quality7lasteee.PDFProduct Diversity in Asymmetric Oligopoly: Is
the Quality of Consumer Goods too Low?¤
Simon P. Andersonyand André de Palmaz
June 2000
Abstract
We analyze an oligopoly model with horizontal di¤erentiation and
quality di¤erences. High quality goods are over-priced and
under-produced. When the market is fairly covered, low quality
products may be pro…table when their social contribution is
negative, leading to too many products in equilibrium. In a
relatively uncovered market, even low quality goods are
under-produced and there may be too few entrants. However, when
…xed costs di¤er across qualities, the market may produce low
quality goods when it should produce high quality ones. The model
is calibrated using market data for yoghurt.
JEL Classi…cation numbers: L11, L13, D43 Key words: Product
di¤erentiation, asymmetric oligopoly, quality, optimum and
equilibrium variety, discrete choice models.
¤We thank two referees and the Editor, Luis Cabral, for helpful
comments that greatly improved the paper. The …rst author
gratefully acknowledges the support of the National Science
Foundation CSBR-9617784.
yDepartment of Economics, University of Virginia, Charlottesville,
VA 22903, USA (
[email protected]).
zThema, Université de Cergy-Pontoise, 33 Bd. du Port, 95100
Cergy-Pontoise, Cedex, France.
1
Firms clearly di¤er in size but theoretical results for asymmetric
price-setting oligopoly are few. Firm costs, abilities and chance
are all factors that lead to asymmetry. Our objective in this paper
is to contribute to the positive and normative economics of
asymmetric price-setting oligopolies.
Most of the theory of market failure under imperfect competition
has ad- dressed the (Chamberlinian) issue of market provision of
variety with symmetric …rms (see especially Spence, 1976, and Dixit
and Stiglitz, 1977). However, asym- metries engender market biases
even for a given number of …rms: production levels can be
suboptimal both in relative and in absolute terms.1 Assuming sym-
metry from the outset shut downs the possibility of relative
production biases, although absolute biases still typically exist
insofar as imperfectly competitive …rms tend to under-produce due
to their market power. Asymmetries also have consequences for long
run performance when the set of produced goods is de- termined
endogenously. By contrast, under symmetry, the only performance
relevant statistic is the number of varieties.
We introduce asymmetries by supposing that …rms produce goods of
(possi- bly) di¤erent qualities and of di¤erent marginal production
costs. In the short run we treat these qualities and costs as …xed.
For the long run analysis, we consider two variants of the model
regarding equilibrium quality determination. In the …rst variant,
each potential …rm is endowed with an immutable triplet (quality,
marginal cost, …xed cost) and each must decide whether to enter the
market. In the second variant, …rms choose a triplet from a …nite
set of di¤er- ent alternatives, but only one …rm can choose any
given triplet.2 These variants correspond to restaurants owned by
chefs of given abilities and restaurants that must choose a chef
from a …nite set of available chefs.3 Which variant is more
appropriate depends on the industry studied. As we will show, both
variants yield equivalent outcomes.
The market and optimal outcomes depend intricately on the menu of
avail- able qualities and costs. Two alternative assumptions
regarding …xed costs allow us to parse the main insights. In the
…rst case, …xed costs are the same for all products. Products
therefore only di¤er by quality and marginal cost. We show that the
market solution involves the products with the highest margin
between quality and marginal cost. In the second case, there are
several product classes with a large number of products in each
class. Quality and costs are the same within each product class,
but …xed costs di¤er across classes. In this case, we show that the
equilibrium involves production of the goods in only one class,
while the optimum may entail the production of the goods in a
di¤erent class.
Our two …xed cost assumptions therefore allow us to distinguish
between 1 Market failures are also provoked by asymmetric
information (e.g. the ”lemons” problem)
and consumption externalities (e.g. buying a car with poor brakes).
We assume that indi- viduals know quality perfectly and that there
are no externalities, so that the social welfare function fully
re‡ects individual tastes. The only possible source of market
failure is due to imperfect competition.
2 The latter is not as restrictive as it may sound. For a product
selection problem, two (or more) …rms would never choose exactly
the same product since to do so would drive net pro…ts to zero
through Bertrand competition.
3 We thank Luis Cabral for the analogy.
2
possible excessive entry (which may occur even when the social and
private ranking of products coincides) and class selection biases
(which a¤ect the type of product in the market, but not necessarily
the number of products). In the …rst case (same …xed costs), if the
market is fairly covered (most consumers are served) too many low
quality products are induced to enter the market because …rms
selling high quality products set their prices too high. In this
case, asymmetries exacerbate the over-entry problem that is usual
in symmetric models.4 If though the market is relatively uncovered,
all outputs can be too small (due to market power and the
consequent oligopoly mark-ups) and too few …rms may enter the
market. We use the second case (di¤erent …xed costs) to highlight
the biases in product class selection. The bias works against high
quality products in the sense that equilibrium may involve low
quality products when the optimum prescribes that high quality
products should be produced.5 In the general case, both biases can
be present.
We use a logit model of asymmetries that has both intuitively
reasonable and restrictive properties. If one …rm produces a higher
quality than another, then its demand is higher if both goods are
priced the same, but not all consumers buy the higher quality
good.6 The larger the degree of horizontal di¤erentiation, the
smaller are the demand di¤erences from quality di¤erentials. One of
the most restrictive properties of our formulation stems from the
Independence of Irrelevant Alternatives property. This means that a
price cut by one product will draw new customers from other
products in proportion to those products’ sales. Models of local
competition (such as the vertical di¤erentiation model) instead
exhibit neighbour e¤ects, so a product competes directly only with
the next highest and next lowest qualities.
We supplement our theoretical contribution with calibration
analysis for a particular industry (yoghurt). It is noteworthy that
only data on market shares are required in order to calculate the
divergence of equilibrium outputs from the optimal outputs, using
the …rst-order conditions for oligopoly pro…t maximisation. These
data also su¢ce to determine the ranking of products in terms of
social desirability, mark-ups, and pro…tability. With extra data on
prices, plus a single parameter value that captures the
heterogeneity of consumer tastes, we can back out values for …rms’
marginal costs as well as product qualities. With this additional
information, we can calculate …rms’ gross pro…ts and hence are able
to put bounds on …rm entry costs. We can then give some indication
as to whether there is excessive or insu¢cient …rm entry in the
market.
Section 1 presents the model with accompanying discussion. Section
2 gives 4 The classic references are Spence (1976) and Dixit and
Stiglitz (1977). In a similar vein,
Deneckere and Rothschild (1992) conclude that the market tends to
provide the right number of products for monopolistic competition.
Anderson, de Palma and Nesterov (1995) show that the market errs
towards over-entry for oligopoly.
5 Dixit and Struglitz (1977) also consider this bias for a special
example: see Section 5. 6 This property contrasts with standard
models of vertical di¤erentiation (see Gabszewicz
and Thisse, 1979, and Shaked and Sutton, 1983), which assume that
all consumers have the same ranking of products when they are
equally priced. In such models, the social optimal solution is
trivial and entails a single good. This excludes discussion about
optimal product variety.
3
the characterisation of equilibrium and shows how market data can
be used to rank …rms in terms of social desirability. Section 3
then shows how market share data can be used to determine optimal
outputs, and illustrates for the yoghurt case. We also argue here
that the market output composition is biased against …rms of high
quality. In section 4, we argue that the market will tend to select
the right …rms if there are no di¤erences in …xed cost, and we show
there can be over-entry or under-entry depending on the degree to
which the market is covered. Section 5 looks at di¤erences in …xed
costs and establishes a bias against products with high quality and
high …xed costs. Section 6 concludes.
1 The Basic Model There are n single-product …rms in the market,
each producing a separate vari- ant of a di¤erentiated product.
Firm i produces a good of quality qi that it produces at constant
marginal production cost ci. The …rms are labelled in terms of
decreasing quality-cost (to be read as ”quality minus cost”
throughout the paper), so that q1 ¡ c1 ¸ q2 ¡ c2 ¸ :::qn ¡
cn.
To analyse the pricing game, suppose that Firm i charges a price
pi; i = 1:::n: The demand side is generated by a discrete choice
model of individual behaviour whereby each consumer buys one unit
of her most preferred good. Preferences can be described by a
conditional (indirect) utility function of the form:
uil = qi ¡ pi + ¹"il; i = 1:::n; (1)
where "il is the realisation of a random variable which can be
interpreted as the match value between consumer l and good i. The
match values in (1) are assumed to be i:i:d. across …rms and
products; the common distribution is the double exponential with
zero mean and unit standard deviation. The parameter ¹ > 0
expresses the degree of horizontal consumer/product heterogeneity.
Each individual draws n match values and then selects the good with
the highest indirect utility. The probability that a randomly
selected consumer chooses good i,(Pi = Probfui ¸ uj ; j = 1:::ng),
is then given by the logit model as (see Anderson et al. 1992, for
further discussion of this model):
Pi = exp [(qi ¡ pi) /¹ ]
nP k=1
exp [(qk ¡ pk) /¹ ] ; i = 1:::n: (2)
Total market size is normalised to 1 without loss of generality, so
Pi is interpreted as the fraction of consumers buying from Firm i
(i’s market share). Recalling that quality qi is associated with a
marginal production cost ci, Firm i’s pro…t is:
¼i = (pi ¡ ci)Pi ¡ Ki; i = 1:::n
4
where Ki is a sunk entry cost. The general version of the model
given in (1) is so complex that only little
headway can be made.7 Some restrictions are needed on the class of
demand functions. We require …rst and foremost that a unique price
equilibrium exists. We also restrict our analysis to individual
unit demand as analysed by Deneckere and Rothschild (1992) and
Anderson, de Palma, and Nesterov (1995). Hence we use a discrete
choice model in which both individual tastes and product quality
di¤er. These models are most suitable for our purposes since they
start from a description of individual behaviour and allow tastes
to be aggregated into a social bene…t function. To the best of our
knowledge, the logit is the only discrete choice model for which a
unique price equilibrium has been shown to exist for the asymmetric
case (see Anderson, de Palma and Thisse, 1992, Caplin and Nalebu¤,
1991, and Milgrom and Roberts, 1990).
2 Characterisation of Equilibrium The logit formulation ensures
that the pro…t functions are strictly quasi-concave, so that the
…rst-order conditions characterise best responses. These conditions
give the logit mark-up formulae as:
pi ¡ ci = ¹
1 ¡ Pi ; i = 1:::n: (3)
This is an (implicit) non-linear system in p1; ::; pn. Using (3),
the equilibrium pro…t reduces to
¼¤ i = p¤
where p¤ i is Firm i’s equilibrium price.8
>From (4), equilibrium gross pro…t is greater the larger is the
price-cost mar- gin. We now show that the pattern of quality-cost
margins, price-cost margins, sales and pro…ts all follow the same
ranking.
Proposition 1 In equilibrium, a …rm with a higher quality-cost
margin has a higher quality- price margin, a higher price-cost
mark-up, a higher output and a higher gross pro…t, and
conversely.
Proof. For all i; k = 1; :::; n, from (2), Pi > Pk <=> qi
¡pi > qk ¡pk; from (3), Pi > Pk <=> pi ¡ ci > pk ¡
ck. Hence Pi > Pk <=> qi ¡ ci > qk ¡ ck. From (4), the
…rms with the highest quality-cost margins will have the highest
gross pro…ts.
7 Even showing that …rms with higher qualities set higher prices
and earn higher pro…ts has proved intractable when there are three
or more …rms.
8 The existence of a price equilibrium is guaranteed from Caplin
and Nalebu¤ (1991) since the double exponential distribution that
generates the logit is log-concave. As noted in Section 1, the
equilibrium is also unique. If all quality-cost margins are equal,
the common price-cost margin is just ¹n /(n¡ 1) , which decreases
with n.
5
This result has a wider applicability than the logit model. As long
as goods with higher quality-price attain higher market shares then
the substantive con- dition is that higher mark-ups should be
associated with higher outputs (from the …rst order conditions).
This condition holds in simple models of both verti- cal
di¤erentiation and (spatial) horizontal di¤erentiation. It also
holds for the CES representative consumer model, as well as for the
extension of the logit model to include outside options
(non-purchase) that we introduce below. More broadly, in a Cournot
(homogenous goods) model with di¤erent marginal costs, it is once
more the low cost …rms that have higher market shares, mark-ups and
pro…ts.
The proposition shows that several di¤erent ways to classify …rms
are equiv- alent, in the sense that any one implies the others. The
di¤erent criteria can be thought of as social desirability (q ¡ c),
attractiveness to consumers (q ¡ p), attractiveness to …rms (p ¡ c)
and market share (P ). This means that mark- ups, outputs, and
hence gross pro…ts follow the same ranking as quality-cost margins,
so p1 ¡ c1 ¸ p2 ¡ c2 ¸ ::: ¸ pn ¡ cn and P ¤
1 ¸ P ¤ 2 ¸ ::: ¸ P¤
n , and so ¼¤
1 ¸ ¼¤ 2 ¸ ::: ¸ ¼¤
n. It is important to realise that high quality products do not
necessarily have the highest sales in this model because they are
also likely to have higher costs. This proposition has a practical
side that we will explore below, since qualities and marginal costs
are not easily observable but market shares are.
Observing market shares and prices enables one to infer something
about both (perceived) qualities and production costs given the
logit structure. For example, if we observe that Dannon yoghurt has
a larger brand share and a higher price than Weight Watchers, we
can infer from Proposition 1 that the quality of Dannon is higher
than the quality of Weight Watchers. Likewise, if we know that
Dannon yoghurt has a larger brand share and a lower price than
Yoplait, we can infer that the marginal cost of Dannon is lower
than that of Yoplait. However, without more information, we cannot
infer the quality di¤erential between these two brands (nor the
cost di¤erential between Dannon and Weight Watchers).
Knowing the value of the heterogeneity parameter, ¹, or a single
cost or quality value, would enable us to determine all of the
unknown qualities, costs and ¹. We provide an example of such
calibration in the next section. Even when this extra information
is not available, it is possible to determine the cost and quality
ranking of products, if we further assume that there is an
increasing relation between cost and quality. For the yoghurt
example, we could then infer that Weight Watchers has the lowest
quality and Yoplait the highest of the three, with unit costs
following the opposite ranking. Indeed, when unit cost is an
increasing function of quality, then we can show that both quality
and cost are ranked in the same way as equilibrium prices -
although equilibrium outputs need not follow the same
ranking.9
9 In the yoghurt example (the data that underlie the example are
discussed below), a fourth brand, Hiland, had the second lowest
price and hence the second lowest quality. Its output was the
lowest of the four brands and hence (from Proposition 1) its
quality-cost and pro…t were also the lowest. Arguably, there are
economies of scale in marketing and distribution, if
6
To see this, consider two products, i and j, such that Pi > Pj ;
if pi ¸ pj then necessarily qi > qj in order to induce the
observed quantity di¤erential in the face of product i’s price
disadvantage. If pi · pj then necessarily ci < cj in order to
satisfy the …rst order conditions (3) that relate high demand to
high mark-ups, and hence qi < qj under the hypothesis that costs
increase with quality. These inequalities imply that high qualities
are associated with high prices (and high costs).
Although this last statement may seem rather obvious, it does
depend on the (seemingly innocuous) assumption that costs are
indeed increasing with quality. However, in speci…c markets, this
need not be true. A chef, a consultant or an artist could be
naturally endowed with special natural skills that are not
associated with high (opportunity) costs. Likewise mineral waters
are of di¤erent qualities which are not correlated with their
extraction costs. In such cases, qualities and costs may not follow
any simple patterns.
3 The Comparison of Optimum with Equilib- rium
We …rst compare the equilibrium allocation to the optimum for n
…xed and describe the bias caused by quality-cost di¤erences.
Consider the …rm with the greatest quality-cost advantage, Firm 1.
Its equilibrium output is
P ¤ 1 =
nP k=1
:
.
:
Comparing the two quantities shows that the equilibrium output is
less than or equal to the optimal output as
nX
nX
which holds since p¤ 1 ¡c1 ¸ p¤
k ¡ck (or p¤ 1 +ck ¸ p¤
k +c1) for all k by Proposition
1. A similar argument shows that Firm n over-produces in
equilibrium since for
not in production, and the local brand (Hiland) may be hurt by
small scale production that raises its marginal cost relative to
the national brands.
7
all k. Likewise, if Firm k over-produces (under-produces), then so
does Firm k+1 (Firm k¡1). The only case when equilibrium and
optimal outputs coincide occurs when all quality-cost di¤erences
are equal, which necessarily holds in the symmetric case. The next
Proposition summarises.
Proposition 2 For the logit model (2), the market equilibrium is
biased against high quality-cost …rms: output of high (low)
quality-cost …rms is too low (high).
Proposition 2 implies that the average quality-cost - that is
quality-cost weighted by demand - is too low. The intuition behind
this result is that high quality-cost …rms use their advantage to
increase their mark-ups at the expense of some market share. These
high prices in‡ate the demands for low quality- cost products, and
so low quality-cost products are over-provided. As we argue later
in this section, once we amend the logit model (2) to allow for the
outside goods, even the low quality-cost goods may be
under-provided if the market is not well covered, though the bias
remains greatest against high quality-cost goods. For the present
though, we retain the assumption of no outside option (fully
covered markets).
A striking feature of the logit model is that the optimum outputs
can be calculated solely from the observed equilibrium outputs. We
…rst show how this can be done, and then give some illustrative
calculations based on market data for the yoghurt industry. The
ratio of equilibrium outputs is given from (2) by:
P ¤ i
P ¤ j
But the expression for the ratio of optimal outputs is:
P o i
P o j
¹
¸ ; i; j = 1; :::; n:
It is the …rst order conditions for the logit oligopoly model that
allow us to link these two ratios, even if costs, qualities and ¹ -
and even prices - are unobservable. The …rst order conditions for
pro…t maximisation, pi ¡ ci = ¹=(1 ¡ Pi), (see equation (3)), imply
that we can replace the standardised cost di¤erence (ci ¡ cj) /¹ in
the expression for the optimum output ratio by:
p¤ i ¡ p¤
:
This observation enables us to write the ratio of optimal outputs
in terms of the equilibrium ones as:
P o i
P o j
8
Before showing how the resulting system can be solved, we pause to
note some properties of the equations (5). First, it is clear that
the optimum ratio coincides with the equilibrium one if and only if
the market equilibrium proba- bilities are the same. Second, the
ranking of the optimum and of the equilibrium probabilities is
always the same, so that P o
1 ¸ P o 2 ¸ ::: ¸ P o
n. However, if P ¤ i and
P ¤ j are di¤erent, the optimum probabilities are more divergent
than the equilib-
rium ones. For example, if P ¤ i > P ¤
j , the extra term on the RHS of (5) is larger than 1, so that P
o
i ± P o
j > P ¤ i
± P ¤
j . This result re‡ects the regression of the market system towards
the mean: there is too much homogeneity because the better products
are over-priced. We should, though, be more careful before we
suggest that the low quality goods are generally over-produced. The
assumption that the market is fully covered plays an important role
in that conclusion, as we discuss further below.
The ratio of market shares can be used to determine the absolute
di¤erence of equilibrium and optimum outputs. These are now
examined. Using the fact that the choice probabilities sum to 1,
the system (5) can be solved to give the optimal choice
probabilities as a function (only) of the equilibrium choice
probabilities:
P o i = P ¤
; i = 1; :::; n: (6)
This expression shows that the equilibrium choice probabilities are
corrected by weights that account for the fact that the equilibrium
prices are not equal to marginal costs (recall the …rst order
condition (3)).
We illustrate using market data for yoghurt, for 4 …rms. The data
are taken from Besanko, Gupta and Jain (1998). These authors
present checkout scanner data collected for two full years from
1986 to 1988 compiled by AC Nielsen from nine stores of a
supermarket chain in Spring…eld, Missouri, USA.
The brand shares were: Dannon (42:82 %), Yoplait (23:05 %), Weight
Watchers (23:91 %), and Hiland, a regional brand (10:22 %). Using
these num- bers as the equilibrium choice probabilities in equation
(6) gives the socially optimal outputs as: Dannon (54.5 %), Yoplait
(18:9 %), Weight Watchers (19:7 %), Hiland (6:9 %). Comparing the
two sets of …gures shows that Dannon should optimally sell 27 %
more than its equilibrium sales level, while Hiland should sell 32
% less.
For the above calculation, we assumed that all individuals must buy
(or, in this context, the relevant market is the set of existing
customers, so there are no gains from enticing customers outside
the group to buy if a …rm cuts its price. The opposite extreme
assumption is that the relevant market is the set of all households
(actual and potential buyers). We can model non purchase as an
outside option with utility U0l = V0l + "0l, (cf. equation (1) and
see Ander- son et al. (1992) for further details on outside
options), so that the purchase
9
exp (Vo /¹) + nP
; i = 1; :::; n; (7)
and P0 is one minus the sum of the purchase probabilities of the n
produced goods. The case already examined arises when the outside
option is extremely unattractive (i.e., V0 low enough, which
implies that the market is fully covered). Note that the …rst order
conditions (3) still hold, as do the expressions for pro…t (4).
More importantly, Proposition 1 still holds, as does the relative
bias against high quality enunciated in Proposition 2. The relation
(5) still holds between all pairs of produced goods, but we must
amend (6) to account for the outside option (no purchase), which
has no associated …rst order condition.10 For the outside good, the
formulae (5) are modi…ed as follows:
P o i
P o 0
1 ¡ P ¤ i
¸ ; i = 1; :::; n:
Since the exponential term exceeds one, all probabilities should be
increased vis-à-vis the no purchase option. This is because all
…rms exercise market power and charge above marginal cost. These
formulae characterize relative and not absolute changes. With an
outside option, the optimum outputs are now given by:
P o i =
:
We can use these formulae for the yoghurt data, taking the no
purchase probability as 83:1% (the number given by Besanko et al.
1998).
10 If the actual demand is very small relative to the potential
demand, the equilibrium ratios are very similar to the optimum
ones.
10
Equil. share of market Opt. share of output Dannon 7:24 % 16:15 %
Yoplait 3:89 % 8:37 %
Weight Watchers 4:04 % 8:70 % Hiland 1:73 % 3:63 %
Outside goods 83:1 % 63:1 %
Table 1. Equilibrium and optimum outputs when the market is not
fully cov- ered
As can be seen from the comparison with the case of no outside
option, the prescription changes rather dramatically. Here we have
the optimal outputs nearly twice the equilibrium ones. This
illustrates the property that all of the goods are under-produced
if the market is relatively uncovered (V0 large enough). As we show
next, the e¤ect that low quality-cost products are over-produced
when the market is fairly covered leads too many …rms to enter the
market at the low end, but there may be too little entry if the
market is not well covered (V0 high).
4 Optimum and Equilibrium with Free Entry The market selection
mechanism for …rms rests on their pro…tability. In a long- run
equilibrium, all existing …rms cover …xed costs while any new …rm
would not. We assume for the benchmark case of this section that
…xed costs are the same for all potential …rms. In Section 5, we
allow …xed costs to di¤er across product classes.
We …rst show that it is always an equilibrium to have the top
products in the market (Proposition 3:1), with the cut-o¤ level
decided by the common level of …xed costs, K. This is also the
equilibrium in a game where …rms choose qualities (Proposition
3:2). A comparison of equilibrium to optimum outcomes shows there
is over-entry for the case in which the market is fully covered (V0
! ¡1, which implies P0 = 0). The usual over-entry problem is
exacerbated when …rms are asymmetric. We then revisit the
over-entry issue for relatively uncovered markets and …nd there may
be under-entry because all products are over-priced. We provide
some illustrative calculations for the yoghurt market in Section
4:2.
4.1 Product Ranking and Cut-o¤ We start with the equilibrium
analysis (recall that the logit model (2) is a special case of (7)
so Propositions 3 apply to (2) as well). We …rst consider the case
when potential …rms are endowed with exogenous qualities and we ask
which …rms will enter.
PROPOSITION 3.1 When all potential …rms face the same …xed cost, K,
there is a long-run equilibrium with …rm-speci…c qualities for the
logit model (7)
11
at which: (i) The …rms in the market are those with the highest
quality-cost margins; (ii) The net revenue of the nth …rm decreases
with n and converges to zero as n becomes large.
Proof. See Appendix 1. The proof of the Proposition 3:1 shows that
it is an equilibrium for the top
quality-cost …rms to enter. This is not always the only
equilibrium. It may be possible that some other set of …rms is in
the market but yet some excluded …rm with a higher quality-cost
cannot pro…tably enter due to the presence of the established …rms
even though it could make more money were it to replace one of the
latter.11
The equilibrium described in Proposition 3:1 is also the unique
equilibrium if there is a large number of …rms and each …rm is free
to choose which product to produce (instead of being immutably
endowed with quality-cost) from an available array of products that
are characterized by quality-cost.
PROPOSITION 3.2 When all potential …rms face the same …xed cost, K,
there is a unique long-run equilibrium for the logit model when
…rms choose product qualities (7). The equilibrium has the
properties of Proposition 3:1.
Proof. See Appendix 2. The proof of Proposition 3:2 is slightly
more complicated than the proof of
Proposition 3:1 in that we have to show that a …rm that switches up
from a lower quality-cost increases its pro…t despite more
strenuous competition from its rivals. While it is obvious that
switching to a higher (hitherto unproduced) quality will increase
net revenue if all other …rms’ prices are …xed, the others will
respond by cutting their prices in face of the increased
competition. This proof shows that this feedback e¤ect is dominated
by the direct quality e¤ect. Thus, the two di¤erent speci…cations
(…rm-speci…c qualities and quality chosen) lead to the same
equilibrium outcome.
The property that the market selects the highest quality-cost
products is not as trivial as it may appear at …rst, at least in
part because a more ”innocuous” low-quality product would elicit a
weaker competitive response from the rivals, and, ceteris paribus,
weaker competitive responses (higher prices) are preferred.
Instead, it is the ”…ghting brands” that the market selects, the
ones that harm the rivals most and draw the most …re in terms of
price competition, but are nevertheless the most pro…table ones.
This property of the model is not shared by the vertical
di¤erentiation model.12 In the vertical di¤erentiation model,
if
11 This is a coordination problem due to the integer constraint on
the number of …rms. With a broader entry mechanism that allows more
sophisticated behaviour, then the equilibrium would frequently be
unique. For instance, we could consider entry with the intention of
levering out less e¢cient incumbents (that make less than the
entrant post-entry). Another mechanism concerns a growing market
whereby those …rms with the most to gain will invest early at the
point in time that would just preempt the less e¢cient …rms from
coming in. In what follows we shall analyse the equilibrium given
in Proposition 3:1.
12 The vertical di¤erentiation model is not a special case of our
framework, but instead
12
we were to assume (as here) that each …rm is endowed with a
quality-cost, then it is not the highest set of products that would
survive in equilibrium because products that are close in quality
engage in tough competition. Rather we would expect substantial
spacing among the tenable qualities.
To recapitulate, the free-entry equilibrium that we shall study
involves the …rst m …rms such that ¼¤
m ¸ 0 > ¼¤ m+1. Hence the mth …rm will enter if (see
(2) - (4)):
¸ K ¹
: (8)
The social desirability of products follows the same ranking as
their prof- itability (it is readily shown that the highest
quality-cost products should be produced), but the optimal number
of …rms, and their outputs, will generally di¤er from the
free-entry equilibrium. As we show in Appendix 3, the equi- librium
over-provides variety when the market is fully covered (V0 ! ¡1).
This result extends the analysis of Anderson, de Palma and Nesterov
(1995) to asymmetric …rms. An important quali…cation is that the
extent of over-entry can be much greater (again, contingent on
fully covered markets) for asymmet- ric quality-costs. Under
symmetry, the number of products is about right for the logit model
(one product too many), while under asymmetric quality-costs,
over-entry can be substantial (see the example in Appendix 3). The
reason for the distortion is that over-pricing of high quality-cost
products increases the demand for low quality-cost ones, and
in‡ates their pro…ts, leading to excess en- try. If though the
market is su¢ciently uncovered, the market solution involves
under-entry. Since asymmetries tend to encourage excessive entry,
under-entry is more prevalent with symmetric quality-costs.
Following Spence (1976), there are two externalities associated
with …rm entry. An entrant does not account for the detrimental
e¤ect on other …rms’ pro…t, but nor does it account for the
bene…cial e¤ect on consumption variety. When the market is
relatively uncov- ered, a new entrant will draw its demand mainly
from the outside good. This means that the negative externality on
other …rms’ pro…ts will be small relative to the positive
externality on consumers’ bene…ts (which also decreases as Vo
increases). The net e¤ect is a positive externality and hence
under-entry. This possibility is substantiated below in the
calibration analysis.
4.2 Calibration of Optimal Entry We provide some illustrative
calculations for the yoghurt market. To do this, we have to …rst
calibrate the unknown costs and qualities of the various brands
(see also Werden and Froeb, 1996). Following our presentation
above, we …rst deal with the case of no outside good.
involves taste heterogeneity over the willingness to pay for
quality.
13
We are unable to determine the value of ¹ from the data available,
and have chosen ¹ = 2, which is the mid-point of the range of
feasible values of ¹. The minimum value of ¹ is 0: then the quality
o¤ered by each …rm is equal to its price (see (2)), as is its
marginal cost (see (3)). The maximum value of ¹ is computed from
the constraint that marginal cost c should be positive. The
…rst-order condition (3) implies that: ¹ · pi (1 ¡ Pi). Using the
data for prices and market shares (Table 1), we get: ¹ · 3:98 (the
binding value is for Weight Watchers). Note that as ¹ increases,
mark-ups increase while quality di¤erentials and marginal costs
decrease.
Given ¹ = 2, marginal costs are then calibrated from the price and
the output data, using the …rst order conditions (3), i.e., ci = pi
¡ ¹ /(1 ¡ Pi) .13
The qualities are calculated using the ratio of the choice
probabilities which leads to an easily calculated expression for
the quality di¤erences: qi ¡ qj = pi ¡ pj + ¹ ln(Pi /Pj ). Since
these equations only determine qualities up to a positive constant,
w.l.o.g. we normalise the quality of the Hiland product so that its
quality-cost is zero. Given the quality-cost data that are
generated in this fashion, we can calculate the optimal prices
taking as a benchmark the equilibrium price of Weight Watchers
yoghurt. Since optimal prices are equal to marginal costs plus or
minus any arbitrary constant,14 all we have done is added Weight
Watchers’ calculated mark-up of 2:63 to all marginal costs.
p¤ Equil. c q q ¡ c Gross ¼ po
Cts/Oz Output Cts/Oz Cts/Oz Cts/Oz Cts/Oz/Csr Cts/Oz Dannon 8:03
42:82 % 4:53 8:67 4:14 1:5 7:16 Yoplait 10:39 23:05 % 7:78 9:84
2:06 0:61 10:41 W.W. 5:24 23:91% 2:61 4:71 2:10 0:63 5:24 Hiland
7:73 10:22 % 5:50 5:50 0 0:23 8:13
Table 2. Calibration of the yoghurt statistics when the market is
fully covered
The pro…t …gures in the Table above are given from equation (4) and
are the gross pro…t …gures per capita in the relevant market.
Assuming that …rms make non negative pro…ts, these numbers give
upper bounds on the …xed costs of the …rms, and can be used to give
some indication as to whether entry is excessive. At one extreme,
the lowest pro…t entrant just makes zero pro…t; at the other
extreme, a potential entrant is unpro…table. We discuss these cases
below. We …rst need to determine the social surplus associated with
various con…gurations of …rms. We shall suppose that the order of
social desirability of …rms follows their quality cost ranking.
This follows for example if …xed costs are the same
13 The values of the calibrated mark-ups, (p¡ c) /c are 0:773,
0:335, 1:01 and 0:405 for Dannon, Yoplait, Weight Watchers and
Hiland, respectively. These mark-ups depend on the value of the
paramter ¹: The average mark-up is 0:631. This value is not far
from the average mark-up of 0:685 obtained by Thomadsen (1999,
Table 2B, p. 26) for fast food.
14 This is only true for the logit model (2); i.e., when Po =
0.
14
across …rms. We discuss other cases in the next section: we do not
address here the possibility that the products selected may be the
wrong ones.
The welfare function for the logit model (2) is given by (see
Anderson et al. 1992):
W (n) = ¹ ln
Kk;
where the …rst term is consumer surplus minus gross …rm pro…ts.
When just Dannon and Weight Watchers are in the market, using the
calibration numbers from Table 2 gives the welfare level (up to a
positive constant that re‡ects the base quality level) as:
W (D;W ) = 2 ln[exp(2:07) + exp(1:05)] ¡ KD ¡ KW
where the numbers in the exponents are simply quality-cost divided
by ¹, for the two products. When we add Yoplait, the welfare level
is:
W (D;W;Y ) = 2 ln[exp(2:07) + exp(1:05) + exp(1:03)] ¡KD ¡ KW ¡
KY
and when we then add Hiland to the group we …nd:
W (D;W;Y;H) = 2 ln[exp(2:07) + exp(1:05) + exp(1:03) + exp(0)] ¡KD
¡ KW ¡ KY ¡ KH :
Taking the di¤erence of the …rst two equations tells us that the
welfare increment from adding Yoplait to the market mix is 0:462 ¡
KY ; whereas the increment from then adding Hiland is 0:142 ¡ KH .
Since the pro…t of Hiland is 0:23, if Yoplait’s …xed cost is not
much more twice this number, Yoplait should be in the market.
However, if Hiland’s …xed cost is close to its pro…t, then it
should not optimally be in the market.
We now reconsider the question of optimal entry under the
alternative hy- pothesis that the relevant market comprises all
consumers and so the probability of choosing the outside option is
83:1% as we discussed in Section 3. The proce- dure for calibrating
the model is basically the same as the one we just outlined for the
case of no outside option, with the following similarities and
di¤erences. First, calculation of marginal cost is still given from
the …rst-order conditions (and there is no such condition for the
outside good) and now because choice probabilities are lower, then
the calibrated values for marginal costs are higher (and mark-ups
correspondingly lower re‡ecting smaller market power from a more
tenuous hold on the market). Second, the expression for quality
di¤er- entials remains unchanged, and since they depend only on
prices (and not on outputs), the quality di¤erences are unchanged
in Table 3 below. The expres- sion for the attractivity of the
outside good is also calculated in a similar ratio form and becomes
V0 ¡ qi = ¡pi +¹ ln(P0=Pi). We again normalise qualities so that
the quality-cost of Hiland is zero.
15
p¤ Equil. c q q ¡ c Gross ¼ po
Cts/Oz Output Cts/Oz Cts/Oz Cts/Oz Cts/Oz/Csr Cts/Oz Dannon 8:03
7:24 % 5:87 8:86 2:99 0:156 5:87 Yoplait 10:39 3:89 % 8:31 10:03
1:72 0:081 8:31 W.W. 5:24 4:05 % 3:16 4:90 1:74 0:084 3:16 Hiland
7:73 1:73 %: 5:69 5:69 0 0:035 5:69 Outside - 83:1 % ¡ 5:71 5:71 -
-
Table 3. Calibration of the yoghurt statistics with an outside
option
The …gures for gross pro…ts per potential consumer are all
signi…cantly smaller with the outside option both because the
mark-ups have declined, but more importantly, because the consumer
base is now larger (this latter e¤ect does not change total gross
pro…ts). This change entails a signi…cant alter- ation to the
welfare analysis. Using the same procedure as before, the welfare
expression with the …rst three yoghurts is:
W (V0;D;W; Y ) = 2 ln[exp(2:85) + exp(1:5) + exp(0:87) + exp(0:86)]
¡KD ¡ KW ¡ KY
where the …rst term under the logarithm is exp(V0 /¹). Adding
Hiland then adds a term exp(0) under the logarithm, along with
subtracting KH. Evaluating these expressions shows that the social
bene…t from adding Hiland (with marginal cost pricing for all …rms)
is 0:07383¡KH . Since the gross welfare gain is almost twice as
large as the gross pro…t (on a per potential consumer basis) we
calculated in Table 3 (0:035), which in turn should exceed the cost
KH , we can conclude that the Hiland …rm is socially desirable.
This …nding shows that the over-entry result breaks down when
outside options are included.
To compute the optimal number of …rms requires guessing what the
next available quality-cost is, and the answers are rather
sensitive to this estimate of an unobservable quantity. If we take
the next possible quality-cost as 2 below Hiland (which is loosely
in line with the quality-cost di¤erence between Hiland and
Yoplait), then we need to add a term exp(¡1) under the logarithm in
the surplus expression, and subtract the corresponding …xed cost.
This exercise yields a welfare increase of 0:027 minus the
corresponding …xed cost. If the latter is in the same range as the
upper bound we used above for Hiland (0:037), then it is not
socially optimal to have more …rms. On the other hand, if we
suppose that there is a large number of …rms waiting in the wings
with quality-costs similar to those of Hiland, then there can be
extensive under-entry.15
15 The welfare gain from m to (m¡1) additional …rms with
quality-cost like Hiland is equal to 2 ln (26:5 +m+ 1)¡2 ln (26:5
+m)¡0:035. Using the upper bound for KH (KH = 0:035), we …nd that
…rst-best optimum requires as many as m = 30 Hiland-type …rms (a
lot of choice for breakfast). This example highlights the
sensitivity of the results to the parameter values chosen for the
new entrants.
16
The above calculations have taken the market qualities as given,
and can thus be viewed as a second-best analysis. However, when
…xed costs di¤er across products, the market solution may also be
‡awed in terms of the qualities chosen by …rms, and not solely in
terms of the number of …rms. This problem seems to be intractable
in general, but the potential biases nevertheless can be
illuminated by considering an extreme possibility whereby products
can be grouped in classes with di¤erent quality-cost and …xed costs
and a large number of potential products within each class. We now
turn to this, problem.
5 The Choice between High and Low Quality- Cost Products
It was assumed in the theoretical part of the previous sections
that …xed costs are independent of the quality-cost produced, and
that there is a limited number of …rms of a given quality-cost. If
…xed costs di¤er across …rms along with quality cost, the market
solution may not provide high quality-cost …rms with high …xed
costs when it ought to. We illustrate this point by now dealing
with the case where …xed costs depend on quality-cost and where
there are a large number of potential …rms at any quality-cost
level. For simplicity of exposition, we present the case of full
market coverage (P0 = 0). We note later how things change with
partial market coverage.
Let there be Z classes of commodity. They di¤er according to …xed
cost and quality-cost. Suppose that nz …rms produce products of
type z (with quality qz) at marginal cost cz. Let Kz be the …xed
cost associated to type z. The social welfare maximand is
W (n1:::nz) = ¹ ln
nzKz;
It is helpful to proceed in two steps to determine the optimal
con…guration. First, if an extra dollar is available to be spent on
set-up costs, allocating this dollar to type j …rms will buy 1=Kj
more of them. The corresponding increase in social welfare (the
social desirability of investment in sector j) is:
@W @nj
1 Kj
¡ 1:
If this expression is largest for Firm j at some vector nz, then it
is largest for all vectors of nz (since the summation term is
common to all types). Hence if a …xed dollar amount is to be spent
on set-up costs, it should be spent in the sector for which this
expression is greatest (there is almost always no solution with
positive numbers of …rms in more than one sector), so that only
type j
17
½ exp [(qz ¡ cz)=¹]
@W @nj
1 Kj
= ¹ nj
1 Kj
¡ 1:
This immediately implies that there should be ¹=Kj …rms of the
optimal type j.17
Now compare this solution with the free-entry equilibrium. To …nd
this, we again treat the number of …rms of each type as a
continuous variable. If there is an equilibrium with several types
producing, the zero-pro…t conditions (4) yield p¤
i ¡ ci ¡ ¹ = K , for all active types. Substituting in the
…rst-order condition (3) and using (2) shows that only type i …rms
will produce, where
i = Arg max z=1:::Z
½ exp [(qz ¡ cz) /¹ ]
Kz (Kz + ¹) exp [¡Kz /¹ ]
¾ (10)
As was the case for the optimum, more than one type will be active
only for a zero measure of parameter values. Equation (10) almost
always has a unique solution i, and there will be (1 + ¹ /Ki )…rms
of this type: let C be the set of types for which (10) holds, then
the numbers of …rms and types should satisfyP z2B
nzKz Kz+¹ = 1.
Clearly (9) and (10) are not equivalent, and the market may provide
the wrong quality. As we show the market equilibrium will not
provide a higher quality than the optimum.
PROPOSITION 4 For the logit model (2) with …xed costs: the
equilibrium product type selected is not of higher quality-cost
than the optimum type.
Proof. >From (9), the optimal quality type, zo, satis…es
exp [(qzo ¡ czo) /¹ ] Kzo
¸ exp [(qz ¡ cz) /¹ ] Kz
(11)
16 The same condition holds in the presence of an outside option,
conditional on being optimal to have the market at least partially
served (which is true unless Vo is too large).
17 If there are several types for which (9) holds, any such type
can be chosen. More generally, if B is the set of types for which
(9) holds, any combination of number and types such thatP z2B nzKz
= ¹ will do. However, this only happens for speci…c values of
parameters so that
there will be almost always one type at the optimum. For Vo …nite,
the optimal number of …rms is given by: Max f0; ¹ /Kj + 1¡ exp [(cj
+ ¹+Kj + Vo) /¹ ]g.
18
for all z. Assume that the equilibrium product ze had a higher
quality-cost. If cannot have a lower …xed cost than zo, since then
zo could not be an optimum. Thus consider a higher quality-cost
product, z, with a higher …xed cost. Multi- plying both sides of
(11) by (Kzo + ¹) exp [¡Kzo /¹ ] and then noting that the function
(K + ¹) exp [¡K /¹ ] is decreasing in K gives
exp [(qzo ¡ czo) /¹ ] Kzo
(Kzo + ¹) exp · ¡Kzo
(Kzo + ¹) exp · ¡Kzo
(Kz + ¹) exp · ¡Kz
¸ :
Comparing the …rst term to the last one leads to a contradiction
since (10) shows that the equilibrium type, ze, cannot exceed zo.
Therefore qze ¡cze · qzo ¡czo .
It is easy to construct examples where the most desirable type from
the social point of view is type 1 and the least desirable type is
Z, but only type Z is produced in equilibrium. If the equilibrium
and the optimum types coincide, the market distortion is small
since there is only a one-…rm di¤erence between equilibrium and
optimum for the logit model. In that sense the major source of
potential bias is in choosing the wrong set of products. To get a
feeling for the nature of the bias, suppose there are two types.
Type 1 has high quality-cost and high …xed cost. The optimum and
equilibrium con…gurations are compared in Fig. 1. We used for
parameter values:
exp h
i = 2 and ¹ = 2:
The values are loosely in line with the parameters of the
calibration, with the two types corresponding to Dannon and
Yoplait, though of course in the current exercise we allow for may
potential …rms of each ”type” and two di¤erent types cannot
coexist.
FIGURE 1. Equilibrium and Optimum Group Selection.
>From the …gure, it is clear that the bias favours low
quality-cost products with low …xed costs. Note that it is not
possible for the low quality-cost type to be selected in
equilibrium if its …xed costs are not strictly lower.
If there are two product classes with the same marginal cost, there
is a bias against the one with the higher quality. If both have the
same quality, there is
19
a bias against the one with the lower …xed cost. Thus the market
could provide many products with high marginal cost and low …xed
cost, when the optimum allocation would have fewer goods produced
at low marginal cost and high …xed cost, with a larger output of
each product.
To give some intuition why the market solution may be wrong,
suppose that all …rms were producing the high quality products, and
that this is socially opti- mal. If one of these …rms switched to
the low quality product, its revenues would go down but so would
its costs. If high quality were optimal, the reduction in net
revenue would fully re‡ect the social loss if all quality-price
margins remained unchanged, and would be greater than the saving in
…xed costs. However, the quality-price margins would not remain the
same. Instead, the remaining high- quality …rms would raise their
prices at the sub-game equilibrium induced from one of their number
switching to low-quality (this is because of a relaxation of
competition). This secondary e¤ect increases net revenues since
goods are substitutes. Thus the revenue loss of the switcher would
be smaller than the social loss, and it may be pro…table to switch
even though it is not optimal. A similar argument applies to all
remaining …rms, and we are left with no …rms at the high quality.
Thus the reason for the market failure is that low quality (or high
marginal cost) leads to less intense price competition.
Dixit and Stiglitz (1977) treat a similar product selection problem
in the context of a CES representative consumer model with two
possible groups of products. The two groups have di¤erent
(exogenously given) demand elastic- ities. Dixit and Stiglitz …nd
that the market solution may be biased against the production of
commodities with inelastic demands and high …xed costs, e.g.
”against opera relative to football matches” (p. 307). Since
inelastic demands are associated with both high revenues and high
consumer surplus, ”it is not immediately obvious whether the market
will be biased in favour of or against them” (p. 306). Dixit and
Stiglitz also claim that their model can be interpreted as one with
heterogeneous consumers, and that in this case the inelastically
de- manded commodities are those intensively desired by a few
consumers.
Our results con…rm the bias they …nd, in a di¤erent, and we would
ar- gue more natural framework. First, consumer heterogeneity is
explicit in our approach; second, our model uses the ”quality”
variables to explain demand dif- ferences across commodities. Even
though the high-quality commodities in our model are intensively
demanded by many consumers, there is still a bias against them. In
this sense our results reinforce those of Dixit and Stiglitz. The
results for opera and football matches extend to three-star and
unrated restaurants.
6 Conclusion We have argued in this paper that a full analysis of
the biases generated by imperfectly competitive markets should
treat asymmetries in costs and demand (as is empirically observed).
Due to the complexity of this issue, we have used a speci…c
functional form (like most of the literature on product selection)
to generate our results. Of course, we feel that our results and
intuition have a
20
wider generality than just for the logit model. Most of the
literature has focused on whether there are too many or too
few
…rms. However, when simple functional forms are used (such as the
symmetric logit, the CES, or a uniform i.i.d. taste density), the
amount of over-entry is small (one …rm or less than one …rm too
many). As we have seen, the amount of over-entry could be large if
…rms face asymmetric demands and if the market is well covered: too
many low-quality …rms enter because high-quality products are
over-priced, and this relaxes competition at the low end of the
market. This provides a counterpoint to the usual belief that such
a large degree of over-entry is speci…c to spatial models (see e.g.
Salop, 1979). A large degree of under-entry is also possible in the
model, and arises when the goods sold are of poor quality relative
to outside options. Finally, when …xed costs depend on quality, the
market can select the wrong product group and this selection bias
may be the major source of ine¢ciency. Previous work has used the
number of …rms as a performance measure, ignoring the biases due to
demand asymmetries. Our results suggest that this may understate
the true extent of market failure in imperfect competition with
di¤erentiated products.
The theoretical model was calibrated with data from the yoghurt
market. The calibration is illuminating, but also serves as a
reminder of the limitations of the model. The analysis is a static
description of pro…t maximising single product …rms and which
simply choose their prices. On the demand side, we have not
explicitly allowed for minority tastes. Nevertheless, since logit
and sim- ilar models are widely used (especially in empirical
applications) it is important to understand their properties and
their implications for market performance.
Clearly asymmetries are important in econometric analysis and
policy stud- ies. The merger simulations used by Werden and Froeb
(1996) are based on …rm asymmetries in costs and qualities, and the
new econometrics of Indus- trial Organisation (see, e.g., Berry,
1994; Berry, Levinsohn and Pakes, 1995; and Goldberg, 1995) allows
for asymmetries both in the costs of production and the
characteristics of goods o¤ered by …rms. Both the merger
simulations and the econometric work use discrete choice models to
describe consumer be- haviour, and are built on previous
theoretical developments in oligopoly theory. Although much of the
earlier theory concerned symmetric cases, the empirical work has
forged ahead in developing asymmetric models. In this paper we hope
to bridge this work back to the underlying theory by looking at the
theoretical properties of the market equilibrium in the presence of
…rm asymmetries.
21
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22
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23
APPENDIX 1: PROOF OF PROPOSITION 3.1.
(i) We know from Proposition 1 that for any set of …rms in the
market, pro…t decreases with the quality-cost margin. We now show
that entry (of any additional …rm) decreases pro…ts of the others
in the market. We know …rst that some …rm k’s market share (or else
the outside option’s share) must decrease following entry (since
the entrant is guaranteed a positive share). By the …rst- order
condition (3) …rm k’s price also falls. Now suppose some other …rm
r’s share rose; so too would r’s price (by (3)). But then the price
change would imply that r is relatively less attractive compared to
k so that the ratio Pr=Pk should fall, contradicting the share
conditions just given. We conclude that all shares must fall; from
(3), so too do prices, and hence so do gross pro…ts. Therefore,
since potential …rms are valued in terms of decreasing
quality-cost, there will be a unique cut-o¤ point such that all
…rms above the cut-o¤ cover their …xed costs and all …rms below the
cut-o¤ point rationally anticipate they will not be able to cover
those costs should they enter. (This argument suggests a simple
algorithm for determining how many …rms enter: add …rms until the
(n + 1)th …rm cannot cover its costs.)
(ii) From Proposition 1, the lowest-ranked …rm earns less than all
others. From the argument of the previous paragraph, a new entrant
(at the bottom of the scale) reduces the pro…ts of all other …rms.
Hence, an (n + 1)th entrant expects a mark-up and a pro…t less than
that of the nth …rm at an n-…rm price equilibrium. Moreover, since
the market share of the (n + 1)th …rm is less than 1=(n + 1), by
Proposition 1, its net revenue goes to zero as n goes to in…nity.
For K > 0, an equilibrium therefore exists with a …nite number
of …rms.
APPENDIX 2: PROOF OF PROPOSITION 3.2.
Assume that some good i is not produced, but a good j > i with a
strictly lower quality-cost is produced. We show that the pro…t of
the …rm producing j rises if it shifts production to i. Let a tilde
denote equilibrium values after the shift. Then we claim that
ep¤
i ¡ ci > p¤ j ¡ cj from (4). From the f:o:c: (3), this
is equivalent to eP ¤ i > P¤
j : Suppose this were not true, i.e.
eP ¤ i · P ¤
j (13)
(since by hypothesis, qi ¡ci > qj ¡cj). Now, since eP ¤ i · P
¤
j ,there must be some …rm k for which eP ¤
k ¸ P ¤ k . From Firm k’s f:o:c:, ep¤
k ¸ p¤ k, and so
qk ¡ epk · qk ¡ p¤ k: (14)
(13) and (14) imply that
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k . Q.E.D.
APPENDIX 3 : THE OVER-ENTRY RESULT WITH NO OUTSIDE GOOD
The welfare function associated to the logit model (2) has the
following form (see e.g. McFadden, 1981, and Anderson et al., 1992,
for a discussion):
W = ¹ ln
Clearly, the incremental social value of an sth …rm is
W (s) ¡ W (s ¡ 1) = ¹ ln ½
s¡1 + exp [(qs ¡ cs) /¹ ] s¡1
¾ ¡ K;
where s¡1 = s¡1P k=1
exp [(qk ¡ ck) /¹ ]. The logarithm term is less than exp[(qs¡cs)=¹]
s¡1
(and approximately equal to this when it is small). Hence the
welfare gain from the sth …rm is less than
¹ exp[(qs ¡ cs)=¹]
s¡1 ¡ K: (15)
We now show that the pro…t of the sth …rm is greater than this
value, and thus that …rms will enter the market even when their net
social worth given by (15) is negative (leading to over-entry).
Using (8), this amounts to showing that
exp [(¡cs) /¹ ] s¡1X
k=1
s) /¹ ] s¡1X
This inequality holds since qi ¡p¤ i ¡cs < qi ¡p¤
s ¡ci, for all i 6= s, by Proposition 1. The discussion above is
summarised by the following result:
For the logit model (2) with asymmetric costs and qualities, there
is excessive entry of …rms in the market equilibrium.
When …rms are symmetric (quality-cost is the same for all …rms),
the number of …rms is approximately the social optimum level (the
extent of over-entry for
25
the logit is just one …rm: see Anderson, de Palma, and Thisse,
1992). With asymmetric qualities and costs, the over-entry problem
can be much more severe. To illustrate the possible extent of the
problem, suppose that marginal costs are zero and ¹ = 1. There are
20 products which have high quality (q1 = ::: = q20 = QH = 4) and
20 products with low quality (q21 = ::: = q40 = QL = 1). Let K =
0:0025. Then it can be shown that the optimum involves only the 20
high-quality …rms, but the equilibrium has all 40 …rms
entering.18
18 The method we used was as follows. First, use the zero-pro…t
equation (2) to …nd the equilibrium price of the low quality goods.
Next, use the two f.o.c.s to …nd the equilibrium price of the
high-quality goods as a function of the (common) number of …rms, n,
of each type, of QH , QL and K. All values of K and n for which QH
> QL are equilibria. Then …nd K and n such that the social
planner is indi¤erent to adding a single low quality …rm to n
high-quality …rms. For n = 20, and QL = 1, this procedure gives
approximatively the values of K and QH in the text.
26