Product choice with economies of scope

Regional Science and Urban Economics 15 (1985) 277-294. North-Holland

PRODUCT CHOICE WITH ECONOMIES OF SCOPE

Simon ANDERSON*

Queen’s University, Kingston, Ont., Canada K7L 3N6

Received June 1984, final version received January 1985

This paper develops a model in which two competing firms on a bounded line each sell two products. Production costs for each firm are lower the closer are its two products. It is shown that there may be anywhere from zero to three possible equilibrium configurations. Equilibrium may entail market segmentation, market interlacing or an intermediate case. The paper contributes to two bodies of literature: (i) Industrial organization. When the line is interpreted as a one-dimensional characteristic space, the model provides an appealing explanation of economies of scope in multi-product firms. (ii) Firm location theory. The model relaxes the unrealistic assumption made in spatial models that each firm has a single outlet and obtains significantly different results.

1. Introduction

Casual empiricism suggests the prevalence of the multi-product firm, yet most of economic theory concerns single product firms. A multi-product firm has several advantages over single product firms. Specifically, it may enjoy economies of joint production and the ability to co-ordinate prices across products. These advantages are greater the more closely related are the firm’s

products. This paper attempts a preliminary analysis of multi-product firms using a

characteristics approach, introducing economies of scope into a Hotelling-

type address model of monopolistic competition.’

*I am indebted to Richard Arnott for extensive suggestions and encouragement. Helpful comments from Ken Lockhart, Louise Czaja and a referee are also gratefully acknowledged. The responsibility for remaining errors is mine alone.

‘Neo-classical value theory defines preferences over available goods. An alternative paradigm, due to Lancaster (1966), defines preferences over the characteristics which are embodied in goods. Although the correspondence is not exact [see for example Archibald, Eaton and Lipsey (1982), henceforth AEL, and Horstman and Slivinski (1982) for discussion], the characteristics approach has been modelled in a spatial framework [see for example Schmalensee (1978)]. Indeed, Hotelling (1929) in the seminal article on the subject, suggested that the bounded geographical line could represent the sweetness of cider, with customers ‘located’ in this characteristics space with respect to their most preferred sweetness, this location being analogous to a geographical address. The term ‘address models’ was first coined by AEL to describe this general approach, encompassing both geographic location and characteristics models. Using the address framework to characterise consumer preferences captures some important aspects of monopolistic competition.

016&0462/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)

278 S. Anderson, Product choice with economies of scope

Recent work by Baumol, Panzar and Willig (1982) (henceforth BPW) has stressed the importance of cost-side considerations as determinants of the range of products offered by a firm. Demand considerations have however been suppressed in their analysis, which involves a non-address framework. The approach taken in this paper provides some insights into the existence of economies of scope and allows analysis of cost and demand interactions as determinants of product location by multi-product firms.

The following example illustrates the class of problem to which the analysis is applicable.

Suppose there is an industry comprised of two motor-cycle manufacturers, each considering production of two model types. The spectrum of possible

motor-cycles can be ranked according to the ratio of horse-power (h.p.) to mileage per gallon (m.p.g.) for each possible model: there is a continuum of possible models. Consumers are differentiated by their most preferred model type as defined by the h.p.1m.p.g. ratio and each consumer purchases the model closest to his ideal type, if indeed there is one sufficiently close. The more distant the model-type offered, the smaller the proportion of consumers purchasing a motor-cycle.2

On the cost side, firms have an advantage in producing a pair of similar models, and this advantage diminishes the more dissimilar are the models. There are several reasons for this. Firstly, two similar models might be

manufactured using similar productive processes, and much of the capital involved in the production run could be used as a joint input, whereas two very dissimilar models might require two distinct assembly lines. Secondly,

the two models may use many of the same components if they are similar, and the firms derive a cost advantage over dissimilar models if such

components can be produced with increasing returns to scale. Other advantages may derive from reduced research and development expenditures. The advantages may also impact on the demand side via more effective advertising as the firm may promote the qualities of the engine.

The problem facing the firms is to locate the model pair in the h.p./m.p.g.

space over which both cost advantages and consumer preferences are defined. In a spatial context, the interpretation of the model involves each firm

operating two outlets or retail stores and there are wholesaling or distribution economies in locating these stores close to each other.

Of particular interest are the resulting equilibrium locations. One possibility is that each firm chooses models such that each model is a closer substitute for a rival model than for its sibling model. A second possibility is that each firm controls a contiguous part of the spectrum. A third alternative

2Here, the demand curve is an aggregate demand curve, summing over a number of consumers at each location where consumers have differing reservation prices. An alternative interpretation involves identical consumers with downward-sloping demand curves.

S. Anderson, Product choice with economies of scope 279

is that one firm ‘straddles’ the other. The first two may be called market

interlacing and market segmentation respectively following Brander and Eaton (1984). The third is a hybrid case. It is noteworthy that the terms interlacing and segmentation are intuitively natural in characteristics space. In the interlacing case the products are located in a real sense between two rivals. The address setting also allows the choice of products from an infinite set of possibilities; the Brander-Eaton analysis assumes only four possible

products. In the next section, the assumptions underlying the formal model are

outlined and discussed. Section 3 presents the analysis of the model and the final section presents the conclusions and suggestions for further research.

2. The model: Assumptions and methodology

2.1. Assumptions

There are two firms in the industry, X and Y. Each firm consists of at most two outlets (the geographic interpretation) or two products (the characteristics interpretation), labelled a and b.

The terms outlet and product will be used interchangeably to denote the

model type. Customers are located along a bounded line of unit length and each unit

interval of the line generates a demand function of the form Q(z) = 2- PJz), where z E [0, l] denotes customer location and Pd(z) denotes delivered price at z. Customers bear transport costs, t per unit distance and firms charge a

parametric mill price which will be normalised to unity. Thus customers purchase from the nearest outlet (if at all) and the

delivered price at a distance z from a given outlet is Pd(z) = 1 + tz. Different parameter values of t are considered in the analysis. Consumers

become less satisfied with the good offered as t increases, and this dissatisfaction is expressed in a lower demand.

Let X,E [O, l] denote the location of X’s first outlet and similarly for Y. The objective is to determine locational equilibria for the two firms.

Firms have zero locational conjectural variations (ZLCV). A pair of outlets may be coincident only if the pair is operated by the same firm, i.e., two opposing outlets may not occupy the same point in space. This precludes sharing of the market. The assumption is not restrictive but greatly simplifies the analysis.

2.2. Costs

The cost structure is the standard one used in address models [see for


example Eaton and Lipsey (1978)] with the addition of a separation cost assumption.3

The novel cost concept introduced in this paper is the separation cost, K,, which is explicitly defined over the distance between two outlets controlled by the same firm. This separation cost can be viewed as a second set-up cost; the cost of setting up at a new location given that the firm is already producing in the market.

The more dissimilar are the two models the higher the costs involved in producing them. The separation cost takes the form: K, = F. (xb-x,) where x, and xr, are the locations of the firm’s two outlets and xt, 2 x,.~

2.3. Discussion

The use of an explicit address model allows simultaneous consideration of cost and demand interdependencies with an infinite number of possible goods. A single firm has a cost advantage over two separate firms if K, > K 1 : this condition implies that there is an economy of scope.5

The behavioural aspects of the model can be analysed in terms of forces of attraction and of repulsion between outlets. The separation cost is a force of attraction between two outlets of the same firm, while the parametric price

acts as a force of attraction between opposing outlets as firms compete for market area. Finally, the fact that transport costs cause revenues to decrease with distance implies that these will be a force of repulsion amongst outlets.

2.4. Methodology

The following notation is used to denote the three basic configurations:

xxyy denotes {x E CO, Y,); y E(x,,, 11);

sFirstly there is the initial set-up cost, K,, which is assumed to be constant across firms and locations. As discussed in Archibald, Eaton and Lipsey (1982), an initial range of decreasing costs - such as is generated by this fixed cost assumption - is crucial to the very existence of firms in a location or characteristics approach. The production of any possible product involves a constant marginal cost, which is again the same across products. Without loss of generality, we can assume marginal cost to be zero by interpreting the price as revenue per unit net of variable costs.

“Thus, this component of set-up costs is a linear function of distance, with K,(O)=O. It is this latter property which will later cause problems for the existence of equilibrium. If we were to introduce the assumption that K,(O) >O, then we should be much more likely to find an equilibrium for any given parameter values. However, the analysis would become correspondingly more complex. Another problem with this cost function is that the new product set-up costs are independent of the initial location. For example, if customers were located (uniformly) over say the space 120, 1201 for horsepower for motorcycles, then it costs the same incrementally to set-up production of a 21 h.p. model given that it is producing one of 20 h.p. as it does to go from 119 to 120.

‘See BPW (1982) for a discussion of this concept.


XYXY denotes {x, E CO, Y,); Y, E ( xa> xb); xb E (Y,> Yb); Yb E cxb, ll)};

XYYX denotes {xa E co, yak Ya c (xa, Yb); yb E (Yay xb)i xb E (Ybr ll);

where 3 denotes the pair (x,,xb) and similarly for y. The three other possible _ orders involve simply relabelling the firms.

The traditional approach to solving the model would use reaction functions; however, the computational difficulties would be considerable.

The analytical technique employed in this paper involves a two-stage approach. The first stage makes use of the concept of a quasi-equilibrium. A quasi-equilibrium is a locational Nash equilibrium where outlets are constrained to lie in a pre-specified order from left to right along the line

segment [0, 11.

Definition 1. A configuration xxyy is said to be a quasi-equilibrium xxyy for given transport cost, t, and separation cost, F, iff

x’: n”($, y’; t, F) 1 TC’(&, y’; t, F) for all x E [0, y,); and _ _

y’: 7cy(x’, y’; t, F) 2 7ty(g’, y; t, F) for all J E(x~, 11,

where nx and 9 are the profit functions of firms X and Y respectively and quasi-equilibrium values are denoted by a prime.

Quasi-equilibria xyxy and xyyx are defined in a similar manner. To eliminate problems arising from the arbitrary selection of starting values, the analysis is restricted to symmetric quasi-equilibria. 6 This ensures a unique quasi- equilibrium for each pair of the parameter values (t, F).

Dejinition 2. A quasi-equilibrium (x’,y’) configuration is said to be a full equilibrium iff

$: ?(x’, y’; t, F) 2 ?(A, y’; t, F) for all {x E CO, 11); and

y’: 7cy(x’, 1’; t, F) 2 TC~(X’, y; t, F) for all {y E [IO, l]}

Profits must also be non-negative for both firms. In order to avoid problems emanating from too high a level of fixed costs, these shall be set equal to zero for the following analysis.

This stage of the analysis requires computation of profits for all possible

61t transpires that this assumption is not restrictive in a number of cases where the market is fully served - the symmetric outcome is the unique one. See the appendix.


relocations. For some parameter values, there are simple arguments to demonstrate non-existence of full equilibrium for a given quasi-equilibrium

configuration. For some others there are simple arguments to show existence. For the ‘grey’ area where we cannot resort to such arguments, a computer

simulation was employed to compare maximum value functions. All three quasi-equilibria are discussed in the text and the simple existence

and non-existence arguments are discussed. The paper gives the full analysis of all three cases in the appendix.

3. Analysis

3.1. Quasi-equilibrium xxyy

The problem facing firm X in this situation (market segmentation) is

max 7+, y; t, F). _

X~KhY*)

P.1)

The’functional form for the profit function is

-(K,$F(x,-xx,)), where

(i) xb 2 x,; (ii) y, 2 xb; (iii) xr 2 0;

and xf is the first point in the market served. Constraints (i) and (ii) ensure

xb~(xa,y,). The formula can be derived with reference to fig. 1. The expression comprises three major components. The first two represent total revenue; it is convenient to separate these into the revenue from a market demand of unity (the first term) minus the loss in revenue due to the elasticity of the demand curve. The last term represents the set-up costs, the second component of which is the separation cost.’

The solution to the problem yields optimal values for firm X’s outlets as a function of firm Y’s location. Solving the symmetric problem for firm Y and finding the restricted Nash location equilibrium yields the quasi-equilibrium values of locations. This analysis is described in the appendix.

7The problem is set up so that there will never be unserved market between xr, and y,; that is, with a spatial monopoly, the internal market boundary is the same point for both firms. This ensures determinate locations for the outlets. The indeterminacy of location values when firms are local monopolies has been noted by Novshek (1980). In the present case, this continuum of equilibria is not a problem as there is no incentive for firms to relocate and there is no entry into the market.

t(xa-Xfl

T REVENUE

S. Anderson, Product choice with economies of scope

I ‘a (X) a

2

283

P=l

- DISTANCE -

Fig. 1. Derivation of the profit function for the market segmentation case, xxyy.

The following notation describes this quasi-equilibrium:

if xb = x,, the situation is labelled p for (self-) pairing;

if xb=ya, the situation is labelled c for clustering;8

if xf =O, the situation is labelled s for served market.

When these equalities do not hold, then a prime denotes the complement

to a label. Thus p’c’s’ describes the situation where outlets are not paired, not clustered and some part of the market is unserved, i.e.,

3.2. Quasi-equilibrium xyxy

This quasi-equilibrium location corresponds to market interlacing. Outlets alternate in ownership. The problem facing firm X is

max 7% y; 4 F), x, E co, y,); xb E (Ym yb)* P.2)

The functional form for this problem is given in the appendix and once again the quasi-equilibrium values and conditions are derived. The possible

sNotice that x,, is not strictly equal to y.. As we have specified that x~E[x,, y,), then a strict interpretation is x,=lim,,, y,-e. Two outlets may not occupy the same point in space. We shall continue to use the equality in order to economise on notation; the meaning should be clear from the context.


cases are again labelled in terms of whether or not firms are clustered with

each other and whether or not the market is fully served. A situation where the internal outlets, y, and xb, are clustered is denoted

by the label ci.

A situation where the external outlets, x, and y,, are clustered with their neighbours is denoted by the label c,.

3.3. Quasi-equilibrium xyyx

This quasi-equilibrium case involves both market segmentation and market interlacing. Firm Y serves a contiguous segment of the market whereas firm X does not. The present framework of analysis allows consideration of this case which is not analysed in the non-address framework of Brander and Eaton.’ In quasi-equilibrium xyyx, the firms face different circumstances. Firm X faces problem P.3

x, E co, y,); xb E (.h,> ll, (P.3)

whereas firm Y faces problem P.4

max 7%, Y; 4 F), (Ya, Yb) E (xm Xb). (P.4) _

The analysis required to solve these problems and the resulting quasi- equilibrium values are again provided in the appendix. Once more some descriptive notation is given:

A label p denotes (self-)pairing of the interior outlets, which are operated by firm Y, and the symmetry assumption implies the outlet is located at y =9. A label c, denotes the clustering of the external X outlets with the Y ones.

A label s denotes served markets as before (note that unserved markets may exist only to the outside of firm X as its outlets are both external) and a prime denotes the complement to a label.

3.4. Full equilibria

It is straightforward to show that any situation involving clustering - where one or more X outlets are located next to an Y outlet, a situation labelled c, c, or ci in the above notation - cannot be a full equilibrium. In each case a simple relocation by one firm will necessarily increase profits.

Likewise, if one firm locates both outlets together (a situation p) then, if all

91t would be possible to adapt the Brander-Eaton analysis to deal with this case by appropriate redefinition of the degree of substitutability between goods.


market is served (situation s), one firm can usurp the other’s whole market

and retain some of its own original market by locating so as to sandwich the first firm’s outlets.

There are also some simple existence arguments. If F =0 (no cost to separation of outlets), then if t> 8 each outlet attains a local monopoly position and there are no benefits to relocating.

Full equilibrium existence is also ensured in quasi-equilibrium xxyy if

some part of the market is unserved: once again each firm enjoys a local monopoly. In configuration’s xyxy and xyyx the existence of unserved markets (situation s’) does not ensure full equilibrium for F>O as outlets are necessarily brought into competition with each other because of the force of attraction of the separation cost.

Residually these are the following cases for which no simple arguments can be made:

(i) for xxyy: p’c’s;

for xyxy: c&is; c#; (iii) for xyyx: p’cks; p’c:s’.

For these cases a computer simulation was run. In each case the maximum value of profits was determined at the quasi-equilibrium values and compared to the maximum profit attainable by relocation of one firm into each possible alternative configuration.

The results of this process are shown in fig. 2, where the axes are labelled Tdzf I/t and f dzf F/t. As the figure shows, there are parameter values for which a full equilibrium attains in configurations xyxy and xyyx but do not constitute a full equilibrium in xxyy; any pair not constituting a full equilibrium in xyxy necessarily is not a full equilibrium in xyyx. Full equilibrium xyxy dominates xyyx, and there is a set of points for which all three configurations are full equilibria. Notice that there is always a full equilibrium in configuration xxyy for t 24 (or T 54) regardless of F. For

high values of F, each firm will choose to operate both possible outlets at the same point in space, and t is suficientl~ large that the firms are not in competition with each other - there is local monopoly with one outlet per firm.

3.5. Asymmetric fill equilibria

The above analysis considers only symmetric full equilibria. By appropriate choice of starting values, it may be possible for values of (x, y) other than the ones given to attain in full equilibrium. This is only possible for a full equilibrium involving unserved markets as any case s which in a full equilibrium is the unique symmetric quasi-equilibrium. It is therefore


I unn t DENOTES FULL EQUILIBRIUM XYXY IN CONFIGURATIONS

I/48 c BXYYX J

I I I 0 l/8 l/4 3/B

f

Fig. 2. The set of full equilibria.

noted that the full equilibria given for case ckc$’ in xyxy and case p’cks’ in xyyx are full equilibria at symmetric locations; other starting values (x, y) will be associated with a different set of parameter values which will sustain-them as full equilibria. It can also be shown that all quasi-equilibria of types ckcfs’ and p’cks’ are unique when locations are expressed in terms of the difference from xr.

The following properties of the model are also noted:

Property 1. Any situation where one firm operates two outlets and the other operates one cannot be a full equilibrium.

Proof: The two possible configurations are xxy and xyx. The two quasi- equilibrium cases which could generate such a scenario are xxyy and xyyx. In the first case the full equilibrium is symmetric for a case s, and for a case


s’, both firms must be equal as they are local monopolies. For a quasi- equilibrium xyyx it is not possible for firm Y to operate just one outlet in a full equilibrium. 0

Property 2. A full equilibrium with two plants operating in the market only attains a configuration xxyy.

Proof: Configuration xxyy is the only one possible for a situation xy where each firm locates both outlets at the same point. This occurs in case PC’S’, i.e., when each firm constitutes a local monopoly and F is sufficiently high. A

local monopoly always constitutes a full equilibrium. 0

3.6. Existence and non-existence offull equilibrium with a circular market

Non-existence of equilibrium in location models is sometimes attributable to the assumption of a bounded market. Eaton and Lipsey (1978) consider firm location along an infinite line and demonstrate existence. However, once we consider a bounded line in the same framework, equilibrium may no longer exist. This non-existence is attributable to the different market conditions facing firms locating nearest the boundary.

It can be shown that existence or non-existence is again dependent on

parameter values and once again there exist multiple full equilibrium configurations. See Anderson (1984) for details.

3.7. Discussion

The model presented in this paper has analysed a special case of a specific linear demand curve, constant consumer density, duopoly and fixed prices.

Two parameters were varied in the model, transport costs and the separation cost. There are certain parameter values for which all three quasi-equilibria are full equilibria and the market is fully served. These cases are interesting as locations are unique in each quasi-equilibrium. The resulting locational patterns can then be compared. It transpires that these patterns are different in each case. Furthermore, this is true even when F = 0. Thus the ordering of firms through space makes a difference as regards equilibrium locations of

outlets when each firm owns more than one outlet. For instance, the location of the first firm in the market, x,, is furthest

from the market boundary in configuration xyxy and closest in xyyx with the xxyy case falling between the two. This result is to be expected as there are different forces of attraction and repulsion between outlets at work in each configuration. This result is then more general than for the specific example presented here.

The key feature of the model is a revenue function which decreases with


distance from an outlet (see fig. 1). This is a general result in a spatial context as long as demand is not completely inelastic and transport costs are not zero. The higher the demand elasticity the quicker revenues fall off over distance. The same is true of higher transport costs and both elasticity and transport costs are seen as forces of repulsion.

Brander and Eaton (1984) also have a problem of multiple configuration equilibria which they were able to resolve by using the notion of a joint Stackelberg leader equilibrium concept. In the present context, such a concept cannot be employed to select one of the full equilibrium configurations where more than one exists as product selection by firms is here a continuous variable.

4. Conclusions

This paper has provided a framework of analysis for competing multi- product firms with economies of scope in an address model. The address framework provides a natural explanation for cost advantages of producing similar products. The notion of similarity can be defined with reference to the

characteristics embodied in the goods, and the more similar the goods produced with respect to their characteristics the greater the cost economies of joint production. This cost economy was characterised in this paper as a linear function dependent on the distance between two products in characteristics space. A clear extension for future research is to use a more general functional form with costs depending on product locations as well as distance between them.

Given exogenous prices, it was shown that there are several possible outcomes to the Nash location game; which occurs depends on the parameter values of the cost economy and consumer preferences. For some parameter pairs the result is non-existence of equilibrium. Non-existence results appear to be a generic problem with simple spatial models, especially when the equilibrium involves prices and location. For other parameter values there are multiple equilibria. The firms may each provide a pair of similar products (market segmentation), each may offer products more similar to its rival’s products than to its own (market interlacing), or a hybrid case is possible (configuration xyyx).

To calculate an equilibrium, reaction functions were solved for a given configuration and then profits compared to those attainable in a different configuration given the Nash assumption on location. This methodology circumvented the need to compute a reaction function of locations (x,,xJ for any possible pair (y,, yb). Despite this, determination of full equilibria still necessitated the use of a computer simulation.

Brander and Eaton (1984) analysed a similar problem of duopoly product location, without cost economies, in a discrete space (a non-address model).


In this model, the choice of configuration was treated as a sub-game perfect equilibrium. In the present model, with prices exogenous, such a solution

concept does not represent plausible firm behaviour. An important research task as regards location models in general is to’ find

a satisfactory way of dealing with the non-existence of a price equilibrium [see d’Aspremont et al. (1979) for a demonstration of non-existence in the Hotelling (1929) model]. Possible solutions suggested to deal with the problem include a doctored Nash equilibrium concept [see Novshek (1980) and Eaton and Lipsey (1978)]; a mixed strategies approach [see Gal-Or (1982)] and uncertainty in consumer demand [see de Palma et al. (1982)].

With an acceptable and tractable solution to the price equilibrium problem many pertinent issues could be addressed. The main question

concerns the social optimality of the market outcome. A market segmentation equilibrium involves cost saving due to economies of joint production, yet firms have some degree of local monopoly power. Interlacing involves greater cost and perhaps less product diversity, yet there is more competitive pressure on prices.

In the present model, the social optimum locations involves firms placed symmetrically around the quartiles in configuration xxyy, yet the market outcome is only compatible with this if (t, F) = (8,O). This sort of behaviour suggests a clear role for government intervention. A policy instrument with a natural characteristic space interpretation is the patent. This may be used to determine how ‘close’ firms may get to each other and mitigate the effects of destructive competition if such is the market outcome. There are clearly many topics in this area for future research.

Appendix

This appendix provides the functional forms and solutions to the problems in the text. The values and conditions for each of the quasi-equilibria are also given. For further details, see Anderson (1984).

A.]. Quasi-equilibrium xxyy

As given in the text, the functional form of firm X’s profit function for problem P.1 is

-_(KO+F’(xb-xx,)), where

(i) xbz.& (ii) y,z&; (iii) x,20.


Although xr, the first point in the market served by firm X, is determined by choice of x,, it is convenient (and equivalent in terms of the analysis) to treat it as a choice variable.

Let k,, k,, k3 be the non-negative Kuhn-Tucker multipliers associated with the above constraints. The first-order conditions are as follows:

drc’/dx, = F - tx, + tx, - k, + t(x, - x,)/2 = 0, (A.1)

d?/dx,, = l/2 - F - t(x, -x,)/2 + t(y, - x,,)/4 + k, - k2 = 0, (A.3

d?/dxr = - 1 + fx, - tx, + k, = 0, (A.3)

where k,(x, - xJ = 0; k,(y, - xb) = 0; k,(xf) = 0. There are 8 possible constellations of values for the ki. However, it can be

shown that k, =O=z-k, = 0, and thus there are 6 constellations of relevance. The equations can then be solved for the multiplier values. The resulting system depends on y,. To solve for the symmetric quasi-equilibrium xxyy, substitute in y, = 1 -x,, and y, = 1 -x,. This process yields table 1.

For any particular case to hold for given parameter values, the conditions 1, 2 and 3 must hold as given. It is also necessary that no preceding case in the table holds. As mentioned in the text, only cases p’c’s, PC’S and p’c’s’ are candidates for full equilibrium. In the first of these, the symmetric quasi- equilibrium is the unique quasi-equilibrium.

Case

PCS

X,

Xb

Table 1

The quasi-equilibrium xxyy.

Xb Xf Cond. 1 Cond. 2 Cond. 3

f 0 1>t 2>t

PC’S Xb 6F>2+t - 4>t

P’CS (@+t) 1

6t 2 3-4F>t 6-4F>t

p’c’s (4F+2+t) (6-8F+3t) o

lot lot 8-4F>t

PC’S’ (t-2) (t-4)

Xb 2t 2t

F>l

p’c’s’ (t-6+4F) (t-2) (t-8+4F)

2t 2t 2t


A.2. Quasi-equilibrium xyxy

Here the solution is required to problem P.2:

max r@, y; 4 F), x, E CO> Y,), x,E(Y,,Yb). (P.2)

This problem can be simplified as the optimal location of each outlet is independent of the location of the other, given the pre-specified ordering of outlets and the linear separation cost. It is analysed as two separate sub-

problems.

Problem P.2a

max 7% y; t, F), x, E CO, Y,), given xb E (y,, 11. (P.2a)

The relevant part of the firm’s problem as regards optimal choice of x, is as follows:

max (x,+Ya-2xf)_P(Xb_x )_t

2 a

(x.-xJZ+(Y,-xd2

(x.3 Xf) 2 1 8 ’ subject to (i) x,20; (ii) y, 2 x,.

Let k, and k, be the Kuhn-Tucker multipliers associated with the above constraints. Note that when k, =O, then we have unserved markets, a situation which, following the text, we shall label s’.

The first-order conditions are

d? -=-l+tx,-tx,+k,=O, dx,

64.4)

drcX -=FF+t-tx,+tx,+ 4 dxa

t(y,-xx,) _k =O 2 )

where k,(x,) = 0 and k,(y, -x,) = 0.

Problem P.2b

max 7@, Y; W), xb E (h ybh given xa E co? Ya). -

The functional form is

max (2xl-xb--L’.)_F~x

2 _x l_t

b a &.x11

64.5)

(P.2b)

subject to (i) x,+y,-2x,20; (ii) x,-y,zO.


Let k, and k, be the Kuhn-Tucker multipliers associated with the above constraints. Let x1 denote the last point in the market served by X.

The first-order conditions are

drcX ---=l+tx,-Lx,-2k,=O, dx,

drc” _=_ dx,

F-$-tx,+tx,-t(x,-yJ/4+k,+k,=O,

64.6)

(A.7)

where k,(x, + y, - 2x,) = 0 and k,(x, - y,) = 0.

Using the symmetry assumption, which in this case is y,= 1 -xb and y, = 1 - xa, the quasi-equilibrium values and conditions are given in table 2.

Table 2

Case Conditions

C&S 1

;4+t+8F) c,qs

lot

f

f

I , (16F+t+6) (9t-24F-2)

c,qs 14t 14t

c,c+ f

I I (-4+t+SF) C&S

2t

f

t

I I, (16F+t-6) C&S

2t

t-8F+2)

2t

2>t;2F+l>t

6-8F>t;8>16F+t

6>12F-t

F>1/2

4F>l

None

Once again for any case to hold, the conditions given must hold and furthermore no preceding case in the table must hold. For c$C,s (the only case where there can be a full equilibrium with served markets) the symmetric quasi-equilibrium is again unique.

A.3. Quasi-equilibrium xyyx

In this case the firms face different circumstances. Firm X faces problem P.3 as given in the text. This problem can be analysed using the algebra from problem P.2a above.


Firm Y faces problem P.4. This will be analysed (to economise on notation) as if an X firm were located between two Y outlets, y, and y,. The problem can be sub-divided into two sub-problems according to whether the market interval is fully served or not. The latter case involves local monopoly and problem P.4a,

max (x1-xr)-F(xI,-x,) (Xa.Xb.X,)

_-t 2(x,-X,)2+(Xb-X,)*+2(X1-Xb)~

il 4

subject to (i) x,-x,20.

Let k be the Kuhn-Tucker multiplier associated with

Let x1 denote the last point in the market served by first point in the market served by X, which is fixed order conditions to the problem are:

drc” --~l++x,-~x,=o, dx,

dn” -= -F-tx,+tx,+t(x,-Xx,)/2-k=O, dXa

(A.9)

dn” ---= ---tx,+tx,-t(x,-xx,)/2+k=0, dX,

(A.lO)

where k(x, -x,) = 0.

(P.4a)

the above constraint. X, and xr denote the arbitrarily. The first-

(A.8)

When the firm is not a local monopoly, it faces problem P.4b

max (xb+Yb-xa-YY,) _Ftx

2 _xl

b a &,.x.$

(Yb-Xb)2+(&-xa)2+2(Xb-X,)2

8 (P.4b)

subject to (i) x,-y, 2 0, (ii) yb - .q, 2 0, (iii) xb - x, 2 0.

Let k,, k,, k, be the Kuhn-Tucker multipliers associated with the above constraints. The first-order conditions to problem P.4b are

drcX --=F-$+t(X,-Xx,)/2-t(x,-yy,)/4+k,-kk,=O, dxa

(A.ll)


d7cX ---=-F+3-t(xb-x,)/2+t(yb-xb)/4-k2+k3=0, dx,

(A.12)

where k,(x,-y,)=O; k,(y,--xJ=O; k,(x,-x,)=0. It transpires that k, = k,.

Invoking the symmetry assumption, in this case xa= I- xb and y,= 1 -y, yields table 3 as the solution to quasi-equilibrium xyyx.

The case in the table where there is a continuum of equilibria is not important as this case can never be a full equilibrium. The case p’cks is a unique quasi-equilibrium.

Table 3

Case Xb Conditions

pc,s

p’c,s

p’c’,s

f f 2>t;2F+l>t

Continuum of equilibria 2>t

(12F+t+4) (5t+l2F-4)

12t 12t 8>12F+t

P&S

p’cbs’

f I F>1/2

(6F - 3 + t/2) ;2t.- 1 + t/2) None

t t

References

Anderson, S.P., 1984, A duopoly model of endogenous product choice with economies of scope, Discussion paper 551 (Queen’s University, Kingston).

Archibald, C.G., B.C. Eaton and R.G. Lipsey, 1982, Address models of value theory, Discussion paper 495 (Queen’s University, Kingston).

d’aspremont, C., J.J. Gabszewicz and J.-F. Thisse, 1979, On Hotelling’s ‘stability in competition’, Econometrica 47, 1145-l 150.

Baumol, W.J., J.C. Panzar and R.D. Willig, 1982, Contestable markets and the theory of market structure (Harcourt-Brace-Jovanovich, New York).

Brander, J. and J. Eaton, 1984, Product line rivalry, American Economic Review 74, 323-334. de Palma, A., V. Ginsburgh, Y.Y. Papageorgiou and J.-F. Thisse, 1982, The principle of

minimum differentiation holds under sufficient heterogeneity, CORE discussion paper no. 8339 (Universitk Catholique de Louvain) and Econometrica, forthcoming.

Eaton, B.C. and R.G. Lipsey, 1978, Freedom of entry and the existence of pure profit, The Economic Journal 88,455469.

Gal-Or, E., 1982, Hotelling’s spatial competition as a model of sales, Economics Letters 9, l-6. Horstman, I. and A. Slivinski, 1982, The foundations of location models of product

differentiation Mimeo (University of Western Ontario, London). Hotelling, H., 1929, Stability in competition, The Economic Journal 39, 41-57. Lancaster, K., 1966, A new approach to consumer theory, Journal of Political Economy 76, 131-

159. Novshek, W., 1980, Equilibrium in simple spatial (or differentiated product) models, Journal of

Economic Theory 22, 3 13-326. Schmalensee, R., 1978, Entry deterrence in the ready-to-eat breakfast cereal industry, The Bell

Journal of Economics 9. 305-327.

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Product choice with economies of scope

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