Monopolistic competition: Some extensions from spatial competition

Regional Science and Urban Economics 19 (1989) 31-53. North-Holland

MONOPOLISTIC COMPETITION

Some Extensions from Spatial Competition*

George NORMAN

Unitaersity of Leicester, Leicester LE I 7RH, UK

Received April 1987, final version received December 1987

The analysis of monopolistic competition by means of a spatial analogy has been advanced considerably by the application of Lancasterian consumption theory. But the new theories of monopolistic competition have neglected the importance of factors such as competitors’ conjectures, collusion, the ‘shape’ of individual demand functions and the existence of equilibrium. This paper addresses each of these factors in the context of the Salop model of monopolistic competition. It is shown that the precise specifications of consumer demand and firms’ conjectural variations carry significant implications for the form and behaviour of equilibrium in monopolistically competitive markets. Apparently perverse comparative statics may be a consequence of collusion, but equally may result from competitive behaviour under perfectly reasonable demand and cost conditions. It is also shown that price discrimination obviates oroblems of existence of oriceelocation equilibrium, and in the limit will lead to the socially optimal number of firms. .

1. Introduction

The analysis of monopolistic competition by means of a spatial analogy has a long intellectual pedigree, going back at least to the work of Hotelling (1929) and Chamberlin (1933). It is only with the pioneering work of Lancaster (1966, 1979), however, that this early work has come to full fruition. The essence of Lancaster’s theory of consumption is the suggestion that consumers’ preferences do not relate to final products as such but to the characteristics embodied in those final products. Thus a direct connection can be drawn between ‘characteristics’ space and ‘geographic’ space, on the basis of which many of the properties of spatial competition can be applied directly to monopolistic competition.

Given the undoubted strength of the spatial analogy, it is perhaps surprising that the ‘new’ theories of monopolistic competition have neglected several elements of competitive behaviour and market structure that are known to be critical in determining the precise form of a spatially competitive equilibrium. This paper concentrates on four such elements:

*I am grateful to two anonymous referees and to the Seminar Group at Washington University, St. Louis for comments on an earlier draft of this paper. Remaining errors are the sole responsibility of the author.

01664462/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

32 G. Norman, Monopolistic competition

(9

(ii)

(iii)

(iv)

alternative formulations of competitors’ conjectures or reaction functions; the feasibility and implications of collusion among initially competitive firms; alternative formulations of the individual demand function; the existence of equilibrium in monopolistically competitive markets.

The formal model with which we work is one developed initially by Salop (1979). Section 2 reviews briefly the main features of that model. The four elements noted above are discussed in sections 3, 4, 5, and 6, respectively, and our main conclusions are presented in section 7.

2. The Salop model: A brief review

The economy is assumed to consist of two industries, one monopolistically competitive with differentiated brands produced using a technology that exhibits decreasing average costs, and the second producing a homogeneous product (the ‘outside’ product) under perfectly competitive conditions. We concentrate on the monopolistically competitive industry, taking the outside good as numeraire.

In order to avoid end-point problems, the market is assumed to be the unit circumference of a circle. There are L consumers distributed evenly over the market, with each consumer’s location being determined by her most preferred brand specification I*. Each consumer is assumed to purchase either one unit or none of the differentiated commodity, determined by preferences, prices, and the distribution of brands in characteristics space. All remaining income is spent on the outside good.

Assume that there are n brands of the differentiated commodity available at prices pi and locations li and that brand i is valued by consumers with most preferred variety I* according to a preference function U(l,,Z*). A consumer with most preferred variety I* will purchase one unit of the product brand Ii that maximizes surplus of utility over price across all brands, provided this surplus is greater than the surplus obtained from consumption of the outside good.

Suppose that consumption of the outside good generates surplus S. Then a consumer with most preferred brand I* will purchase one unit of the differentiated product Ii satisfying

max [U(Z, I*) -pi] 1 S. I

(1)

This can be put into a transport cost framework if preferences are given by

U(li,l*)=u-c~zi-l*~=U-c& (2)

G. Norman, Monopolistic competition 33

I 0 Quantity

Fig. 1. The perceived demand curve.

where li = [Ii - I*1 is the shortest arc length between Ii and I*. Eq. (1) can be written

max [( 24 - S) - pi - c;l,] 10. (3)

Using this formulation the effective delivered price is pi + CL, and the effective reservation price is

u=u-s>o. (4)

It should be noted in passing that this approach is directly analogous to the analysis of f.o.b. pricing. Consumers of differentiated product li are paying the mill price pi plus the loss of surplus CJi which ‘looks like’ transport costs with transport cost rate c and distance li. The full price pi + cAi is equivalent to an f.o.b. mill price plus transport costs.

The maximising problem (3) gives rise to a perceived demand function for the ‘representative’ brand. This function has three distinct regions as illustrated in fig. 1: ‘monopoly’, ‘competitive’ and ‘supercompetitive’ regions.

Assume that the n brands in the monopolistically competitive industry are evenly spaced and that the brands neighbouring the representative brand are sold at price p. The monopoly and competitive demand for the representative brand are given respectively by

q” = 2yu - p)/c (PZPA

qc = L( p + c/n - p)/c (PI ZP1Pzh (6)


where p1 is the price at which the market radius (X) of the representative brand just touches those of its neighbouts.’ Thus, p1 is given by

p1 +cx=u=p+c(l/n-x) (7)

from which

p1 =2v-p--cc/n.

The market radius X is such that

Z=(P+c/n-pp,)/2c. (9)

Given Bertrand-Nash conjectures regarding the reactions of neighbouring firms to a price cut of the representative brand, the slopes of the monopoly and competitive regions of the demand curve are respectively

sl( D”) = - c/2L, (10)

sl(D’) = -c/L. (11)

Demand is more elastic in the monopoly region than in the competitive region. This may appear surprising in the context of traditional (spaceless) microeconomic analysis, but is perfectly reasonable in the differentiated product (spatial) context of this analysis. A monopolist will capture more new consumers from a reduction in price than will a firm subject to competition from neighbouring firms.

Eqs. (5)49) define a family of demand curves for the representative brand, determined by the prices and locations of neighbouring brands. Now assume that total production costs are identical for all brands of the differentiated product and given for the representative brand by

C=F+mq, (12)

where F =fixed production costs and m = marginal production costs. This is a function used extensively in spatial analysis and captures the economies of scale that are assumed to exist in the monopolistically competitive industry. Three symmetric zero profit equilibria (SZPE) can be identified, determined by the set of parameters {f, m, u, c, L}. These are illustrated in fig. 2, where monopoly, kinked and competitive equilibrium prices are denoted by sub- scripts m, k and c respectively.

‘The supercompetitive region of the demand function arises when price of the representative brand is set sufficiently low (at psp,) that the representative brand undercuts the immediately neighbouring brands. We shall ignore this region of the demand function in subsequent analysis.


pad p:pp& p:b*c 0

monopoly L/n 0

kinked L/n 0

competitive L/Xl

Fig. 2. Market equilibrium.

The SZPE satisfies two conditions. Marginal revenue (less than or) equal to marginal cost, and price equal to average cost:

dp P+qdqSm, (13)

p=m+F/q. (14)

Assume that the equilibrium has no gaps. Then sales of each brand in the SZPE are

4 = Lln, (15)

and the various equilibria are given by

Monopoly:

Kinked:

Competitive:

pm = m + c/2n,,

n, = -5 J(cLIF),

v-m=,/(2cF/L), (16)

pk = v - c/2n,, F

,/(2cF/L)<v-m5~~(rFfL),

t nk+c/2nk=v-m,

pC = m + c/n,, n, = ,/(cLIF),

v-mzi,/(cF/L).

(17)

(18)

At a competitive equilibrium, comparative statics are ‘normal’, but at a kinked equilibrium they are ‘perverse’, e.g., at a kinked equilibrium price rises with a reduction in production costs or the ‘value’ (c) of production


differentiation, and with an increase in reservation price (u) or market size

(L).

3. Alternative competitor reactions

An important assumption underlying the derivation of the demand curve in fig. 1 is the Bertrand-Nash assumption that neighbouring firms do not react to a reduction in price of the representative brand, even when their market areas overlap. In examining the implications of relaxing this assumption, we assume that the SZPE will always be either a kinked or competitive equilibrium. Profit for the representative firm is then

W,P,n)=2L(p--m)Z--F=L(p--m)(p--p+c/n)/c-F. (19)

Now let us assume that the representative firm believes that the slope of its neighbours’ reaction functions (at least in the vicinity of the SZPE) is dp/dp=6 5 1. We rule out the possibility of competitive price undercutting (6 > l), and assume identical reactions on the part of the two neighbouring firms. It will also be noted that we mix the concepts of reaction function and conjectural variations. This is acceptable within our essentially static approach; a dynamic version of the model would, of course, have to specify reaction functions more clearly: see also Friedman (1983, ch. 5). Finally, we assume that the SZPE exists. We shall consider in section 6 why non- existence of equilibrium can be a problem in these price-location models.

3.1. 6<1

The first-order condition (13) for the profit maximising price is

p~m+c/n(l-6) (6 # 1). (20)

Assume that (20) holds as an equality, i.e., that we have a competitive equilibrium. Then (20), (14) and (15) give the competitive equilibrium:

PE=m+c/n,(l--6) (6 z l), (21)

n, = J(cL/F( 1 - 6)) (6 # 1). (22)

This equilibrium is illustrated in fig. 3. Price rises and product variety increases the greater is the representative firm’s conjecture with respect to the reactions of its neighbours to a cut in price. This is consistent with the analysis of f.o.b. pricing in spatial competition [Greenhut, Norman and Hung (1987)]. The greater is the conjecture 6 the lower is the elasticity of the


0 L/n

Fig. 3. Competitive equilibrium and competitor reactions.

representative firm’s perceived demand function and so the higher is the equilibrium mill price. Put another way, the more nearly that neighbouring firms are expected to match any price reduction, the less the representative firm will attempt to reduce price in order to expand its market area.

The range of parameter values for which the competitive equilibrium holds is2

v-m>3+F/L(1-6)) (6#1). (23)

It follows that the kinked equilibrium holds when

~~2cFIL)~u-m<~~(c~/L(I-6)) (6#1). (24)

As is to be expected from fig. 3, the greater is the conjecture 6 the more likely is it that a kinked equilibrium will arise.

One extreme case is of some interest. If 6= - 1 then the range (24) collapses to the empty set. This case is, in fact, what has been termed Greenhut-Ohta (GO) competition in the spatial competition literature [Capozza and Van Order (1978)]. The representative firm believes that its neighbours’ prices on the market boundaries are known, fixed values: any

2See Salop (1979, p. 148, fn. 7) for the method by which this range is derived.


0 L/n Fig. 4. SZPE: Greenhut-Ohta competition.

price decrease by the representative firm is expected to give rise to a matching price increase on the part of its neighbours.3

With GO conjectures the kink in the representative firm’s perceived demand function disappears. But competitive entry leading to the SZPE drives this perceived demand function down as in fig. 4 until equilibrium is reached at point E.

3.2. 6 = 1: Liischian competition

The analysis of section 3.1 explicitly excludes the case where 6 = 1. This case has been termed Loschian competition in the analysis of spatial competition: the representative firm believes that its neighbours will exactly match any price cuts in order to retain their existing customers.

With Loschian competition, once price has been reduced to p1 (fig. 1) any further price cut will be exactly matched. Since each consumer is assumed to buy only one unit of the differentiated product it follows that price reductions below p1 will lead to no increase in sales. The supercompetitive region of the demand curve will disappear, leading to the family of demand curves of lig. 5. Demand is perfectly inelastic once the representative brand’s price is reduced to the point at which its market touches that of its neighbours.

It follows, of course, that the kinked equilibrium is the only equilibrium - since the average cost curve cannot be expected to be vertical. In addition, comparative statics will always be perverse at a Loschian equilibrium: a result that is familiar in spatial competition.

3This is, as noted by Capozza and Van Order (1978), a somewhat forced interpretation of GO competition.


AC

L/n Fig. 5. Demand curve: LBschian reaction.

3.3. Consistent conjectures

One question that sections 3.1 and 3.2 leave unanswered is just what might constitute a ‘reasonable’ or ‘consistent’ conjectural variation.4

‘A conjectural variation is consistent if it is equivalent to the optimal response of the other firm at the equilibrium defined by that conjecture.’ [Perry (1982, p. 197).]

‘For the conjectural variation S to be consistent it must be equivalent to (the) local equilibrium response of the other firms at the overall symmetric equilibrium.’ [Perry (1982, p. 199).]

Identification of the consistent conjecture in our free entry monopolistically competitive industry proceeds as follows [Perry (1982)]. Given n equally spaced firms, and any price p charged by the representative firm, we can identify the optimal pricing policy for either of the representative firm’s neighbours as a function of the neighbour’s price (p)), the price p, and the number of firms n. Denote the pricing equation P(p,p,n). In addition we have the zero-profit condition for each firm, generated by setting (19) to zero. This gives the pair of equations:

P(P, P, 4 = 0,

WP, P, 4 = 0. (25)

Differentiated with respect to p,p and n5 gives the system

(26)

4See, for example, Capozza and Van Order (1980), Ulph (1983), Geroski (1981) and Perry (1982) for analysis of consistent conjectures in various settings.

5As usual, n is treated as a continuous variable.


where, for example, P, = 6Pf6p. Solving for dp/dp gives

dp _ $,I& - P,ZZ p dp - P,ll, + P,II, ’

and for the conjectures to be consistent:

(27)

Solution of (27) and (28) (see appendix) gives one stable consistent conjecture:

6 = 1 - ,/( l/2) ~0.293. (29)

As might have been expected, the consistent conjecture is intermediate between the Bertrand-Nash and Loschian conjectures. Consistent conjectures, therefore, slightly increase the possibility of a kinked equilibrium as compared with Bertrand-Nash conjectures: from (24) and (29) this range is now

,/(2cF/L)~u-m~l.610,/(cF/L). (30)

Compare this with the range of eq. (17).]

4. Collusion

The analysis of section 3 identifies a zero-profit equilibrium in the monopolistically competitive industry. One question that then arises is whether the firms that populate the industry in the SZPE could improve their profitability by tacit collusive agreement, even given the free entry condition implicit in the Salop model.

This question has been addressed by MacLeod, Norman and Thisse (1987) in the context of a Capozza-Van Order (1978) model. Their analysis can be applied directly to the Salop model.

MacLeod et al. model a tacit entry deterring collusive (EDC) equilibrium as a two-stage process. The first stage identifies the SZPE of section 3 above, and so identifies the firms that will be party to the collusive agreement. In the second stage the firms in the SZPE can move from the competitive price that characterises the SZPE [eq. (21)] to the optimal Lijschian price FL, for that market area, given by


$

”

0 L/n

Fig. 6. EDC equilibtium.

L/n

pL. = v - cLJn,, (31)

where n, is given by (22). This two-stage process is illustrated in fig. 6. The competitive equilibrium is E and the EDC equilibrium is E’.

Each firm earns excess profit in the EDC equilibrium: leading to the threat of entry by new firms. There is also the possibility common to all collusive agreements that some members of the collusion will try to cheat on their neighbours. MacLeod, Norman and Thisse show, however, that there exist rational and credible threats on the part of incumbents that will deter entry and prevent cheating and so maintain the EDC equilibrium.

All that is needed to deter cheating is that cheating can be easily detected and that non-cheaters can react ‘quickly’ to cheating firms. The first condition is likely to be satisfied in the monopolistically competitive industry since each firm has ‘few’ neighbours. The second condition will be satisfied so long as prices can be changed ‘quickly’ [see MacLeod (1985) for a more precise statement].

Entry deterrence is similarly achieved unless the potential entrant incurs significant sunk costs in entering the market. In this latter eventuality it may well not be rational on the part of incumbents to attempt to drive out the new entrant by competitive price cutting.’

One exception to the profitability of the EDC is worth noting. If the SZPE is a kinked equilibrium, then collusion will leave this equilibrium unchanged and so will not improve the profitability of the incumbent firms: consider, for example, collusion in the context of the Lijschian equilibrium of fig. K8 In a

6With one exception to be discussed below. ‘The same conditions apply as those that deter cheating: easy detection of an entrant and

speedy reaction to the entrant. Since the EDC equilibrium also has implicit in it the assumption of single product firms it is not subject to the criticisms of Judd (1985) who argues convincingly that entry may be all too easy in the multi-product case.

8This has led some authors, e.g. Capozza and Van Order (1978) to view the Liischian equilibrium as a purely collusive equilibrium.


more general sense, the greater the conjectural variation 6 that characterises the initial competitive stage, the lower will be the subsequent returns to collusion.

This should not be particularly surprising. At a kinked equilibrium each firm is behaving as a local monopolist. Market areas just touch and price is set at the optimal monopoly price for the appropriate market area. Nor should it be thought, however, that this is the best possible collusive agreement. Consider, for example, a case in which firms could collude prior to entry: a situation that might arise if firms in an existing market consider developing a new market. The global profit maximising collusive agreement would be characterised by

Pd = v - c/2n,,,,,

ncoll = $ J(cLIF). (32)

Comparison with (15) and (16) indicates that a prior collusive agreement will contain fewer brands than the EDC collusive agreement.

5. Elastic individual demand

One of the limitations of many recent analyses of product differentiation is the assumption that each consumer buys exactly one unit of the differentiated commodity (if, of course, the differentiated commodity is bought at all). It is a simple matter (as Salop indicated) to relax this assumption and consider the case of rather more elastic individual demand. What may not be so obvious, however, is that when this is done the comparative statics of the SZPE can be ‘perverse’ even if the SZPE is not a kinked equilibrium; the qualitative nature of the comparative statics will be determined by the competing firms’ reactions and the precise form of the individual demand function.

Assume that individual demand by a consumer with most preferred brand 1* for output of brand li of the monopolistically competitive industry is

qi(li,I*)=f(pi+Cii) if pi+Cli<pj+Clj for all j#i

and pi + di < V, (33)

=o otherwise,


Table 1

Analogy between geographic space and characteristics space.

Characteristic space equivalent to Geographic space

Consumer density L Product price P Marginal cost m

Value of product Differentiation ‘Distance’ Full price Market radius Fixed costs

c li p+cL, 27 F

Consumer density D Mill price m Marginal cost c

Transport cost Distance Delivered price Market radius Fixed costs

t r m+tr

f”

where u is the reservation price and f’ ~0.~ Rather than continuing to use the Salop approach to analyse this case, we shall apply a technique first developed by Capozza and Van Order (1978) and generalised by Greenhut, Norman and Hung (1987, Ch. 3). In doing so, we need to make more explicit the analogy between geographic space and characteristics space. Table 1 summarises the main elements of that analogy.

Zero-profit equilibrium in the geographic model requires

n(R,m): 2D(m-c) Iff(m+tr)dr-j=O, 0

and the optimal pricing equation islo

W,m): m( l-&)=(X

(34)

(35)

where e,(R,m) is the elasticity of the perceived aggregate demand function with respect to mill price:

R

m J f’(m + tr) dr aR f(m+tR)

e&R,m)= -e-----m-

J f(m + tr) dr amaf( . m + tr) dr 0

(36)

9A linear demand function of the form q(li, I*) = max [a - b(p, + clJ, 0] will arise, for example, if the individual utility function is of the form [Love11 (1970)]: U =(a/b-cli)qi-qf/2b+s and is maximised subject to the budget constraint R =piqi+s, where s denotes consumption of the outside good, and a/b is the reservation price o.

“See Greenhut Norman and Hung (1987) for details. A similar model is discussed in Greenhut et al. (ld85).


(a) less ccmvex than (b) more convex than

P

PL

'BN

PGO

negative exponential

P

PL

PBN

%c

negative exponential

PL

'BN

pGO

0 %L 50

i 0 %L %I ir

'BN 'BN Fig. 7. SZPE - elastic demand.

The term JR/am is the representative firm’s expected radius that will follow any reduction in mill price. determined by the representative firm’s conjectures (6)

increase in market Clearly, aR/am is

regarding the price response of its neighbours. In particular, 6 =0 implies aR/am = - l/24 6 = 1 implies aR/am = 0, and 6 = - 1 implies aR/am = - l/t. Perceived aggregate demand is more elastic and mill price lower the smaller is 6.

Substituting from table 1 gives the appropriate equilibrium conditions for the SZPE in the monopolistically competitive industry:

n(%p): Zip-m) jf(p+ci.,)dl,-F=O, 0

(37)

R&wP(l-&)=o. (38)

The SZPE is illustrated in fig. 7 for two classes of individual demand functions and a number of possible conjectural variations: eq. (37) is represented by the zero-profit locus (ZPL).

Consider first the case in which individual demand is more convex than a negative exponential. l1 Then fig. 7(b) indicates that the comparative static effects of a change in market conditions will be ‘normal’, since the price equations slope upward in the (p,X) plane. An increase in fixed costs or a

“The negative exponential demand function is q=ae -‘I. Demand is more (less) convex than a negative exponential if f”.f>( <)(f’)‘. A linear demand function is less convex than a negative exponential.


2 1

‘BN PL)

4N

(F1 < F2)

13g08. Comparative statics - less convex demand. x

reduction in market size, for example, will move the ZPL to the right and so increase price and reduce product variety.

This is to be contrasted with the less convex case (which includes the linear case). Comparative statics are ‘normal’ for GO conjectures, ‘perverse’ for Laschian conjectures (even though we consider only a non-kinked equilibrium), and may be normal or perverse for intermediate conjectures such as Betrand-Nash (BN) - see fig. 8. In the Bertrand-Nash case, comparative statics will be perverse unless entry to the monopolistically competitive industry is ‘substantial’: such that the ZPL cuts the price equation to the left of the turning point B in the pricing equation. More generally, the pricing equation P, is higher in the (p,X) plane and its turning point lies further to the left the greater is the conjectural variation 6. The more that the representative firm believes its competitors will react to any price change in this less convex case, the higher will be the equilibrium price, the greater will be product variety, and the more likely is it that a decrease in costs or increase in market size will lead to an increase in price.

The analysis of this section and of section 3 can be used to give a standard competitive interpretation of observed apparently non-competitive pricing behaviour in markets such as those for ready-to-eat breakfast cereals and physicians’ services. In the former market, Scherer (1979) noted that

‘(the) average level of prices rose in real terms as a consequence of new product introductions’.

In the latter market, Pauly and Satterthwaite (1980) found that


‘the zero order correlation between physicians per capita and various measure of physician fees tends invariably to be positive’.

Scherer attributed the increase in prices of breakfast cereals to, presumably tacit, collusion on the part of sellers in order to maximise their joint profits. Pauly and Satterthwaite used concepts such as increasing monopoly and additional search costs with increased variety to explain the apparent anomaly in the pricing of physicians’ services.

It is clear from the analysis of sections 4 and 5 that collusion can indeed be expected to lead to an increase in prices. l2 Further, since the collusive equilibrium is Loschian, an increase in product variety can be expected to increase prices, provided, of course, that demand is less convex than a negative exponential. It is equally clear from the analysis of this section and of section 3, however, that price may increase with increased product variety even in the absence of seller collusion or consumer search costs (our model presumes perfectly informed consumers). A kinked equilibrium or costly entry and non-convex demand will generate the same results.r3

It is interesting in this respect that the costs of introducing new brands of breakfast cereal tend to be relatively high - estimated by Scherer at $4.4 million per product in the early 1960s - and that demand appears to be strongly concave - Scherer suggests an indirect demand function P= 1 -0.001Q3. Thus even with Bertrand-Nash conjectures equilibrium is likely to lie on the downward sloping part of the pricing schedule (fig. 7). Greenhut et al. (1985) argue that similar considerations may well apply in the market for physicians’ services.r4

6. Price-location equilibrium and price discrimination

We have assumed throughout this paper that a SZPE exists. Implicit in the analysis is the assumption of relatively costless relocation of brands in the monopolistically competitive industry. It is well known, however, that in this type of model a joint price-location Nash equilibrium can be identified only under restrictive conditions - see Jaskold Gabszewicz and Thisse (1986). Even introducing rather ad hoc additional restrictions such as the ‘no mill price undercutting’ assumption proposed by Eaton (1972) does not, in general, solve the existence problem - MacLeod (1985).

‘*The EDC equilibrium with elastic demand can be identified using the same two-stage process discussed in section 4. It will lie on the Loschian price line. -

I3 See Benson (1985) for a similar argument, and Greenhut et al. (1985) for a more detailed analysis of the application of the techni&es discussed in this paper to the’pricing of physicians’ services.

t4Greenhut et al. (1985) also showed that the same analysis can be applied to any service industry in which the full price paid by the consumer is distinct from the net price received by the producer. But in service activities to which entry is less costly or restricted, e.g., plumbing, television repair, ‘conventional’ price effects can be expected to arise from competitive entry.


If it is assumed that product relocation is costly, the existence of the SZPE is restored.15 Presumably, equilibrium in this case should be modelled as a sequential process, in which case the analysis of Prescott and Visscher (1977) can be applied directly. Products will be located in the monopolistically competitive industry such that no new product can enter profitably between any two existing, neighbouring products.

Where product relocation is relatively costless, non-existence of a joint price-location equilibrium remains a problem unless firms in the monopolistically competitive industry are allowed to price discriminate among consumers. MacLeod, Norman and Thisse (1988) discuss such a discriminatory equilibrium.

For price discrimination to be feasible in the context of product differentiation

‘the firm now supplies customers with a band of differentiated products instead of a single product.. . . Transport cost is no longer interpreted as a utility loss, but as an additional cost incurred by the firm in adapting the basic product to the customers’ requirements.. . so long as product design is under the control of the producer.. . he need not charge the full cost of design change. Discriminatory pricing is feasible.’ [MacLeod et al. (1988, p. 443).]

In order to facilitate comparison with our earlier analysis, assume that the marginal cost of product redesign is constant and equal to the consumers’ valuation of product differentiation. The marginal cost to a producer with basic product (location) I* in supplying a consumer with most preferred brand Ii is then

m+c~li-l*~=m+cl~i. (39)

Equilibrium is modelled as a two-stage game using the Selten (1975) concept of sub-game perfect Nash equilibrium. The Nash equilibrium price schedule that emerges is one identified initially by Hoover (1937), and is illustrated by the heavy line in fig. 9 for the inelastic demand schedule discussed in section 2. The representative firm sells each of its product variants at the reservation price or at its neighbour’s marginal cost, whichever is lower. Thus, producer i supplies all consumers in the region [a,b]. Given that all producers face identical production costs and are equally spaced lli --al = Iii - bl = 1/2n.

Profit to the representative firm is given by

ISEaton and Lipsey (1978), Prescott and Visscher (1977) and Rothschild (1976) discuss the implications of costly product relocation.

48

,Price

G. Norman, Monopolistic competition

I I I I I

'i-1 a

‘i ? _ b

‘i+l

Fig. 9. Price equilibrium - discriminatory p:cing.

&~+cA))dA+j(c,n-2cl)dl 0 f

where 2 and X are as illustrated in fig. 9:

2 = max (0, (m + c/n - u)/c),

X = 1/2n.

(40)

(41)

Two extreme cases can be considered:

(i) Case 1: ii-=0 In this case market parameters are such that firms in the monopolistically

competitive industry are sufftciently closely packed as to eliminate the ‘flat’ portion of the pricing schedule in fig. 9. Profit for the representative firm is from (40) and (41):

II’ = cL/2n2 - F. (42)

The SZPE will, therefore, contain ni firms such that

nj = $ ,/(cL/F). (43)

Comparison with eq. (16) indicates that the discriminatory SZPE will contain


the same number of firms (and so brand ranges) as the non-discriminatory monopoly equilibrium.

The parameter values necessary to give 2 = 0 are such that

.?=O=sm+c/n-050. (44)

Substituting from (43) gives

i=O=w--mzJ(2cFfL). (45)

This is just the condition that must be met in the non-discriminatory case for entry to be feasible and for any monopolistically competitive equilibrium to be established [eqs. (16)-(18)].

(ii) Case 2: i=i=l/2n In this case only the flat portion of the pricing schedule in fig. 9 applies.

Profit for the representative firm is

112 = cL/4n2 - F (46)

from which we derive

n,f = i ,/(cL/F). (47)

This equilibrium holds for parameter values

v -m = J(cF/L). (48)

Eq. (47) is, in fact, the socially optimal number of firms identified by Salop (p. 152), while eq. (48) is the lower bound on the optimality condition necessary for non-negative social surplus.

As might be expected, discriminatory pricing extends the range of parameter values for which production is feasible: a SZPE with O$a 5% will be secured by parameter values such that ,/(cF/L) 2 v-m 5 J(2cF/L). A non- discriminatory SZPE would not be feasible for this parameter range. On the other hand, the discriminatory SZPE contains fewer firms than the non- discriminatory SZPE. The highly competitive nature of the (Nash equilibrium) discriminatory pricing policy imposes quite powerful entry constraints.

The comparative statics of the discriminatory SZPE are ‘normal’. An increase in market size or reduction in fixed costs, for example, will increase


the equilibrium number of firms and lead to some reduction in price. It is worth noting, however, that price reductions are localised: the greater the number of firms in the SZPE the more discriminatory will be the equilibrium pricing schedule.

7. Conclusion

This paper has extended the original Salop analysis of monopolistic competition by incorporating a number of elements that are known to be of central importance in the analysis of spatial competition. In particular, we have shown that the precise specifications of consumer demand and firms conjectural variations carry significant implications for the form and behaviour of equilibrium in monopolistically competitive markets.

Perverse comparative static effects from changes in market conditions are shown to arise both as a result of collusive agreements and as a consequence of not particularly restrictive competitive demand and cost conditions: concave demand and/or significant initial entry costs are sufficient. This should be neither surprising nor disturbing. As we have indicated, applied research in pricing policy has identified a remarkable diversity of behaviour in a wide range of industries. The temptation has been to design an individual ‘explanation’ for each case. This approach involves all sorts of obvious dangers. It also suffers from the more fundamental criticism that the theoretical foundations which give rise to the need for such exceptional explanations may themselves be flawed. In our view this criticism is, indeed, valid. The observed, apparently perverse pricing behaviour is taking place in spatial markets, or markets that can be modelled quite closely using a spatial analogy. The reader should by now have realised that perversity, or more properly diversity, characterises standard, profit maximising behaviour in spatial markets. We can, in other words, use standard (spatial) theory to analyse the apparently perverse observed behaviour.

It is, of course, the case that the introduction of ‘space’ to microeconomic theory leads to a situation in which very few general policy conclusions can be drawn from a simple investigation of market conduct. Once again, this should be neither surprising nor disturbing. It simply implies that policy conclusions cannot be based on simple measurement of a few variables. Detailed study of market structure almost on a case-by-case basis is necessary before, for example, we can claim conduct to be collusive or monopolistic [Norman (1983)].

A final set of conclusions is also interesting in the doubt it casts on conventional policy prescriptions. A major problem with models in which firms are denied the power to discriminate between consumers is that there exists, in general, no price-location equilibrium. Various suggestions have been made regarding ways in which existence can be restored, but there is


always the disturbing feeling that firms in such volatile markets have a strong incentive to collude. Allowing firms the option of discriminating between consumers extends the range of parameter values for which market supply is feasible (as might be expected) and resolves many of the non- existence problems. It is tempting to suggest, in other words, that price discrimination may limit collusion. At the same time price discrimination leads to an equilibrium containing fewer firms than characterises non- discriminatory pricing: in the limit, price discrimination will lead to the socially optimal number of lirms.

Appendix

Assume n firms equally spaced with the representative firm charging price p and neighbouring firms charging price ij. Market radius for the representative firm is

x: p+cx=p+c(l/n-.f),

2 = p - p + c/n)/2c. (‘4.1)

Profit for the representative firm is

I7=2L,(p-m)?,

and the first-order condition for profit maximising price is

j7-p+cc/n+(p-m)(6- l)=O (A.3)

where

6 = dfildp.

Profit for the neighbouring firm(s) is

IT=(P-m)L(l/n-_f)+(jY--m)Li, (A.4)

where this firm’s neighbour is assumed to charge price p, and

i=(p-p+c/n)/2c.

Assume all firms have identical conjectures:

dpjdp = dpldp = 6.

(A.5)

(A.@


The optimal pricing condition on (A.4)

policy for the neighbouring firm is the first-order

P(p,p,n): l/n-x+~+(p-m)(6-l)/c=0.

The zero-profit condition is

I7( p, p, n): 2L( p - VI)% = 0.

(A.7)

(A.8)

From (A.l) and (AS)--(A.8):

P, = 112~; P, = - l/n’;

LIP = 2L(X - (p - m)/2c);

Further, from (A.l) and (A.3):

P, = (6 - 2)/c

II,,= -Up-m)/n2; J7,=qp-m)/c. (A.9)

Z-(p-m)/2c= -(p-m)6/2c, (A.lO)

dP dp=

P,Il, - P,Il,

- Ppzz, + PJ-I, .

Substitute from (A.9) and (A.lO) to give

dP_ dP

-1-26_d

26-6

The consistent conjecture is, therefore, such that

f(d): 262-46+1=o; 6=lkl/J2.

Stability requires [Perry (1982)]

df(6)/d6<0*46-4<0*6<1.

Hence 6 = 1 - l/J2.

(A.1 1)

(A.12)

(A.13)

(A.14)

Benson, B.L., 1985, Increasing product variety and rising prices: From ready-to-eat cereals to physicians’ services, Mimeo. (Montana State University, Bozeman, MT).

Capozza, D.R. and R. van Order, 1978, A generalised model of spatial competition, American Economic Review 68. 896908.


Capozza, D.R. and R. van Order, 1980, On competitive reactions in spatial monopolistic competition, Mimeo. (University of British Columbia, Vancouver, B.C.).

Chamberlin, E.H., 1933, The theory of monopolistic competition (Harvard University Press, Cambridge, MA).

Eaton, B.C., 1972, Spatial competition revisited, Canadian Journal of Economics 5, 268-278. Eaton, B.C. and R.G. Lipsey, 1978, Freedom of entry and existence of pure profit, Economic

Journal 88,455-469. Friedman, J., 1983, Oligopoly theory (Cambridge University Press, Cambridge). Geroski, P.A., 1981, Rational conjectures and entry, Mimeo. (University of Southampton,

Southampton). Greenhut, M.L., C.S. Hung, G. Norman and C.S. Smithson, 1985, An anomaly in the service

industry: The effect of entry on fees, Economic Journal 95, 1699177. Greenhut, M.L., G. Norman and C.S. Hung, 1987, The economics of imperfect competition: A

spatial approach (Cambridge University Press, New York). Hoover, E.M., 1937, Spatial price discrimination, Review of Economic Studies 4, 182-191. Hotelling, H., 1929, Stability in competition, Economic Journal 39, 4147. Jaskold Gabszewicz, J. and J.-F. Thisse, 1986, Spatial competition and the location of firms: in:

J. Jaskold Gabszewicz, J.-F. Thisse, M. Fujita and U. Schweizer, eds., Location theory (Harwood Academic Publishers, Chur).

Judd, K.L., 1985, Credible spatial preemption, Rand Journal of Economics 16, 1533166. Lancaster, K., 1966, A new approach to consumer theory, Journal of Political Economy 74,

132-157. Lancaster, K., 1979, Variety, equity and efficiency (Basil Blackwell, Oxford). Lovell, M.C., 1970, Product differentiation and market structure, Western Economic Journal 8,

12&143. MacLeod, W.B., 1985, On the non-existence of equilibria in differentiated product models,

Regional Science and Urban Economics 15, 245-262. MacLeod. W.B.. G. Norman and J.-F. Thisse, 1988. Price discrimination and equilibrium in

monopolistic competition, International Journal of Industrial Organization 6, 429446. MacLeod, W.B., G. Norman and J.-F. Thisse, 1987, Competition, tacit collusion and free entry,

Economic Journal 97, 189-198. Norman, G., 1983, Spatial pricing with differentiated products, Quarterly Journal of Economics

XCVIII, 291-310. Pauly, M.V. and M.A. Satterthwaite, 1980, The effect of provider supply on price, in: The target

income hypothesis and related issues in Health Manpower Policy (U.S. Department of Health, Education and Welfare, Washington, D.C.).

Perry, M.K., 1982, Oligopoly and consistent conjectural variations, The Bell Journal of Economics 13, 197-205.

Prescott, E.C. and M. Visscher, 1977, Sequential entry among firms with foresight, The Bell Journal of Economics 8, 378-393.

Rothschild, R., 1976, A note on the effect of sequential entry on choice of location, Journal of Industrial Economics 24, 313-320.

Salop, S., 1979, Monopolistic competition with outside goods, The Bell Journal of Economics 10, 141-156.

Scherer, F.M., 1979, The welfare effects of product variety: An application to the ready-to-eat cereals industry, Journal of Industrial Economics 28, 113-134.

Selten, R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4, 25-55.

Ulph, D., 1983, Rational conjectures in the theory of oligopoly, International Journal of Industrial Organization 1, 1-18.

Documents

Monopolistic competition: Some extensions from spatial competition