost m, for price equal

to or below marginal cost necessarily is a losing strategy in the presence offixed costs. Referring to (16) and (18), supercompetitive behavior is not profitable,since the equilibrium piice is no greater than m + c/n. Similarly, if marginalcosts are increasing, as with (/-shaped AC curves, then the market-capturingprice lies below the minimum AC price. H owever, if marginal costs are decreasing,then such price cuts may be profitable and cause nonexistence of a SZPE.4. Comparative statics As the exogenous technological or demand parame ters {F,m,v,c,L} vary,the equilibrium price-variety pair also changes. These changes may be calculatedfrom the equilibrium values in equations (16)-(I9).D Competitive equilibria. The comparative statics at com petitive equilibria arestraightforward and traditional. Substituting (19) into (18) we have

Pc=m + J^ (22)(23)

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SALOP / 149As fixed costs (F ) rise, price rises and equilibrium variety falls. Changes inmarginal costs (m ) are fully shifted onto con sum ers; equilibrium variety remainsthe same. Surprisingly perhaps, changes in the net valuation v have no effect onthe equilibrium. The market is competitive enough that aggregate demand is un-affected by changes in this demand price.As the value of product d ifferentiation (c) falls, prices fall and varie ty falls.As market size (L) rise s, prices fall and variety rises. As c/L decreases, demandbecomes more elastic and price moves toward marginal cost. Thus, the ratio c/Lis the relevant measure of monopolistic product differentiation in the model.For t/-shaped average costs, perfect competition obtains when c/L = 0, for thenevery brand faces a perfectly elastic demand function.

D Kinked equilibria. The comparative statics at kinked equilibria are allperverse . An increase in eitherfixedor marginal costs lowers p rices. This is illus-trated diagrammatically below as a movement from E to ' . Intuitively, costincreases reduce the equilibrium number of brands, allowing the remainingbrands to further exploit scale economies. This is a very striking result. If theincrease in costs is interpreted as an excise tax levied on the industry, then theincidence of that excise tax is negative at the kinked equilibrium. In terms ofconsumer welfare, the lower price is offset by the decline in variety, of course.However, it is shown in the next section that consumer welfare does rise fromthe tax, even if the proceeds of the tax are ignored.

It should be emphasized that this perverse reaction to a cost increase is along-run response that results from the exit of marginal firms. In the short run,there is no reaction at all. Since the marginal revenue curve is discontinuous atthe kinked equilibrium, a small change in marginal costs induces no priceresponse. Thus, the industry responds to a small marginal cost increase asfollows. In the short run , prices and quan tities do not chang e, though profits fallbelow normal (zero). These losses induce some firms to exit, resulting in higherdemand for those that remain. This increased dem and allows the remaining firmsto better exploit scale economies, resulting in decreased long-run prices.

An increase in the valuation (v) raises price and variety, as illustrated by themovement from E to ". As may be seen from Figure 8, price rises by more thanthe increase in valuation, as scale economies are lost. As with cost inc reases, thewelfare effect of this increase in valuation is also perve rse; it may be shown thatconsumer welfare falls. Interpreting the increase in valuation as arising frominformative advertising and the cost increase as the cost of that informativeadvertising, the valuation effect lowers welfare, while the cost effect raiseswelfare.

Decreases in c/L, arising from either an increase in market size (L ) or adecrease in the value of product differentiation (c), raise prices in equilibrium,as illustrated in Figure 9 as a movement from E to E'. As with the othercomparative statics discussed, this result is the reverse of what occurs in thecompetitive equilibrium configuration.

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150 / THE BELL JOURNAL OF ECONOMICSFIGURE 8COM PARAT IVE STATICS OF (v, m .F ); KINK ED EOU ILIBRIUM

L/n

market, this may be tested by comparing the condition under which a segment ofthemarket will be served by a monopolist (the profitability condition) with thecondition under which service yields positive net surplus (the optimalitycondition).Suppose each brand prod uces up to the point wh ere the net benefit to themarginal consumer, who is at a distance x* from the brand serving him, iszero. Then, the net social benefit per brand of serving the entire circularmarket is:

B = 1L \ (v - ex - m)dx - F, (24)

FIGURE 9COMPARATIVE STATICS OF c/L: KINKED EOUILIBRIUM

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SALOP / 151where the marginal consumer's benefit is given by

V - ex* - m = 0. (25)Substituting (25) into (24) and integrating, we have

B = (V - m)^ - F, (26)cand this surplus is nonnegative (B ^ 0) if and only if the following optimalitycondition is satisfied;

/ (27)On the othe r hand, a monopolistic firm will choose to serve any segment of

the market only if its profits are nonnegative,'*' where profits for the monopolyportion of the demand function are given bynm= (v - ^ q -m\q - F . (28)

Maximizing (28) with respect to q, we haveq^ = ~(v - m). (29)c

Substituting (29) into (28), profits are given byn^ = -^(v~ m)^ - F. (30)

Profits are nonnegative (n^ ^ 0) if and only if the following profitabilitycondition is satisfied;J ^ (31)

Comparing the optimality and profitability conditions, we see thatprofitability is sufficient but not necessary for optimality. All markets servedshould be served, but not vice versa.D Optimal vs. equilibrium variety. Given that the entire circular m arket shouldbe served, the optimal price-variety pair may be compared with the equihbriumprice-variety pair. A tradeoff between price and variety exists because of th escale economies present in production.If n firms operate and serve the entire un it-circumference market, then themarginal consumer travels a distance '/in and a consumer located at x ^ */mobtains a surplus in excess of marginal cost of u - m -ex. Since there are Lconsumers per unit distance and 2n intervals of length '/i, total surplus isgiven by

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152 / THE BELL JOURNAL OF ECONOMICSIntegrating, we have:

W = (v - m - -]L - nF. (33)\ 4 n JThe interpretation of (33) is the following. Since (i) the marginal consumertravels a distance of V.in in product space, (ii) the consumer who travels theshorte st distance (zero) obtains his most preferred brand , and (iii) L consumersare distributed uniformly in product spa ce, the average distance travelled is 14at an imputed cost of c per unit. Then the average net surplus per consumer isV - m - c/4n. Total fited costs are nF. Maximizing (33) with respect to thenumber of brands n to find the optimal price-variety mix, we have

(34)Com paring this optimum to the possible equilibria given by (17), (19), and (21) wehave n* < < fc < n^. (35)That is , optimal variety is less than equilibrium variety for this circular m arket, ifthe market should be served.This result of too m any brands is not robust, but rather depends crucially onthe distribution of consumers and preferences. As Spence (1976) and others"have pointed out, the optimum depends on the difference between the averagesurplus and the surplus of the marginal consumer relative to fixed costs; thevalue of adding an extra brand (and respacing the others) effectively con-verts marginal consumers to average ones, at fixed cost F.'^

Graphically, the comparison ofthe equilibrium with the optimum may bemade as follows. The planning problem in (33) is equivalent to maximizingaverage consumer welfare minus price W(n,p) subject to the price equal toaverage cost breakeven constraint.'^ Since the average consumer travels adistance !4 in product space, we have1 cmax W(n,p) = v - p (36)4 n

Fsubject to p = m -\- n. (37)Equation (36) defines linear indifference curves in (p,L/n) space with slope of-I4(c/L) while (37) expresses the constraint. As illustrated in Figure 10, asmaller value of 5 expresses a higher surplus v - S. Then, the optimum lies atthe point where the average cost has slope equal to -V4(c/L), whereasequilibrium lies at the point where the average cost curve has slope between-^(c/L) (for monopoly equilibrium) and -c/L (for competitive equilibrium).

A graphical representation of the optimum vs. equilibrium price-varietypair may be used to show the welfare effects of the comparative statics at

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SALOP / 153FIGURE 10EQUILIBRIUM VS. OPTIM AL VA RIETY

SLOPE = -1/4 c/L

SLOPE - - (L/n) 2

the kinked equilibrium. (See Figure 11.) Fo r exam ple, an increase in costs fromAC to AC that lowers prices will improve welfare, since the slope ofthe in-difference curve (-V4C/L) is flatter than the slope of the monopoly demandcurve {-V2C-/L). Thus, movements down the demand curve represent higherwelfare as illustrated below by comparing 5 to 5 ' . S imilar analysis will show thatFIGURE 11DERIVA TION OF OPTIMAL VA RIETY

SLOPE = -1/4 c/L

SLOPE - 1/2 c/ L

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154 / THE BELL JOURNA L OF ECONOM ICSFIGURE 12DERIVATION OF THE p(L/n) FUNCTION

p(L/n)

an increase in valuation (v ) and an increase in market size (L) lowers consumerwelfare.Finally, we can determine the optimal number of brands given monop-olisticaUy competitive pricing. Spence (1976) shows that for his partialequilibrium model, the market solution is optima l. For the circular model studiedhere, the optimum is either the market equilibrium or complete monopoly. Wemay prove this as follows. First we derive the Nash equilibrium price fordifferent (exogenously given) numbers of symmetrically spaced brands. Wedenote this relationship by p(L/n).^'* Thcp(L/n) function is illustrated in Figure 12as EE'M, where we assume the SZPE (point E) is competitive. The completemonopoly equilibrium is labeled M, and E'M is a portion of the monopolydemand curve.The consumer welfare maximum could then be found by placing indiffer-ence curves, which have slope -i4(c/l), in Figure 12. It is clear that theoptimum must lie at E or M. If the SZPE were kinked, say at E\ then p(L/n)would be the portion 'Af, and the optimum would lie at the com plete monopolypoint. Thus, for the circular industry, optimal entry policy is either free entryor entry restricted to the point of each brand having a complete monopolymarket.

6. Conclusions In the example studied he re, explicit attention has been paid to the role asecond industry (outside goods) plays in determining the properties of monopolisti-

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SALOP / 155nonuniform preferences across more complicated and realistic product spacesor different technologies across brands.The major contribution of the approach taken here, which was first notedby L erner and S inger (1937), is to provide a rationalization ofthe kinked dem andcurve in terms of symmetric "Nash" conjectural variations. Previous ration-alizations by Sweezy (1939) and others were based upon asymmetric com-petitive responses to price increases and decreases. This paper goes beyondLerner and Singer by deriving the industry equilibrium and analyzing itsproperties.

The industry equilibrium model displays conventional properties whenequilibrium occurs at a Chamberlinian tangency away from the kink. On theother hand, the properties of equilibria occurring at the kink are perverse.In the short run, industry prices do not adjust to small cost changes, as notedby Sweezy in his model. It is only through the process of entry and exit thatthe industry adjusts to cost changes. Moreover, at the kinked equilibrium, thelong-run response to a cost increase is exit by some brands followed by adecrease in industry prices, as remaining brands be tterexploit scale econom ies.

The short-run price rigidity of the kinked equilibrium accords with casualempiricism. However, the long-run properties are more difficult to confirm orreject, since they depend on longer run entry adjustments. Moreover, theactual symmetric example analyzed entails the abstract and unrealistic assump-tions of uniform preferences around a circular product space and identical costfunctions and valuations among competing brands. While these assumptionsconsiderably simplify the theoretical analysis, they make empirical confirmationmore difficult.

Other shortcomings of the approach taken here are the zero-profit andcostless relocation assumptions we have made. Because the technology ischaracterized by an indivisible fixed cost, the number of brands must beinteger-valued. Therefore, free entry need not lead to a zero-profit equilibrium,as originally pointed out by Kaldor (1935) and analyzed by Eaton (1976).An interesting "deterrence" equilibrium concept built on these foundationshas been explored for a circular market by Hay (1976) and Schmalensee (1977)and for other spatial markets by Prescott and Visscher (1977). In a deterrenceequilibrium sequential entrants locate in such a way that no new entrant wishesto locate in the interval between two firms. As a result, the deterrenceequilibrium has half the number of brands as the SZPE.

In the model analyzed here, the deterrence equilibrium configuration isgenerally that point in the p(L/n) curve with the number of brands n equal to\6 . the number at the SZPE.'^ If the SZPE is competitive, the deterrenceequilibrium may be competitive, kinked, or at the monopoly point; whichequilibrium occurs depends on the particular parameter values. Hay showedthat if the deterrence equilibrium is competitive, prices are higher than at theSZP E. H owe ver, kinked or m onopoly deterrence equilibria may result in lowerprices than the competitive SZ PE. Finally, if the SZ PE is kinked, the d eterrenceequilibrium lies at the monopoly point and entails lower prices and unserved

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156 / THE BELL JOURNAL OF ECONOMICSIn closing, it should be reemphasized that the exact results derived heremerely refiect the example used. That there is excess variety at equilibrium isnot robust. The kink appears robust as the number of dimensions of productspace increases. However, as Archibald-Rosenbluth (1975) and Weiss (1977)

point out, equilibrium may not exist with higher dimensional product spaces.ReferencesARCHIBALD. G . C . AND ROSENBLUTH, G . "The "New" Theory of Consumer D emand andMonopolistic Competititm." Quarterly Journal of Economics, Vol. 89 (1975).CHAMBERLIN, E.H. The Theory of Monopolistic Competition. Cambridge, Mass.: Harvard Uni-versity Press, 1931.D IX IT, A.K . AND STIGLITZ, J.E . "Mon opolistic Competition and Optimum P roduct D iversity."The American Economic Review, Vol. 67 (1977).EATON, B.C. "F ree Entry in One-D imensional Models: Pure Profits and Multiple E quilib ria."Journal of Regional Science (1976).GRUBEL, H.G. "Wastes of Monopolistic Competition." (Based on a lecture by J. Tobin.) YaleUniversity, 1%3.HOTELLING, H . "Stability in Comp etition," Economic Journal, Vol. 39 (1929).KALDOR, N . "M arket Imperfection and Excess Capac ity." Economica NS. 2 (1935).LANCASTER, K . "Socially Optimal Product D ifferentiation." The American Economic Review,Vol. 65(1975).LERNER, A.P. AND SINGER, H.W. "Som e N otes on D uopoly and Spatial Competition. Journalof Political Economy, Vol. 45 (1937).PRESCOTT, E . AND VISSCHER, M . "Sequen tial Location D ecisions among Firms with F oresigh t."The Bell Journal of Econom ics, Vol. 8 (1977).ROBERTS, J. AND SONNENSCHEIN, H. "On the Foundations of the Theory of Monopolistic

Competition." Economeirica, Vol. 75 (1977).SCHMALENSEE, R . "En try D :terrence in the Ready-to-Eat Breakfast C ereal Indus try." The BellJournal of Economics, Vol. 9 (1978).SPENCE, A.M. "Product Selection. Fixed Costs and Monopolistic Competition." Review ofEconom ic Studies, Vol. 43 (1976).SWEEZY. P.M. "D emand under Co nditions of OVvgopoly." Journal of Political Economy, Vol. 47(1939).WEISS, A. "Spatial Competition with Two D imensions." Unpublished m anuscript. BellLaboratories, 1977.

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