Variety Competition in Retail Food Markets

Stephen F. Hamilton∗

University of Central Florida

April 12, 2004

Abstract

Retailers generally sell more than one product. This paper considers a modelof retail oligopoly in which firms jointly select the range of products to offer andtheir prices (or alternatively quantities). The retail market equilibrium is robustto the selection of varieties-and-prices or varieties-and-quantities as its strategyspace. In the symmetric equilibrium, the range of product variety offered byeach retailer decreases with retail margins, but product variety also falls withretail entry. Relative to the socially optimal resource allocation, product varietyis undersupplied in both the free-entry and the no-entry case.

JEL Classification:Keywords: Product differentiation; oligopoly; monopolistic competition

∗Department of Economics, P.O. Box 161400, University of Central Florida, Orlando, FL 32816(407) 823-4728, Fax: (407) 823-3269, email: shamilton@bus.ucf.edu.

1 Introduction

Firms in the retail sector generally sell more than one product. One reason for

this is that providing multiple consumption goods at a single retail location reduces

transaction costs between producers and consumers in the exchange of finished goods.

When consumers purchase multiple goods at a time, as in the case of a supermarket,

the ability to engage in multiple transactions at a single retail facility can generate

economies of scope that reduce consumer transportation costs. When consumers

purchase only a single good at a time, as in the case of a shoe store, the ability

to compare across a larger breadth of product varieties can reduce consumer search

costs.

To date, much of the literature on multiproduct firms has centered on the production-

side motivations (see, e.g., Baumol, Panzar, and Willig (1982)). These models tend

to focus on the properties of cost functions and do not consider strategic interac-

tions. Although it has been widely recognized since at least the work of Spence

(1976) that increasing the breadth of consumer products can have similar effects in

utility functions, consumption-side motivations for multiproduct firms have received

remarkably little attention. Instead, models that consider consumer utility effects

from increasing product variety tend to focus on retail environments in which each

firm produces a single brand. This removes from consideration the possibility that

the range of products offered by a retailer can be used as a strategic instrument to

generate store traffic.

This paper considers a model of retail competition in which consumer demand for

retail goods depends on the menu of products offered at each firm. Consumers can

choose to shop at any number of retail locations, and retailers compete to acquire

store traffic by jointly selecting product menus and prices.

The model has two types of product differentiation. First, the product menu a

given retailer is comprised of differentiated brands. This is a necessary condition for

a retailer to sell multiple brands in the market equilibrium. Second, the retailers

1

themselves also differ, even when they sell otherwise identical brands. If this were

not the case, retail margins would be competed away, and retailers would have little

incentive to incur the fixed set-up costs of introducing new brands. Product differen-

tiation has been formulated two ways in the literature: (i) through location models,

which follow Hotelling (1929), Lancaster (1975), and Salop (1979) by viewing prod-

ucts to be differing either in geographic or in characteristic space; and (ii) through

representative consumer models (e.g., Chamberlin (1933), Spence (1976), and Dixit

and Stiglitz (1977)), which consider competition by all brands simultaneously for each

consumer. This paper synthesizes the two approaches. Consumers make discrete

choices regarding which retail store to enter. However, once consumers are inside a

particular retail store, all brands compete for each representative consumer.

An important question in markets with differentiated products is the comparison

between equilibrium product diversity and the socially optimal resource allocation.

When firms sell multiple products, diversity depends not only on the total number

of firms, but also the breadth of products offered by each firm. The model isolates

these two effects and compares the private and social outcome in each dimension.

A small number of papers has examined strategic considerations in a multi-market

oligopoly setting. Raubitschek (1987) uses a CES benefits function to derive product

demands and considers a two-stage process in which a centralized manager for each

firm selects the number of products to offer in the first stage, then each brand is

managed independently by an individual agent in the second stage, as in the stan-

dard monopolistic competition equilibrium. This approach addresses an important

element of strategy between firms in the provision of varieties, but ignores the coor-

dination of pricing decisions across brands within each firm. Moreover, she considers

brand differentiation at a single consumption point, and does not consider hetero-

geneity in retail location. Anderson and de Palma (1992) consider a nested logit

demand model in which the products sold by each firm are closer substitutes for each

other than they are for products sold by rival firms. An attractive feature of the

nested logit demand system is that it yields closed-form solutions for all decision vari-

2

ables. However, logistic demand has the somewhat restrictive feature that aggregate

demand for retail goods is independent of the breadth of retail products available

to consumers. This removes all but the purely strategic “business-stealing” motives

for product introduction, and implies that colluding retailers would never provide

any product variety at all. The present model departs from theirs by formulating

a discrete process for consumer store selection and encompassing a more general de-

mand system that includes an outside alternative to products in the retail category.

Despite these differences, a special case of the model with a logarithmic subutility

function produces qualitatively similar outcomes.

In the next section, the symmetric oligopoly and monopolistically competitive

equilibria are calculated. In Section 3, the model equilibrium in each case is demon-

strated to be robust to the selection of prices or quantities as the strategy space

for retail competition (per brand). The welfare analysis is presented in Section 4.

Section 5 considers the special case of a logarithmic subutility function and derives

closed-form expressions for the equilibrium prices, for the breadth of products offered

by each firm, and, in the free-entry case, for the equilibrium number of firms.

2 A Model of Retail Competition

This section specifies a model of multi-product retail competition with heterogeneous

retailers. Each retailer purchases an (endogenous) number of wholesale products

from a competitive wholesale market and sells the products to consumers in the

retail market. Retail competition is localized in the sense that the retailers differ in

terms of their proximity to consumers, either in physical space or in characteristic

space. For example, consumers may differ in their preferences for customer service,

product layout, cleanliness, and convenience. Following Salop (1979), each retailer

is represented as a point on a circle of unit length. The strategic rivalry between

retailers is to acquire store traffic, which is measured (continuously) as the number of

customers who choose to visit a particular store at the equilibrium ranges of products

and prices.

3

Two types of market equilibrium are studied. The first is a non-cooperative

Nash equilibrium in product breadth, m, and prices, p ∈ (0,m]. The second is a

non-cooperative Nash equilibrium in product breadth, m, and quantities, x ∈ (0,m].For reasons of analytical convenience, the number of products is treated in both cases

as a continuous variable and only symmetric equilibria are examined. In such an

equilibrium, each retailer takes as given the prices p and varietiesm (or the quantities

x and varieties m) chosen by other retailers and selects its own p and m (or x and

m) to maximize profits. The model suppresses from consideration the problem of

retail location choice. Instead, following Salop (1979), it is assumed that whatever

the number of retailers happens to be, they are equally spaced around the circle.

The analysis considers both symmetric retail oligopoly and monopolistically com-

petitive equilibria. The symmetric retail oligopoly equilibrium takes as given the

number of firms. The monopolistically competitive equilibrium endogenously deter-

mines the number of firms by allowing for the possibility of free entry and exit in the

retail sector.

Retailers pay an exogenous and identical wholesale price of w for each prod-

uct purchased in the wholesale market. This abstracts from Ramsey pricing issues.

Moreover, because each wholesale product is independently priced at the wholesale

level, it is not possible for retailers to realize economies of scope in production by

varying the mix of retail goods offered to consumers.

Consumers in the model are identified by points on the circle corresponding to

their most preferred set of retail market characteristics. Consumers are uniformly

distributed around the circle with a constant density per unit length and this density

is assumed to be sufficiently large that consumers at each location can be represented

by an aggregate utility function. This aggregate utility function is separable between

the retail brands, x ∈ (0,m], and all other goods, x0, and is symmetric in the senseof Spence (1976). Specifically,

U(x,m, x0) = u

µZi∈m

xθi

¶+ x0, (1)

4

where 0 < θ ≤ 1 measures the degree of substitutability between brands.1 The

brands are perfect substitutes when θ = 1 and product diversity is more valuable to

consumers for smaller values of θ. Throughout, the sub-utility function, u¡Ri∈m x

θi

¢is

assumed to be strictly concave. This requires 0 < E and 1 − θ + θE > 0, where

E = −(u00/u0) Ri∈m xθi denotes the elasticity of the marginal subutility function (inabsolute terms). In addition, to ensure aggregate demand for all products is non-

decreasing in the number of varieties (m), attention is confined to circumstances in

which E ≤ 1.One interpretation of this formulation is that the utility function of the repre-

sentative consumer in (1) aggregates over several distinct categories of consumers,

each with different preferences for brands. If consumers within each category are

uniformly distributed around the circle, then the benefits function of the representa-

tive consumer at a given point on the circle is identical to that of the representative

consumer at any other point. Thus, individual consumers may differ in their tastes

for retail brands, but, if so, the implicit assumption is that consumers with particular

brand preferences do not agglomerate near particular retailers.

For each brand, inverse demand is given by

pi = θu0(y)xθ−1i , i ∈ (0,m] (2)

where y ≡ Ri∈m xθi and u0(y) ≡ ∂u/∂y, and pi is the retail price of brand i. Equations

(2) implicitly define the demand functions, xi(p,m), i ∈ (0,m], which can be used torecover the indirect utility function of the representative consumer,

v(p,m) = maxxu

µZi∈m

xθi

¶−Zi∈m

pixi. (3)

Before turning to the consumer’s choice of retail store, it is helpful to catalogue a

few results for later use. Notice that the representative consumer maximizes utility

with respect to the consumption of brands, but does not choose the available range

of brands. The retailer selects the number of brands to stock, and this makes m1Use of this utility specification to assess equilibrium product variety is also attributed to Dixit

and Stiglitz (1977).

5

exogenous to consumers. The following lemma states an envelope result for retail

brand choice.

Lemma 1. The effect of product variety on indirect utility is ∂v(p,m)/∂m =¡1−θθ

¢pmxm.

Proof. Differentiate (3) with respect to m and make use of the envelope theorem

to get∂v(p,m)

∂m= u0(y)xθm − pmxm. (4)

To complete the proof, evaluate inverse demand (2) for brand i = m, and make this

substitution into (4). ¥

Consumer utility increases with the range of retail products on offer. This is

because an increase in retail product variety facilitates better matches between con-

sumers and brands.

As will be made clear in a moment, localized retail competition for store traffic

involves an element of monopoly pricing and product selection. In the present setting,

much unlike the formulation in the spatial oligopoly models that follow Hotelling

(1929) and Salop (1979), each consumer has a continuous demand function over

product varieties and prices. The model retains the familiar elements of localized

competition: prices (and varieties) are selected to attract marginal consumers into

the store. The model also produces a familiar element of monopoly behavior: prices

(and varieties) are selected to maximize the acquisition of rents from inframarginal

consumers who are already there. This latter dimension is suppressed in oligopoly

models with localized competition that assume consumers to have unit demand. The

following lemma establishes a useful symmetry result for pricing and variety decisions

by a multi-product monopolist.

Lemma 2. In a multi-product monopoly equilibrium with symmetric prices (pi = p),

the optimality conditions for p and m are the same when symmetry is imposed prior

to maximization or after deriving first-order conditions.

Proof. The monopoly outcome does not depend on whether prices or quantities are

taken to be the strategic instrument. Consider a monopolist that selects symmetric

6

quantities (xi = x) and the number of brands (m). Define the symmetric inverse

demand function to be

ps = θu0(mxθ)xθ−1. (5)

The proof has two parts. For part (i) it is necessary to show that

p+ x∂ps∂x−w = ∂

∂xi

µZi∈m

(pi − w)xi¶.

Consider, first, the left-hand side of this expression. The left-hand side of this

equation reduces to

p+ x∂ps∂x− w = θp(1−E)− w. (6)

Next, substitute the inverse demand system (2) into the right-hand side of the ex-

pression and factor to get

γ ≡Zi∈m

(pi − w)xi = θu0 (y)Zi∈m

xθi − wZi∈m

xi.

Differentiating this with respect to xi yields

θ2u0(y)xθ−1i + θ2u00(y)xθ−1i

µZi∈m

xθi

¶− w.

and the proof of part (i) is complete upon collecting terms.

For part (ii), it is necessary to show that

x

·(p− w) +m∂ps

∂m

¸=

∂γ

∂m.

Proceeding similarly,

x

·(p− w) +m∂ps

∂m

¸= x [p(1−E)− w] = ∂γ

∂m. ¥ (7)

These results are not surprising. A multi-product monopolist selects its range of

product variety and prices in a manner that fully internalizes all cross effects.

Now consider the problem of retail oligopoly. The demand facing each retailer is

derived as follows. Taking the consumer choice of retailer as given for the moment,

and fixing both prices, p, and varieties, m, the representative consumer who chooses

to shop with the retailer selects the consumption bundle, x, to maximize utility

7

in (1). This yields indirect utility in (3). Aggregate demand facing the retailer

depends on the decision made by the representative consumer at each point on the

circle regarding where to shop. Let t denote consumer transportation cost per unit

distance. A consumer who is at distance δ ∈ (0, 1) from the representative retailer

could achieve surplus of v(p,m) − δt by purchasing from that retailer. Suppose

there are n retailers located about the circle. For consumers located on the interval

0 ≤ δ ≤ 1/n between the representative retailer and its nearest neighbor, the surplusavailable by purchasing from the best alternative to the representative retailer is

v(p,m) − t (1/n− δ), where v(p,m) is indirect utility evaluated at the prices and

product varieties of the (n − 1) rivals. Let δ∗ denote the location of the consumer

who is indifferent between these two alternatives. δ∗ solves

v(p,m)− δt = v(p,m)− t (1/n− δ) ,

and doing so yields

δ∗(p,m) =1

2n+1

2t[v(p,m)− v(p,m)] .2 (8)

All consumers located at a distance of δ ≤ δ∗ prefer to shop with the representative

retailer. Notice that the location decision by consumers regarding where to shop in

(8) depends on the indirect utility received from consuming the entire basket of retail

goods. As alluded to above, the consumer’s problem can be viewed as a two-stage

game in which consumers select retail stores to minimize the total cost of procuring

goods in stage 1, then enter stores to shop in stage 2. For the set of consumers

attracted to the representative retailer by its menu of product variety and prices,

the retailer acts as a multi-product monopolist. Nonetheless, retail prices and the

provision of varieties in the oligopoly equilibrium are tempered by the endogeneity

of store traffic. The range of product variety and prices set by a retailer to attract

consumers into the store cannot be raised subsequently upon consumer entry.

2This formula and the ones that follow hold only in the range v(p,m)− t/n < v(p,m) < v(p,m)+t/n. To avoid outcomes where an equilibrium may fail to exist, it is assumed throughout that theseconditions are always met.

8

The aggregate demand for brand i sold by the representative retailer, counting

consumers on either side, is

Xi(p,m) = 2δ∗(p,m)xi(p,m)

where xi(p,m) ≡ argmaxu(x,m) −Ri∈m pixi is demand for brand i. Aggregate

demand facing the representative retailer for its entire menu of brands is X(p,m) =

2δ∗(p,m)Ri∈m xi(p,m). This is simply the product of the number of customers,

2δ∗(p,m), and the demand of each customer for the range of available brands,Ri∈m xi(p,m).

Suppose the retailer pays a wholesale price, w, and a fixed set-up cost, f , to

market an individual brand. Brand-specific fixed costs may include promotional

expenses, inventory costs, staffing expenses, and the opportunity cost of the shelf-

space.3 The total cost to the retailer of selling brand i in the retail market is

thus c(p,m) = wXi(p,m) + f , and the total cost of selling m brands is C(p,m) =

2δ∗(p,m)wRi∈m xi(p,m) +mf .

Profits for the retailer are assumed to be quasiconcave in p and m. A necessary

condition for this is E < 1. In addition, profits are assumed to decline with retail

entry, which requires E > (1− θ)(1−E).4

For the representative retailer, retail profits are

π(p,m) = 2δ∗(p,m)Zi∈m

(pi − w)xi(p,m)−mf. (9)

Dropping arguments for notational convenience, the first-order necessary condition

with respect to pi is

−xit

µZi∈m

(pi − w)xi¶+ 2δ∗

µ∂

∂pi

µZi∈m

(pi −w)xi¶¶

= 0, i ∈ (0,m]3In general, these fixed marketing costs could be modeled as f(m). Here, we assume this function

to be linear, so that the promotional cost for each brand is constant. In the case of a bindingconstraint on retail shelf space, the opportunity cost of allocating shelf space to a given brand willgenerally increase with the number of brands, f 00(m) > 0. Nonetheless, the results in this case donot differ qualitatively from those presented below. The specification f(m) = fm can be viewed asthe opportunity cost of allocating retail shelf space in the long-run.

4When 1 < E, a brand introduction contributes less to a retailer’s sales than the amount cannabal-ized from existing brands, so that increasing the number of brands reduces aggregate demand facingthe retailer. When E < (1− θ)(1−E), equilibrium prices (and profits) increase with retailer entry,which prevents convergence of the model to the free-entry equilibrium. As described below, theseassumptions limit the permissible values of the variety elasticity of demand, εm = −(∂x/∂m)m/x,to the range 1/2 < εm < 1.

9

where use has been made of Roy’s identity. Notice that the strategic interaction

between retailers is subsumed entirely by the term δ∗. In the symmetric retail

equilibrium, pi = p = p and m = m, this condition reduces to

1

n

µx+ (p− w)∂xs

∂p

¶=(p−w)mx2

t, (10)

where use has been made of lemma 2 in deriving the term on the left-hand side.

This condition has an intuitive interpretation. The term x + (p − w)∂x/∂p ismarginal revenue less marginal cost (per brand) for sales to the representative con-

sumer. A monopoly retailer would set this term to zero. Under oligopoly, this

term is positive —the retailer sets the oligopoly price below the monopoly price

level— because the term on the right-hand side of (10) is positive. Monopoly prices,

which serve to extract maximal rent from inframarginal consumers, now serve also

to drive marginal consumers away. In equilibrium, the retailer serves 1/n con-

sumers. A small increase in the price per brand augments the retailer’s profits

on sales made to these inframarginal consumers and provides the marginal private

benefit of (x+ (p− w)∂x/∂p) /n. A small price increase also reduces the num-

ber of consumers who choose to shop with the retailer, ∂v(p,m)/t∂pi = −x/t < 0,

and this decreases the retailer’s sales of all brands by mx2/t and reduces profits by

(p−w)mx2/t. Equation (10) equalizes these margins.The remaining first-order necessary condition for a profit maximum is

pmxmt

µ1− θ

θ

¶µZi∈m

(pi − w)xi¶+ 2δ∗

µ∂

∂m

µZi∈m

(pi − w)xi¶¶− f = 0,

where use has been made of lemma 1 in deriving the first term. In the symmetric

equilibrium, pi = p = p and m = m, this condition is

1

t

µ1− θ

θ

¶(p− w)pmx2 + (p− w)

n

µx+m

∂xs∂m

¶= f. (11)

A small increase in product variety increases utility for the representative consumer,

and this generates additional store traffic of ∂v/t∂m = (1− θ) px/θt. Sales to these

marginal consumers augment profits by ((p− w)mx) (1− θ) px/θt. The second term

10

in (11) is the effect of an additional brand on the rents earned from retail sales made to

inframarginal consumers. An additional brand creates profits of (p−w)x directly bystimulating consumers to purchase it, but a portion of these sales come at the expense

of cannibalizing the retailer’s sales of existing brands, (p−w)m∂x/∂m < 0. In net,

the additional brand increases sales to inframarginal consumers by (x+m∂x/∂m)/n

and augments retail profits by (p−w)(x+m∂x/∂m)/n. The marginal private gain

from introducing an additional brand sums the value to the retailer of increasing sales

to marginal and inframarginal customers, and equation (11) states that this be equal

to the marginal private cost of introducing the brand, f .

The provision of retail product variety has strategic implications under oligopoly.

To see this, notice that a monopoly retailer would set (p−w)(x+m∂x/∂m)/n = f .

Under oligopoly, a larger range of product variety than this is provided, because the

first term in (11) is positive. The proliferation of brands has additional value to the

retailer when doing so increases store traffic.

The price per brand, pe, and number of brands, me, in the symmetric oligopoly

equilibrium is determined by the simultaneous solution of (10) and (11). The sym-

metric, monopolistically competitive equilibrium (pm, mm, nm) is determined by

these two equations and the free-entry condition, which states that profits in (9) are

zero. This equation is given by

m

n((p− w)x− nf) = 0. (12)

2.1 An Invariance Result on Strategy Space

Before examining the retail market equilibrium in greater detail, it is worthwhile

to establish an equivalence result between the outcomes under price- and quantity-

setting forms of oligopoly. To do so, consider the case in which retailers jointly select

the number of brands and the sales level per brand.

Under quantity competition, the representative retailer selects a quantity for

each brand and market-clearing prices are determined as in the Walrasian model.

A consumer who decides to shop with the retailer obtains the net utility level,

11

u¡Ri∈m x

θi

¢ − Ri∈m pixi. Evaluated at the market-clearing prices in (2), this can

be written

bu(x,m) ≡ uµZi∈m

xθi

¶− θu0

µZi∈m

xθi

¶Zi∈m

xθi .

A consumer located at bδ on the interval 0 ≤ bδ ≤ 1/n between the representative

retailer and its nearest neighbor could achieve surplus of bu(x,m)− bδt by purchasingfrom that retailer and bu(x,m)− t³1/n− bδ´ by purchasing from the neighbor, where

bu(x,m) is utility evaluated at the quantities and variety level of the (n − 1) rivals.The location of the consumer who is indifferent between these two alternatives, bδ∗,is given by bδ∗(x,m) = 1

2n+1

2t[bu(x,m)− bu(x,m)] .

Profits for the retailer are

π(x,m) = 2bδ∗ Zi∈m

(pi − w)xi −mf, (13)

where pi = pi(x,m) is the inverse demand function for brand i given by (2).

The first-order necessary condition with respect to xi is

1

t

∂bu(x,m)∂xi

µZi∈m

(pi −w)xi¶+ 2bδ∗µ ∂

∂xi

Zi∈m

(pi − w)xi¶= 0, i ∈ (0,m].

In the symmetric equilibrium, xi = x = x and m = m, use of (2) and lemma 2 gives

1

n

µp+ x

∂ps∂x− w

¶=x

t

∂ps∂x

((p− w)mx) . (14)

The interpretation of (14) is analogous to (10). The term p+x∂p/∂x−w is marginalrevenue less marginal cost for sales to the representative consumer. A monopoly

retailer would set this term equal to zero. Under oligopoly, this term is negative

—sales per brand are higher than in the monopoly case— because the term on the

right-hand side of (14) is negative. Stocking a higher quantity of each brand reduces

the market-clearing retail prices, and this attracts marginal consumers to the store.

A small increase in quantity above the monopoly level reduces the retailer’s rent

acquired from inframarginal consumers by (p+x∂p/∂x−w)/n. Nonetheless, doing sois an attractive proposition to the retailer, because a small increase in quantity raises

12

consumer’s surplus, which attracts marginal consumers to the store, ∂bu(x,m)/∂xi =−x∂p/t∂x > 0, thereby increasing the retailer’s sales of all brands and augmentingretail profits by −x∂p/t∂x ((p− w)mx). Equation (14) states that the marginal

private gain of increasing sales to new customers equal the marginal private cost of

increasing sales to existing customers.

The remaining first-order necessary condition for a profit maximum is

1

t

∂bu(x,m)∂m

µZi∈m

(pi − w)xi¶+ 2bδ∗µ ∂

∂m

Zi∈m

(pi − w)xi¶= f.

In the symmetric equilibrium, xi = x = x and m = m,

x

t

µp

µ1− θ

θ

¶−m∂ps

∂m

¶((p−w)mx) + x

n

µ(p− w) +m∂ps

∂m

¶= f. (15)

Providing an additional brand increases utility for the representative consumer, and

this generates additional store traffic of ∂bu(x,m)/∂m = x[p(1 − θ)/θ −m∂p/∂m]/t

and augments profits by x[p(1 − θ)/θ − m∂p/∂m] ((p− w)mx) /t. Providing an

additional brand also increases retail sales to each of the 1/n inframarginal consumers.

The additional brand increases profits directly by (p − w)x through its sales, butsales of the new brand also crowd out consumption of existing brands, and this

reduces the equilibrium prices, mx∂p/∂m < 0. The marginal private gain from

introducing an additional brand is the sum of the increased profits from marginal

and inframarginal consumers. In (15), this is set equal to the marginal private cost

of brand introduction, f .

Proposition 1. The retail oligopoly equilibrium is robust to the selection of its

strategy space.

Proof. Consider first the outcome under quantity competition. The equilibrium

quantity of sales per brand is found by substituting (6) into (14) to get

(1− θ + θE) (p−w)mnpx+ (θ(1−E)p− w) t = 0.

Next, substitute (7) into (15) to get

(1− θ + θE) (p− w)mnpx2 + ((1−E)p− w) θtx = θfnt.

13

Solving these equilibrium conditions simultaneously yields output per brand, which

is

xe =θfn

(1− θ)w. (16)

Substituting (16) into either equilibrium condition gives a single equation in m:

θ (1− θ + θE) fmn2pe(pe − w) = (1− θ)(w − θ(1−E)pe)tw, (17)

where pe ≡ p(xe,m) denotes inverse demand (5) evaluated at the output level in

(16).

To complete the proof, it is necessary to show that the equilibrium under price

competition involves identical output per brand and an identical number of brands as

arise under quantity competition. To characterize the outcome under price competi-

tion, define the demand elasticities associated with (5) to be εp = −(∂xs/∂p)p/xand εm = −(∂xs/∂m)m/x. Use of the implicit function theorem on (5) gives

εp = 1/(1 − θ + θE) and εm = Eεp.5 Making these substitutions into (10) and

(11) yields, after some manipulation,

(w − θ(1−E)p) t = (1− θ + θE) (p− w)mnpx,

and

(1− θ)(p− w)x [(1− θ + θE) (p−w)mnpx+ θt(1−E)] = θfnt (1− θ + θE) ,

respectively. Equating these conditions gives xe in (16). Substitution of (16) into

either equilibrium condition yields (17), which completes the proof. ¥

Price competition and quantity competition result in equivalent equilibrium out-

comes for retail prices, varieties, and profits. In a single product model with localized

competition, Hamilton (2003) decouples the element of aggregate commitment from

the choices of quantity-setting firms and demonstrates that the oligopoly equilibrium

5The variety-elasticity of demand (per brand), em, is negative, because adding an additionalbrand cannibalizes consumers from existing brands. Nonetheles, it is straightforward to verify thataggregate demand facing the retailer for its entire product range is at least weakly increasing in m(i.e., 1 > em).

14

is impervious to the selection of prices or quantities as the strategic choice variables

of firms. The same result obtains here. The reason is that, although market-clearing

prices are determined at the representative consumer level as in the Walrasian model,

retailers behave as monopolists at this level of aggregation. The oligopoly rivalry

between retailers is to attract marginal consumers into the store, and this can only be

done by offering a menu of products and prices that provide consumers with greater

surplus than they can acquire elsewhere from rival retailers. The oligopoly equilib-

rium is robust to the selection of its strategy space, because it makes no difference

whether this surplus is offered to consumers in terms of prices or quantities.6

3 Comparative Statics

The comparative statics effects properties of the oligopoly and monopolistically com-

petitive equilibria can be derived, in each case, through a two-step process. For

the oligopoly case, implicit differentiation of (17) with respect to the parameters

α = (w, f, t, n) gives the comparative statics effects on the retail provision of product

variety. These values, together with the equilibrium output level in (16), can then be

used to recover the equilibrium price effects. It is assumed in deriving all comparative

statics results that the elasticity of the marginal subutility function, E, is constant.

This is formally equivalent to considering a class of subutility functions with constant

elasticity of substitution, although the magnitude of any effects on E are likely, in

general, to be quite small, particularly for retail product categories characterized by

a large number of brands. The calculation of all comparative statics effects under

oligopoly are provided in the appendix (and the results are presented below in Table

I).

Under monopolistic competition, the comparative statics can be determined from

(10), (11), and the zero profit condition (12) as follows. From (12), output per brand

is x = nf/(p−w). From (10) and (11), output per brand must also satisfy (16), andequating these expressions gives the equilibrium price per brand under monopolistic

6It is straightforward to verify that quantities are strategic complements in this case.

15

competition,

pm = w/θ. (18)

Substitution of this value into either (10) or (11), m can be written as a function of

the number of firms as

m(n) =θEt

(1− θ + θE) fn2. (19)

Use of this expression with (16) in (18) gives a single equation in n:

θ2x(n)θ−1u0(m(n)x(n)θ) = w. (20)

Implicit differentiation of this equation yields the comparative static effects for nm,

whereupon use of these effects in (19) gives the comparative statics effects for mm.

These calculations are derived in the appendix.

Let N = nm denote total product variety in the retail market. The comparative

statics results for both the oligopoly and the monopolistically competitive cases are

presented in Table I.

Table IComparative statics resultsEndogenous variables

Oligopoly Monopolistic competition

p m N p n m NExogenous w + (−, 0)1 − + (0,+)1 (−, 0)1 (−, 0)1variables f (0,+)1 − (−, 0)1 0 (0,+)1 − −

t + − − 0 + − (−, 0)1E + − − 0 + − −n − − − ––— ––— ––— ––—

Notes:1. zero if E = 1

Under oligopoly, total retail output, mx, always moves inversely with the retail

price level. An increase in the cost parameters (w,f , or t) raises equilibrium prices

and reduces aggregate output. An increase in E, which makes demand per product

less elastic, raises equilibrium prices and reduces retail output. Retail entry increases

retail output, but narrows retail margins.

16

The provision of product varieties also tends to be inversely-related to the equilib-

rium prices. There are two reasons for this. First, among inframarginal consumers,

increasing the breadth of retail products reduces demand for individual brands.

Brand proliferation increases aggregate demand for retail products by shifting con-

sumer expenditure into the product category and away from the outside alternative,

but demand per brand nonetheless falls with each brand introduction. Retailers

respond to lower demand per brand by reducing prices. Second, among marginal

consumers, decreasing prices and increasing product variety are now complementary

techniques for raising consumer’s surplus, and so that retailers tend to lower prices

and increase product breadth in conjunction when attempting to steal business from

rivals.

The elasticity of the marginal subutility function, E, plays an important role

in these effects. To see this, consider the effect of an increase in the magnitude

of E. A larger value of E reduces the price-elasticity of demand (per brand), ep,

which provides retailers with an incentive to raise retail margins. A larger value

of E also reduces the variety-elasticity of demand (per brand), em. Because the

introduction of an additional brand contributes to total category demand for the

retailer as x(1 − em), the term 1 − em = (1 − θ)(1 − E)ep measures of the degreeto which an additional cannibalizes sales from existing brands. Larger values of E

reduce a retailer’s incentive to proliferate varieties. In the extreme, when E = 1,

the introduction of an additional brand has no effect on total category demand,

as consumption of the additional brand is exactly offset by the cannibalization of

existing brands. In this case, an increase in the equilibrium retail margin (e.g., in

response to a decrease in the wholesale price level, w) does not affect the retailer’s

incentive to provide more varieties. For smaller values of E, there is some product

cannibalization, but it is not complete, and a decrease in the wholesale price level

increases retail margins and stimulates retailers to increase total sales through the

proliferation of additional brands. Similarly, an increase in the set-up cost per

product,f , which has the direct effect of reducing product variety, has no implication

17

for total retail sales when E = 1. In this case, a decrease in the range of products

offered does not reduce retail sales, and, accordingly, product-specific fixed costs are

unrelated to retail margins.

It is interesting to note that the effect of a decrease in transportation cost and

the effect of an increase in the number of firms are not isomorphic as they are in a

single-product model with localized competition. Here, a decrease in transportation

costs causes retailers to proliferate varieties, whereas retail entry reduces the range

of products on offer. Retail entry reduces the distance between any two adjacent re-

tailers on the circle, which reduces transportation costs (on average). The reduction

in transportation costs draws consumer expenditure into the product category and

away from the outside good, and this tends to increase the breadth of product vari-

eties offered by each firm. The reduction in transportation costs also increases the

competitive pressure on retailers, which reinforces the downward pressure on retail

prices and the upward pressure on variety provision. But retail entry also creates

an additional effect unassociated with the decline in transportation costs. The addi-

tional retailer changes the total level of variety available to consumers. The varieties

offered by the entrant increase aggregate demand in the product category, but de-

mand per retailer decreases (precisely opposite the outcome following a decline in

transportation costs). Retailers respond by reducing the range of products offered

to consumers.

The market provision of variety, Ne, increases with the breadth off products

offered per firm when the number of retailers is fixed. When n increases, the product

range of existing retailers falls, but new varieties are provided by the entrant. In

net, entry reduces total product variety in the market.

Under monopolistic competition, a change in each of the various parameters has

qualitatively similar effects on product breadth per retailer and on total variety pro-

vision as those which occur under oligopoly. When entry is endogenous, the number

of products offered by each retailer is inversely related to the equilibrium number of

firms. This is because the monopolistically competitive price in (18) is independent

18

of all parameters except the wholesale prices. Changes in market conditions that

lead to retail price increases under oligopoly (e.g., higher consumer transportation

costs) now precipitate the entry of new retailers, and entry occurs until the original

retail price level is restored. The entrants increase the product variety available to

consumers, but incumbent firms reduce their product breadth, and the net effect is

that total variety provision decreases.

It is interesting to note that higher retail costs stimulate retail entry. To see why

this is so, notice that the breadth of products provided by incumbent retailers, mm,

decreases as retail costs increase. This reduction in the equilibrium product range

decreases the total set-up costs necessary to introduce a menu of products, mmf ,

which makes retail entry more attractive.

4 Welfare Analysis

Aggregate welfare in the economy is taken to be the sum of consumer surplus and

producer surplus. This is given by

W (x,m,n) = u

µZi∈m

xθi

¶− w

Zi∈m

xi − nmf − t

4n.

The first term is consumer utility gross of transportation costs. The second and

third terms are wholesale costs and total product set-up costs, respectively, and the

final term is consumer transportation costs (the average traveling distance between

consumers and brands is δ = 1/4n).

The socially optimal x, n, and m are defined by the three first-order conditions,

Wxi : θu0(y)xθ−1i = w, i ∈ (0,m] (21)

Wm : u0(y)xθm = wxm + nf, (22)

and

Wn :t

4n2= mf. (23)

In (21), the retail price of each brand is set equal to its wholesale price in the social

allocation. With identical wholesale prices for all brands, this implies symmetric re-

19

tail prices, pi = p, and symmetric sales, xi = x, for all brands. In (22), the marginal

contribution of an additional brand to consumer utility is set equal to the cost of re-

tailing it, which is the sum of the wholesale cost and the fixed set-up cost of providing

the brand at each of the n retail stores. Equation (23) defines the optimal number of

retailers. An additional retailer on the circle reduces consumer transportation costs,

which provides the marginal social benefit of t/4n2. The marginal social cost of an

additional retailer is the fixed set-up cost the retailer must incur to market the social

menu of brands, mf .

The social equilibrium can be characterized as follows. Making use of the symme-

try condition in (21) and (22), these equations can be combined to yield the optimal

output per retailer per brand, which is

x∗

n∗=

θf

(1− θ)w. (24)

Notice that output per retailer per brand in the socially optimal resource allocation

(24) is identical to that which obtains in the market equilibrium (16). This feature

facilitates the comparison of variety provision in the market equilibrium and the

socially optimal resource allocation. The market provides excessive output for each

brand when nm > n∗ and otherwise produces too little output per brand.

4.1 Welfare comparison under oligopoly

In this subsection, the level of product variety in the oligopoly equilibrium is com-

pared with the level that would be socially optimal, given a fixed number of retailers.

Three distinct effects, in general, cause the equilibrium and the optimal allocations

to diverge:

First, an individual retailer does not account for the profit reduction at other

retailers as it increases its product breadth. This is the standard business-stealing

externality. The private benefit of introducing an additional brand is comprised

of rents earned from inframarginal consumers —the additional brand increases the

total retail purchases of each customer— and rents earned from marginal consumers

—greater product variety draws customers into the store. The business-stealing effect

20

refers to the failure of a retailer to account for the lost profit of its rivals when

attracting a marginal consumer to the store. This creates a tendency for the excessive

proliferation of retail brands.

Second, the private benefit of providing an additional brand to inframarginal

consumers generally differs from the social benefit. Each retailer is a monopolist

over its inframarginal customers and, when deciding whether or not to introduce a

new brand, the retailer considers only the marginal contribution to profit, p − w.Yet, providing an additional brand generates consumer surplus as well, p(1−θ)/θ, sothat the social benefit of an additional brand, p/θ − w, exceeds the private benefit.This causes the retailer to undersupply variety.

Third, an additional brand cannibalizes sales from existing brands offered by

the retailer. This shifts demand inward for each product and reduces retail prices.

Unlike the standard outcome of product entry in a Chamberlinian model, the ef-

fect of brand introduction is fully internalized by each retailer over its inframarginal

consumers. Nevertheless, brand introduction creates an externality in the retailers

competition for marginal consumers. Introducing an additional brand reduces the

retailer’s prices, and this causes rival retailers to respond by reducing their retail

prices. The proliferation of brands by a retailer promotes retail price competition.

Retailers ignore this socially beneficial effect of brand introduction, and this causes

product variety to be undersupplied in the market.

To see which effects dominate, it is necessary only to compare retail prices. For a

fixed number of brands in the private and social allocation, ne = n∗ = n, output per

brand in the market equilibrium (16) is identical to that in the social optimum (24).

The comparison between the social optimum and market equilibrium is summarized

by:

Proposition 2. Given an exogenous number of firms, product variety is undersup-

plied in the market equilibrium.

Proof. Non-negativity of retail profits in the market equilibrium implies that the

equilibrium price per brand must satisfy θpe ≥ w. Noting that pe > p∗, it follows

21

immediately by the implicit function theorem on (2) that m∗ > me. ¥

Firms undersupply varieties in the market equilibrium. The intuition for this

result is that retailers in the oligopoly equilibrium exercise a degree of market power

and introducing a smaller number of brands supports higher retail prices. Oligopoly

retailers reduce prices from the monopoly level to attract marginal consumers, but,

as long as the retailer retains a positive share of inframarginal consumers, prices are

not reduced to marginal cost. Similarly, oligopoly retailers increase product variety

to attract marginal consumers, but do not fully exhaust the gains from doing so.

It is worthwhile to contrast this result on the market underprovision of variety

with that of Salop (1979), who finds the market provides excessive diversity when the

range of products offered by each firm is exogenous (i.e., m = 1). Here, when the

number of firms is fixed, the opposite occurs. Variety is underprovided. What re-

mains to be seen is the comparison of outcomes when both variables are endogenously

determined.

4.2 Welfare comparison under monopolistic competition

In this subsection, the market equilibrium and socially optimal resource allocation are

compared in the free-entry case. To the extent that the number of firms differs in the

market allocation from the socially optimal number, retail prices can exceed wholesale

prices in the market equilibrium for two reasons. Monopolistically competitive firms

either provide insufficient product breadth (mm < m∗) or provide too little output

per brand, which, in turn, would imply by (16) that too few retailers exist, nm < n∗.

In Anderson and dePalma (1992), the market underprovides variety. However,

the variety elasticity of demand is unit valued with logistic demand, which minimizes

product diversity in the market. Because the socially optimal level of product diver-

sity is independent of E, moreover, this implies that product variety is more likely

to be undersupplied in the case of logistic demand than for any other specification

of consumer utility. Nonetheless, the comparison between the social optimum and

market equilibrium in the CES case (0 < E ≤ 1) reveals:

22

Proposition 3. For the class of CES subutility functions, the market solution

provides too many firms (nm > n∗) and insufficient product breadth per firm (mm <

m∗). Total product variety is undersupplied (Nm < N∗).

Proof. See the appendix.

The intuition for this result is straightforward. As in Salop (1979), the equi-

librium price under monopolistic competition exceeds the social price, and this pro-

vides the incentive for excessive retail entry. Retail entry places downward pressure

on market prices under oligopoly, so that the monopolistically competitive price,

p = w/θ, can be maintained either by reducing output per brand or by reducing

product breadth. By (16), output per retailer per brand, x/n, is constant, and it fol-

lows that output per brand must rise proportionately with entry. As a consequence,

the range of products offered by each retailer must fall. Because (inverse) demand

is more elastic with respect to output than with respect to product breath, the range

of products offered by each retailer must contract by more than output expands for

price to be maintained. Total product variety always falls as a consequence of retail

entry.

In general, an outcome in which the market provides excessive product variety is

possible only if nm < n∗. In this case, entry is deterred at supernormal retail prices

only if incumbent retailers excessively proliferate products (mm > m∗). However,

product breadth is more attractive in the private market when demand is relatively

inelastic with respect to variety provision (i.e., em is small), and this elasticity is

bounded from zero by the requirement that price must fall with retail entry (1/2 <

em). This limits the value of product breadth to retailers, and, at least in the CES

case, generates an insufficient supply of products in the market.

5 Special Case: Log Subutility

A particularly convenient special case of the model arises when the representative

consumer has a logarithmic subutility function, u(y) = ln y. In this case, E = 1, and

the introduction of an additional brand has no effect on total consumer demand for

23

retail products (i.e., em = 1). Consumption of the additional brand is exactly offset

by the cannibalization of existing brands, so that the total size of the retail market

is independent of the selection of product breadth. Thus, the only motivation for

retail brand introduction in (11) is to acquire marginal consumers, and this facilitates

closed-form solutions to the model.

For the case of logarithmic utility, inverse demand for brand i is given by

pi =θxθ−1iRi∈m x

θi

.

In the symmetric case, this reduces to

p =θ

mx. (25)

and the first-order necessary conditions for a profit maximum are

1

n

µ1−

µp− wp

¶¶=

θ

t

µp−wp

¶, (26)

andθ

mpt(1− θ) (p−w) = f. (27)

Under oligopoly, the equilibrium is given by the solution to (26) and (27). Sub-

stituting (26) into (27) gives

pe =w(θn+ t)

θn, (28)

and

me =θ(1− θ)

(θn+ t)f. (29)

When consumers have logarithmic utility, a decrease in the wholesale price level

has no effect on the retailer’s incentive to provide more varieties and the retail prices

are independent of the per product set-up cost. These effects are also derived by

Anderson and dePalma (1992) for the case of a nested logit demand system. As in

their model, both prices and varieties decrease with the entry. A notable difference

here is that the degree of heterogeneity across retailers (t) and the degree of hetero-

geneity within products (θ) interact with both prices and variety provision. Greater

24

retailer heterogeneity increases retail margins in their model, but has no effect on

product breadth. Here, retail heterogeneity reduces the range of products offered

by each firm, because consumers respond to higher retail prices by shifting consump-

tion from the retail product category to the outside alternative. Retailers respond

by providing fewer brands. Similarly, increased product heterogeneity raises retail

prices by drawing consumption into the product category from the outside good.

Interestingly, product heterogeneity has an ambiguous effect on the range of prod-

ucts offered by retailers. For relatively homogeneous brands (θ ≥ 1/2), retailers

respond to greater brand differentiation by increasing the breadth of retail products,

but for more highly differentiated brands (θ < 1/2), the effect of brand differentia-

tion on product proliferation is unclear. In general, the number of brands offered

by each retailer decreases in response to greater product heterogeneity whenever

(1 − 2θ)t > nθ2, but otherwise increases. The intuition for this is straightforward.

With logarithmic utility, the total quantity of retail goods each customer purchases

is independent of the number of products retailers offer. Providing a larger range

of products serves only to generate store traffic, and the retailer can extract greater

rents from inframarginal consumer only by increasing retail margins. Consequently,

as greater heterogeneity (either between brands or between retailers) raises total cat-

egory demand, retailers respond by increasing retail prices. This creates offsetting

incentives in the competition for marginal consumers. Each consumer attracted to

the store now pays higher retail margins, which makes brand proliferation attractive,

but higher prices dampen sales per brand, which lowers the incentive to pay the fixed

start-up cost of introducing additional brands. When consumer transportation costs

are relatively small, aggregate consumption on the circle is relatively large and, at

the same time, consumers can readily switch between retailers and are more sensitive

to relative prices and varieties. For sufficiently small values of t, retailers respond to

increased product heterogeneity by proliferating brands.

Total variety in the market is proportional to output per firm under oligopoly,

25

and total retail output is

χe =θ2n

w(θn+ t).

Not surprisingly, total retail output decreases both with product heterogeneity and

with retailer heterogeneity. Total output is inversely-related to retail prices, as only

prices —and not product breadth— have consequences for retail sales.

Retail profits are given by

πe =θ(t− (1− θ)n)

n(θn+ t). (30)

Notice that retail profits are independent of industry costs. By (29), total fixed cost,

mef , is constant in the logarithmic case, so that a doubling of f exactly halves the

product range, with no consequence to retail sales. A change in the wholesale price

level reduces total retail sales, but the decrease in sales is exactly compensated in

profits by the increase in retail prices.

Retail profits increase with the degree of retailer heterogeneity, t, but decrease

with the degree of product heterogeneity, θ. For smaller values of θ, products are

highly differentiated, and product variety has greater value to consumers. Retail

margins rise, which reduces retail sales, but nonetheless makes generating store traffic

more desirable. Increasing the range of product varieties is now a more powerful

instrument to generate store traffic, so that retailers tend to respond to a decrease in

θ by proliferating retail goods. Because the proliferation of retail goods serves only

to cannibalize sales of existing brands, profits consequently decline.

The monopolistically competitive equilibrium is derived by equating profits to

zero in (30). This yields the equilibrium number of firms,

nm = t/(1− θ).

The number of firms is independent of the fixed set-up cost per product, f , increasing

in the degree of heterogeneity across firms, and decreasing in the degree of heterogene-

ity across products. It is interesting to note that an increase in E has an equivalent

effect on reducing the price elasticity of demand as a decrease in θ. Nonetheless,

26

for more general specifications of the demand conditions, an increase in E induces

firm entry. The reason for this is that an rise in E increases the variety elasticity

of demand, whereas a decline in θ decreases the variety elasticity of demand. In

general, entry is accommodated to a greater degree in the retail market when the

effect of an entrants brands increase total consumption in the product category, as in

the case of a decrease in θ, rather than arising entirely from brand cannibalization,

as in the case of an increase in E.

Substituting the equilibrium number of firms into (28) and (29), respectively,

yields the equilibrium prices

pm = w/θ,

and varieties

mm = θ(1− θ)2/ft.

The effects of the various parameters on prices and varieties are qualitatively identical

to those which emerge under oligopoly.

Total retail output is given by

χm =θ2

w,

and product diversity in the retail sector is

Nm = θ(1− θ)/f. (31)

The degree of retail product variety in the monopolistic competition equilibrium de-

pends negatively on product-specific fixed costs. The effect of product heterogeneity

is ambiguous. Increased product heterogeneity induces retailer exit, but can generate

offsetting incentives for brand proliferation among incumbent firms.

Now consider the socially optimal resource allocation in the logarithmic case.

When consumers have logarithmic utility, the market demand for each brand is x =

θ/pm. At the social price level in (21), this implies x = θ/mw. Making this

substitution into (24) gives

N∗ = (1− θ)/f. (32)

27

Total product variety in the socially optimal resource allocation (32) exceeds the

amount of variety provided under monopolistic competition (31) in the case of loga-

rithmic utility.

Making use of (32) in (23) identifies the optimal number of firms,

n∗ =t

4(1− θ).

and product variety per firm

m∗ =4(1− θ)2

ft.

Relative to the socially optimal resource allocation, excessive entry occurs in the

monopolistically competitive equilibrium, but the product breadth of each retailer is

too narrow.

6 Conclusion

This paper has considered a retail oligopoly model with localized competition between

retailers for store traffic. Consumers make discrete choices regarding retailers, but

once retail stores are selected, all brands compete for each representative consumer.

A robust oligopoly equilibrium emerged, regardless of whether the model was specified

as Nash in varieties and prices or as Nash in varieties and quantities.

In a single product retail environment, oligopoly firms fail to account for the

detrimental effect of a price reduction on the profit of existing firms. This is the

familiar business stealing externality which reduces prices in the oligopoly equilibrium

below the level that maximizes joint profits. In a multi-product retail environment

with an endogenous number of goods, this externality has a second dimension, as it

is now possible to steal business two ways, by reducing prices or by increasing the

range of products available to consumers. Under oligopoly, retailers seek to generate

store traffic by increasing the breadth of retail products they offer, and this increases

equilibrium product ranges above the level that maximizes joint profits. Nevertheless,

oligopoly firms undersupply product variety vis a vis the social optimum.

28

In the free-entry case, greater entry occurs in markets where the set-up cost of

providing retail products is large. Large product set-up costs increase the cost of

retailing a given brand, but, in equilibrium, incumbent retailers provide a smaller

range of products, and the net effect is to facilitate entry. The free-entry equilibrium

is characterized by excessive retail entry, but insufficient product breadth at each

retailer, and an underprovision of overall product diversity.

7 References

References

[1] Anderson, S. P. and A. de Palma. 1992. Multiproduct firms: A nested logit

approach. Journal of Industrial Economics 40(3), 261-76.

[2] Baumol, W. J., J. C. Panzar, and R. P. Willig. 1982. Contestable Markets and

the Theory of Industrial Structure (San Diego: Harcourt Brace Jovanovich).

[3] Chamberlin, E. H. 1933. The Theory of Monopolistic Competition (Cambridge:

Harvard University Press).

[4] Dixit, A. K. and J. E. Stiglitz. 1977. Monopolistic competition and optimum

product diversity. American Economic Review 67(3), 297-308.

[5] Hamilton, S. F. 2003. Walrasian rent and the equivalence of price- and quantity-

setting outcomes under oligopoly, in manuscript.

[6] Hotelling, H. 1929. Stability in competition. The Economic Journal 37(1),

41-57.

[7] Lancaster, K. 1975. Socially optimal product differentiation. American Eco-

nomic Review 65(4), 567-85.

[8] Raubitschek, R. S. 1987. A model of product proliferation with multiproduct

firms. Journal of Industrial Economics 35(3), 269-79.

29

[9] Salop, S. 1979. Monopolistic competition with outside goods. Bell Journal of

Economics 10(1), 141-56.

[10] Spence, A. M. 1976. Product selection, fixed costs, and monopolistic competi-

tion. Review of Economic Studies 43(2), 217-36.

30

8 Appendix

8.1 Comparative Statics

8.1.1 Oligopoly

To derive the comparative statics effects under oligopoly, substitute (16) into either

optimality condition to get (17), and define me as the solution to

Ω(m) ≡ θ (1− θ + θE) fmn2pe (pe − w)− (1− θ)(w − θ(1−E)pe)tw = 0, (33)

where pe = p(xe,m). Note that this implies (w − θ(1−E)pe) > 0.The comparative statics results for m follow by use of the implicit function theo-

rem on (33). We have, after slight manipulation,

ϕ ≡ ∂Ω

∂p= θ (1− θ + θE) fmn2p+ (1− θ)

w2t

p> 0.

From our earlier definition of the demand elasticities,

∂p

∂m=−pEm

,

∂p

∂x=−px(1− θ +Eθ).

It follows that

∂Ω

∂m=1

m[(1− θ)(1−E)(w − θp)tw − θE (1− θ + θE) fmn2p2] < 0.

where the inequality holds upon noting that π ≥ 0 if and only if θp ≥ w.Proceeding similarly with the exogenous parameters yields

∂Ω

∂w= −θ(1−E)pϕ

w≤ 0,

∂Ω

∂f=− (1− θ + θE)

f

·θfmn2p(w − θ(1−E)p) + (1− θ)

w2t

p

¸< 0,

∂Ω

∂t= −(1− θ)(w − θ(1−E)p)w < 0,∂Ω

∂E=−θ(1− θ)(p− w)wt

1− θ + θE< 0

∂Ω

∂n= −θ (1− θ + θE) fmnp(w − θ(1−E)p)− θ(1− θ)(1−E)(p− w)tw

n< 0.

31

The comparative statics effects on me follow immediately by the implicit function

theorem.

To derive the price effects, write the symmetric (inverse) demand function for

each brand as pe = θu0(m∗xθe)xθ−1e . Differentiating this equation with respect to the

various parameters yields

∂pe∂w

= (1− θ + θE)p

w− pEm

µ∂m

∂w

¶> 0,

∂pe∂f

=−(1−E)(1− θ)2(w − θ(1−E)p)tw

mf (∂Ω/∂m)≥ 0,

∂pe∂t

= −pEm

µ∂m

∂t

¶> 0

∂pe∂E

= −pEm

µ∂m

∂E

¶> 0

∂pe∂n

=(1− θ) (E − (1− θ)(1−E)) (w − θ(1−E)p)ptw

mn (∂Ω/∂m).

Retail entry causes equilibrium prices to decline if and only if E > (1− θ)(1−E).

8.1.2 Monopolistic Competition

To derive the comparative statics under monopolistic competition, define nm as the

solution to (20). Implicit differentiation of this expression gives

∂nm∂w

=θ(1−E)n

[E − (1− θ)(1−E)]w,

∂nm∂f

=(1− θ)(1−E)n

[E − (1− θ)(1−E)] f ,

∂nm∂t

=En

[E − (1− θ)(1−E)] t∂nm∂E

=(1− θ)n

(1− θ + θE) [E − (1− θ)(1−E)] .

Use of these results in (19) gives

∂mm∂w

=−2mn

µ∂nm∂w

¶,

∂mm

∂f=− [E + (1− θ)(1−E)]m[E − (1− θ)(1−E)] f ,

32

∂mm∂t

=− [E + (1− θ)(1−E)]m[E − (1− θ)(1−E)] t

∂mm

∂E=

−(1− θ)m

[E − (1− θ)(1−E)] .

8.2 Proof of Proposition 3

Let E = 1 − σ, where 0 < σ < 1. The condition on E that E > (1 − θ)(1 − E)implies 1 − σ > σ(1 − θ). (Alternatively, 1/2 < em when 1 − 2σ + θσ > 0.) The

equilibrium price in the market is given by the inverse demand function

p = θσmσ−1xθσ−1. (34)

From (19), the number of brands in the monopolistic competition equilibrium is

mm =θ(1− σ)t

(1− θσ) fn2m. (35)

Making use of (35) and (16) in (34), the number of firms in the monopolistic compe-

tition equilibrium solves

θ2σ

µ(1− θ)w

θf

¶1−θσ µ(1− θσ) f

θ(1− σ)t

¶1−σn1−2σ+θσm = w.

Proceeding similarly with the social problem, the socially optimal number of firms

solves

θσ

µ(1− θ)w

θf

¶1−θσ µ4ft

¶1−σ(n∗)1−2σ+θσ = w.

Equating wholesale prices implies

n∗ = θσ

1−2σ+θσ

µ1− θσ

4(1− σ)

¶ 1−σ1−2σ+θσ

nm.

For the first part, notice that 1 − 2σ + θσ = 1 − σ − σ(1 − θ) is positive by the

assumption that prices decline with entry. Suppose nm ≤ n∗. By (8.2), this impliesthat

1 ≤ θσµ1− θσ

4(1− σ)

¶1−σ.

Clearly, this can hold only if 4(1− σ) ≤ 1− θσ. But this implies

3(1− σ) ≤ σ(1− θ),

33

which contradicts the requirement that 1− σ > σ(1− θ). Therefore nm > n∗.

The remainder of the proof is constructed as follows. From (35), the market

provision of variety is

Nm =θ(1− σ)t

(1− θσ) fnm.

From (23), optimal product variety is

N∗ =t

4fn∗= 0.

Comparing these expressions and making use of (8.2), varieties are oversupplied in

the market equilibrium (i.e., Nm ≥ N∗) if and only if

θ1−σ+θσσ(1−θ)

µ1− θσ

4(1− σ)

¶≥ 1.

The first term on the left-hand side takes less than unit value. Hence, Nm ≥ N∗ canonly hold when 4(1−σ) ≤ 1−θσ. But this was shown to contradict above. Therefore,Nm < N

∗. The final result on product breadth per firm follows immediately by the

definition of N = nm. ¥

34