FIRM SIZE INEQUALITY AND MARKET POWER
August 1998
Philippe Barla
Université Laval
Département d'économique
Pavillon J-A de Sève
Québec, Québec
Canada, G1K 7P4
(E-mail: [email protected])
ABSTRACT
In this paper, we reexamine the relationship between performance and concentration in the
light of modern oligopoly theory. More specifically, we examine the link that exists between firm size
inequality (FSI) and market power. Traditional theory predicts that market power should be higher in
markets where FSI is high. Using a model with capacity constraints and endogenous conduct, we show
that the market power-FSI relationship is in fact more complex. We show that two effects are at play
leading to a U-shaped relationship between market power and FSI. Another implication of this model
is that prices should be more unstable in markets where firms are asymmetric in size.
In the second part of this paper, we test these predictions on data for the U.S. airline industry.
We estimate a fare equation for a panel of 400 routes. We first show that using traditional measures of
market concentration such as the Herfindahl is restrictive. We then show that there is indeed a Ushaped relationship between FSI and prices holding costs constant and that prices are more unstable in
markets where FSI is high.
JEL classifications: L13 (Oligopoly and Other Imperfect Markets),
Transportation)
L93 (Air
_________________________________________________________
I would like to thank C. Constantatos, Linda Khalaf, Randal Reed, Yves Richelles and the seminar
participants at the Université de Montréal, the participants at the 14 Journées de Microéconomie
Appliquées in Marrakech and the First Conference of the ATRG in Vancouver, two anonymous
referees and the Editor for their helpful comments. The usual disclaimer applies.
1
Firm Size Inequality and Market Power
I. Introduction
There is a long tradition in industrial organization of examining the link
between market structure and performance. One aspect that has received much
attention is the effect of concentration on market power. The traditional paradigm
attributed to Bain (1951) predicts that market power should be higher in concentrated
markets. Bain's original insight is that concentrated markets are more likely to
sustain collusion. Hundreds of empirical studies have tried to test this idea. Proxies
of market power (like price-cost margin) have been linked to measures of
concentration such as the C4 or the Herfindahl. These indexes aggregate two aspects:
the number and size distribution of competitors. It is generally agreed that a good
index of concentration must decline with the number of firms and increase with the
level of inequality among firms. The Herfindahl, for example, respects these two
conditions. These conditions and the predicted positive link between market power
and concentration mean that a market with unequal firm size should be characterized
by higher market power than markets with firms of equal size (the number of
competitors being fixed). For example, price-cost margin should be lower in a
market where two firms share the market equally (Herfindahl=0.5) than in a market
where one firm has 70% and its rival 30% of the market (Herfindahl=0.58). It is the
validity of this relationship that we will review here both theoretically using modern
oligopoly theory and empirically using data for the U.S. domestic airline industry.
The traditional predicted positive link between firm size inequality (FSI) and
market power is based on oligopoly models that assume conduct among firms is
exogenous (for example, firms are assumed to behave à la Cournot). In recent
oligopoly theory, conduct is endogenous. In supergame models for example, tacit
collusion may be sustained by the threat of punishment in case of deviations. In such
a setting, it is not clear if collusion is more likely when firms are equal or unequal in
size .
2
In section 2, we provide some background on the relationship between market
power and FSI. In section 3, we develop a theoretical model of firms competition
with capacity constraints and endogenous conduct. We show that the impact of FSI
is more complex than the usual link obtained by traditional theory. We show that
when firms are symmetric in size, tacit price coordination is in fact easier to sustain
than when firms are asymmetric in size. However, when FSI is high meaning that
one firm dominates the market, the presence of capacity constraints leads to a positive
relationship between FSI and prices as in the traditional theory. The combination of
these two effects (conduct shift and dominance) leads to a non-monotonic U-shaped
relationship between FSI and market power. Another prediction of this model is that
since collusion is easier when FSI is low, one should expect prices to be more stable
in markets where firms are symmetric in size. In section 4, using airline data, we
empirically test the relationship between market power and FSI. For a sample of 400
routes (or markets), we obtain a non-monotonic U-shaped relationship between firm
size inequality and prices (controlling for cost). We also find that prices are more
stable in markets where FSI is low. These results suggest that contrary to what
traditional theory predicts, tacit price coordination may in fact be easier in markets
where firms are symmetric in size. In section 5, we examine the conclusions,
potential extensions and policy implications of these results.
II. Firm Size Inequality and Market Power: Background
The link between market power and concentration has been central in
industrial organization. The traditional paradigm predicts that concentrated markets
are more likely to experience severe market power distortions. Bain's original idea is
that concentrated markets are more likely to be plagued by collusive behavior.
Following Bain's work, literally hundreds of studies (see Schmalensee 1989) have
examined the correlation between measures of performance and indexes of
concentration. It is interesting to note that most of these studies have used static
oligopoly theoretical models as a basis for their empirical modeling while Bain's idea
is essentially dynamic.
3
The Cournot model has been the basis for many empirical specifications. For
example, one can easily obtain the following relationship in a Cournot setting (under
some conditions on costs, see Waterson 1984):
Π
HERF
=
R
η
[1 ]
with:
Π: the industry aggregated profit
R: the industry aggregated revenue
HERF: the industry Herfindalh
η: the demand elasticity
Here, conduct among firms remains constant (firms behave à la Cournot whatever the
structure). The positive correlation is due to the externality that exists among firms.
As one firm's share increases, the importance of this externality decreases leading to
higher prices. The same type of relationship can be obtained using other hypothesis
concerning the firms' conduct (for example Stakelberg, see Waterson 1984).
To measure concentration, several measures have been used in the empirical
literature such as the concentration ratio C4 or the Herfindahl.1
Encaoua and
Jacquemin (1980) have proposed a list of axioms that a "good" index should satisfy.
An index should aggregate, in some way the number and the size distribution of firms
in the market. Different indexes put different weights on these two aspects. But
following Encaoua and Jacquemin, all indexes should be such that as the number of
firms in the market increases concentration decreases while concentration increases
as firm size inequality increases. These conditions are respected for the Herfindahl.
Indeed, this index may be expressed as:
cv 2 +1
HERF =
N
1
[ 2]
The Herfindahl is defined as:
HERF = ∑ s 2i
i
where si is firm i relative size (for example
the productive capacity in the market).
si corresponds to firm i share of consumers or its share of
4
with:
cv: the coefficient of variation (standard deviation of the share distribution divided by
the mean)
N: the number of firms
This decomposition shows the respective contribution of the number of firms and the
firm size distribution to the Herfindahl. Indeed, 1/N is the value of the Herfindahl if
all firms have the same market share. The effect of firm inequality is then measured
2
by cv /N .
Relationships like [1] and [2] imply that market power should be lower in
markets where two firms have the same size (shares of 50%-50%) than in a market
where firms are unequal in size (for example with share of 70% and 30%). They also
impose a structure on the relative effect of the number of firms and their relative sizes
2
(using the Herfindahl for example implies that the impact of cv /N and 1/N are
identical). It is the validity of this traditional view that we review in this paper.
III. FSI and Market Power When Conduct is Endogenous
Traditional static models predict that market power should increase with FSI.
In this section, we develop an example where conduct among firms is endogenous
and examine the link between the maximum sustainable price and FSI.
Let us suppose that two firms have capacity k1 and k2. These capacities are
fixed and exogenous. The marginal cost for both firms is zero up to capacity and
infinity beyond capacity. The market demand is linear D(p)=1-p. For each period,
firms simultaneously quote a price. The horizon is infinite. The demand for firm i
has the following structure:
q i = D ( pi )
if pi < p j
qi = si D( pi )
if pi = p j
qi = Max[0 , 1 − k j − pi ]
if pi > p j
5
The demand has the usual structure with efficient rationing. Usually in this type of
model, if firms charge the same price they split the market in some undetermined
fashion. Following Davidson and Deneckere (1990), we assume that a firm's market
share when both firms charge identical price will depend positively upon its capacity
share.2 More specifically, at equal prices, we consider a firm's market share to be:
liα
k iα
si = α
=
(li + l αj ) ( k iα + k αj )
α ≥1
[3]
with:
si : market share of firm i
li : capacity share of firm i
ki
(ki + k j )
The above rule is inspired from studies of the airline industry where it represents the
familar S-curve.3 In essence, it implies that consummers view a firm's capacity as an
aspect of quality. At equal price, the firm with a larger capacity share will attract
proportionally more consumers. We reproduce this relationship for α=2 in figure 1.
The restrictive aspect of this formulation is that consumers are unwilling to pay for
that quality. Its advantage, beside allowing a link with the empirical analysis of
section 4, is that it allows us to concentrate on pricing while ignoring issues related to
product differentiation. In any case note that using alternative sharing rule (such as
for instance, 50-50) would not alter our main conclusion.
2Davidson
and Deneckere use the following sharing rule:
Min( k i ,1)
si =
( Min( k i ,1) + Min( k j + 1))
Note that 1 is the maximum quantity that can be demanded. In their paper, they examine the choice of
capacities and the existence of excess capacity. In our example, we consider capacity exogenous and
examine how the maximum collusive price is affected by FSI.
3 This relationship has played a significant role in the deregulation debate. When firms could not
compete in prices (prices were set by the C.A.B.), they were attracting travelers by competing in terms
of quality (mostly flight frequency). Since deregulation, firms can also compete in terms of prices.
Yet capacity competition may remain an important aspect, especially if there is collusion on prices.
Empirical evidence suggests that collusion in terms of capacity may be harder to sustain than collusion
6
In the following, we assume that k2 ≤ k1 (firm 2 is the small firm). We also
assume, to simplify the analysis, that α = 2. In this setting, firms can tacitly maintain
market power (p > 0) by threatening to return permanently to the static Nash
equilibrium if its rival deviates from the collusive price.4 The firm decision can be
summarized by its incentive constraint that compares the gains through cooperating
or deviations. For firm i, cooperation will be optimale if:
1
IC i = (Π id − Π ic ) − ( Π ic − Π ip ) ≤ 0
r
[4]
with:
Π id : one period profit if firm i deviates from the collusive price
Π ic : one period profit if firm i cooperates
Π ip : one period profit for firm i in the static Nash equilibrium
r: the interest rate (thus the discount factor is δ=1/(1+r))
In this setting, we examine the price pattern when firm 2 gets smaller relative
to firm 1 (k2 gets smaller while k1 is fixed). We restrict ourselves to one case where
k1=1 (so that firm 1 is not capacity constrained).5 There are several ways to measure
firm size inequality in this model. In the following, we present the results using a
Gini coefficient of capacity shares li as this is the type of measure that will be use in
the empirical part of this paper.6 The Gini coefficient is defined as the double of the
area between the diagonal and the Lorenz curve. This curve links the proportionate
number of firms and the proportionate share. The Gini coefficient of shares may be
written as (see Waterson 1984):
Gini =
( N + 1 − 2 ∑ i li )
i
N
on prices (see Scherer 1980). In our example, collusion on capacity is not an issue since capacity is
exogenous.
4Note that more severe punishment may be possible (see Abreu 1986). However, as there exists
asymmetry among firms, the optimal structure of the most severe punishment may be difficult to
characterize (the simple stick and carrot structure may not be optimal in this setting).
5For
k1 ≥ k2 ≥ 1, one can easily show that collusion can be sustained for only k2 ≥
r k1. For 1≤ k2
r k1, no collusion is possible and thus p=0.
We could also use the Herfindahl to measure the inequality among the firms as in this setting the
number of firms is fixed.
<
6
7
Note that the firms are ordered by decreasing size (i=1 is the largest firm, i=2 is the
second largest etc). To advoid that the value range of Gini depends on N, we replace
in the denominator N by (N-1). In this case, the value of this coefficient is comprise
between zero and one: a value of one corresponds to a situation where one firm has a
monopoly while a Gini equals to zero means that all the firms have the size. Note
that in our case, the Gini is simply equal to (2 l1 -1).
Since Levithan and Shubik (1971) (later extended by Kreps and Scheinkman
1983), we know that the static Nash equilibrium may involve mixed strategies when
there are capacity constraints. Indeed, a firm may not only undercut its rivals, it may
also want to deviate up and act as a monopolist on its residual demand. In the case
we are examining, the static Nash equilibrium will be in mixed strategy. Each firm i
chooses its price based on a cumulative distribution function Fi defined on the
support [pL , pH]. This distribution can be determined in our example and thus the
expected prices, market shares and the price variances can be computed (see
appendix 1).
In the static equilibrium, we obtain the traditional relationship between market
power and firm size inequality: the expected prices are higher in markets where firms
have unequal size. Note however that this relationship is here explained by the
presence of capacity constraints. In figure 2, we represent the relationship between
the expected price of firm 1 and the Gini (the same pattern holds for firm 2).
Tacit collusion may be sustained in this setting by the threat to return to the
static Nash equilibrium in case of deviation. The incentive constraints are now given
by:
IC1 = ( p (1- p) - s1 p (1- p) ) - 1r (s1 p (1- p) - π 1N ) ≤ 0
IC 2 = (p Min [k 2 ,(1- p)] - p Min[k 2 ,s 2 (1- p)]) -
1
r
[5]
(p Min[k 2 ,s 2 (1- p)] - π 2N ) ≤ 0 [6]
with
π 1N = 0.25 (1 − k 2 ) 2 , the static Nash equilibrium profit of firm 1 (see appendix 1).
π 2N = p L k 2 , the static Nash equilibrium profit for firm 2 (see appendix 1).
8
In Appendix 2, we show that IC2 ≤ 0 implies IC1 ≤ 0 for collusive prices that yield a
total profit higher than the firm 1 profit in the static Nash equilibrium.7
The
conditions for collusion to be sustainable and the determination of the maximum
collusive price are presented in appendix 3.
We summarize the type of equilibrium as a function of Gini and r in figure 3.
The results are similar in nature to those of Davidson and Deneckers (1990). For
(Gini,r) ∈ Z1, the monopoly price (pm=0.5) is sustainable. For (Gini,r) ∈ Z2, no
collusion is possible, the only equilibrium being the static Nash equilibrium. Finally,
for (Gini,r) ∈ Z3, the monopoly price is not sustainable though p* (0<p*<pm) is.
This means that for r > 1, no collusion is possible for any value of Gini. For r ≤ 1,
there exist GL and GH such that for Gini ≤ GL, the monopoly price is sustainable.
For Gini > GL, no collusion can be maintained and for Gini ∈ ]GL,GH] some
collusion is possible (p* < pm). The intuition for this last case is the following: in
this interval, a price lower than pm limits the incentive of the small firm to deviate (its
deviation gains are limited by its capacity, (1-p) > k2) while, its gains in cooperation
are not affected by its capacity (as s2 (1-p) ≤ k2).
As an example, we show in figure 4 the pattern of the maximum sustainable
price for r = 0.23 as a function of the Gini coefficient of capacity shares. The
relationship FSI-price is then in this setting quite different than the one obtained
using traditional static models.
Here two effects are at play.
When firms are
symmetric in terms of size (Gini is low), collusion is possible for a discount factor
sufficiently high. When firms become unequal, collusion becomes more difficult to
maintain for the same discount factor. At some point collusion become impossible.
But as inequality further increases, a dominance effect resulting from the capacity
constraints leads to higher prices.
Another implication of this model that can be tested on data is the fact that
prices should be more volatile in markets where FSI is high. In these markets, the
equilibrium is indeed more likely to be in mixed strategies with firms cutting each
7This
means that it is the smaller firm that deviates from the collusive outcome. Note that this result
depends upon the sharing rule. Indeed, if we assume that consumers split evenly between the two firms
9
other in equilibrium. In figure 5, we show the relationship between firm 1 price
variance and the Gini coefficient for r = 0.23. When collusion is sustainable (for low
values of Gini), the price variance is zero while when collusion is not possible, the
equilibrium is in mixte strategies leading to price instability.8 As the markets tend to
a monopoly situation (Gini going to one), the price variability declines towards zero.
VI. FSI and Prices: an Empirical Evaluation
In this section, we test the effect of FSI for the airline industry.
Since
deregulation in the late seventies, numerous empirical studies have analyzed pricing
in this sector. It has been shown that higher concentration on a given route (usually
measured by the Herfindahl) means higher prices (for example, Bailey et al.,1985).
For most of these studies, the approach has been to estimate a price equation such as:9
Log ( Pirt ) = F (C , D, S ).
The average price charged by airline i on route r at time t is explained by variables
that reflect the cost (C), demand (D) and market structure (S) conditions. The market
structure variables usually include the route Herfindahl, the airline's market share and
variables that reflect airport dominance like market share at the endpoint airports and
the average endpoint airport concentration. We adopt this approach and review
specifically the link between prices and FSI. We want to examine whether there is a
U-shaped relationship between prices and FSI holding cost conditions constant as
predicted by the model in section 3. We also want to test if prices are more stable in
markets where firms are symmetric in size.
when they charge identical prices, we can show that it is the incentive constraint of the large firm that
matters in this case.
8 Note in a model where capacities is endogenous, Staiger and Wolack (1992) also show that markets
where collusion is difficult, the equilibrium will more likely be in mixed strategies leading to price and
market share instability.
9A few studies (see for example Berry 1990 or Brander and Zhang 1990) have adopted a structural
approach. However, given the difficulties of identifying the parameters in this setting, they do not
allow conduct to depend on market structure.
10
The Data and Sample
The main source of the data is the Origin and Destination Survey data bank 1A of the U.S. Department of Transportation. This data bank is a 10% random sample
of all the tickets used on U.S. airlines. We use the data for the second quarter for the
years 1988 to 1993. We restrict our sample to the 400 largest domestic routes. A
route is defined as a city pair.10 A complete description of the data can be found in
Appendix 4.
The Empirical Model
We estimate the following equation:
Log ( Pirt ) = αi + αr + αt + X irt β + θirt
whereα i , α r and α t are fixed effects at the airline, route and time level respectively.
These control for all the unobservables at these levels. For example, αi may control
for factors such as carrier reputation, fleet structure (assuming these aspects are
relatively stable over time), αr controls for the route length, the type of travelers
(business or leisure oriented routes) andα t controls for factor price changes (common
to all airlines) or macro economic conditions.
We include in X irt variables that have been identified in other studies as
being significant in explaining prices. These include: a cost indicator for an airline
at a specific time since no route specific cost data are available (COST is the average
operating cost per-passager-mile for airline i at time t), the number of destinations
served by the airline from both endpoint airports (N-DEST), the average airline
market share at the endpoint airports (AIRPORT-MS), the average Herfindahl at the
endpoint airports (AIRPORT-HERF).
In Appendix 4, we report the detailed
definition of each variable and some descriptive statistics.
10Multiple
airports in the same city are aggregated.
11
To control for market structure, we first include the carrier market share on
the route in terms of passagers (MSR), since it has been shown (see notably Evans
and Kessides 1993) that a firm with high market share may be able to charge higher
prices (perhaps as a result of product differentiation). We then include variables to
control for the two aspects of market concentration ie, the number of firms and FSI.
These variables are defined below.
Estimation and Results
Before examining the link between prices and FSI, we show how using
traditional measures of market concentration may be restrictive. First, we estimate
equation [3] with the route Herfindahl (HERF) to control for market concentration
(table 1, model 1). 11 The Herfindahl is defined as the sum of MSR squared of the
airlines operating on the route. This is the measure used in most airline pricing
studies. We find a positive and significant impact of route concentration on average
prices which conforms with other studies in this industry. An increase in HERF from
0.4 to 0.6 (about one std increase from the sample mean) leads to a price increase of
about 7.5%. This effect is somewhat larger than the estimate effect in other studies.
As Barla (1997) has shown this difference results from the sample structure that
includes only large markets.
Indeed, it appears that the effect of the route
concentration is larger in dense markets.
In model 2, we decompose the Herfindahl in its two elements, following
equation [2] that is cv 2 / N and 1/N.
We allow the coefficients on these two
variables to be different. The results indicate that both have a significant effect on
price but that the coefficient for cv2 /N is statistically different than for 1/N.12 Using
the Herfindalh index is thus restrictive.
11
Following Bailey et al. (1985), Call and Keeler (1985), Morrison and Winston (1995), we decide to
treat the variables that capture the route competitive structure as exogenous. Findings goods
instruments may indeed be difficult notably for the variables measuring FSI. Furthermore, given the
mounting evidence on the difficulties of inference when using weak instruments (see Staiger and Stock,
1997), we prefer treat the price equation as a reduce form.
12We obtain a F-value of 50.78 with (1, 11 492) degree of freedom.
12
In model 3, we examine the impact of FSI on prices. We measure FSI by a
Gini coefficient. Unfortunately, data on firm capacities are not included in the O&D
data, we therefore use the market share of passengers as a proxy for firm size.13 In
this setting, the Gini coefficient is defined as:
GINI rt =

1 
 N + 1 − 2∑ i MSRirt 

( N − 1) 
i
where i=1 corresponds to the largest firm, i=2 to the second largest firm etc.
As we want to examine the possibility that there exist a non-monotonic relationship
between prices and FSI, we also include the square of the Gini (GINI 2 ) . To control
for the effect of the number of firms, we include 1/N and its square (1 / N) 2 .
All these variables have a statistically significant effect on prices and as
predicted by the theoretical example in section 3, it appears that FSI has a nonmonotonic U-shaped relationship with prices. Two firms that equally share a market
(50-50 implying GINI=0) can charge an average market price 4.06% higher than two
firms that share the market 35-65 (i.e., GINI=0.3).14 If, however, a firm dominates
the market with 85%, leaving 15% to its rival (GINI=0.7), the average market price
will be 10.18% higher than in the market with the 35-65 market share structure. This
shows that the effect of firms inequality is indeed more complex than usually
assumed in the traditional view.
The number of rivals on a route lowers the average price. From N=2 to N=3
the average price would decline by about 0.57% while from N=3 to N=4, the average
price would decline by about 6.4%. The impact of the number of competitors is more
important than in other studies (Evans and Kessides 1993). Once again the sample
structure that includes large size markets explains this difference.
13
In the theoretical example of section 3, there is indeed always a positive link between a firm market
share and its capacity share.
14Note that these values are computed taking into account the changes in GINI and MSR.
13
In model 3, we use all the airlines on the route to compute GINI, even airlines
that have a very small market share.15 These small competitors may however not
necessarily serve the route on a regular basis and therefore should not be used to
compute GINI. In model 4, we compute GINI excluding airlines that have a market
share less than 1%. 16 The same pattern appears for GINI and N.17
All the other explanatory variables have a significant effect on prices
consistent with other studies. A higher cost (COST) increases the average price.
Dominance at the endpoint airports (AIRPORT-MS) is a source of market power
while the degree of concentration at the endpoint airports (AIRPORT-HERF) has a
negative impact (this is similar to Borenstein 1989). Finally, a carrier route market
share (MSR) is also a source of market power.18
So far, we have assumed that the error term has the usual property (distributed
iid Normal with finite variance). This is however restrictive. We might expect the
error term among carriers on the same route at the same time to be correlated. We
should also expect some serial correlation for an airline on a route. Ignoring these
aspects may affect efficiency of the estimates. We reestimate model 4 with the
following error structure:
θirt = µrt + ρθirt −1 + εirt
with:
µrt ∼ N(0,σ µ2 )
εirt ∼ N(0, σ ε2 )
15
Note that if the observations of airlines with market share of less than 1% are used to compute GINI
and N, they are not used as observation in the regression.
16 The market share of the other airlines are adjusted so that the sum of the market share of the airlines
that are used to compute GINI is still 1. The U-shaped pattern also remain true if we eliminate the
airline that have a market share of less than 3%.
17 Note that we also tried measures of FSI based on the coefficient of variation. In this case, the
pattern FSI-price is following: prices decreases as FSI for very low value of the coefficient of variation
(this effect is however not significant), then they slowly increases as the coefficient of variation further
increases. This is consistent with the theoretical model of section 3. Indeed, when we examine the
relationship price-coefficient of variation, collusion is possible only for a small range of low value of
the coefficient of variation. In fact, this measure of FSI pull most observations toward the extremes.
18Note that the results on the effect of FSI on prices are unaffected if we estimate the price equation
without MSR as an explanatory variable.
14
The error term for two carriers on the same route at a particular time is
correlated through a random effect µrt. The serial correlation is modeled as an AR(1)
process.19 In table 2, we reproduce the OLS and GLS estimates. As we have seen in
the theoretical model, there may also be some heteroskedasticity as price variability
may be more important when FSI is high. We report in table 2, the standard error
corrected for heterostedasticity. We find that the results using GLS are relatively
close to the OLS results even if we clearly reject the hypothesis that there is no
random effect (ie, we reject H0:σ µ2 = 0).20
As we have seen in section 3, if price coordination is easier when firms are
symmetric in size and if capacity constraints plays a role, one may expect prices to be
more stable in markets where FSI is low. We examine this possibility by using a
Glesjer-type test for the presence of heteroskedasticity and to test its source.21 We
find that the variability of the price residual significantly increases with FSI. It would
appears then that indeed prices are more stable in markets where firms are symmetric
in size.
Alternative Specifications
To test the robustness of the results we tried alternatives specifications. First,
the results appear robust to changes in functional form which we have tried. Second,
as the route fixed effect (α r ) controls for a lot of the variations across observations,
we examine how the results changes when we exclude this effect. In this case
however, we include the distance between the two endpoint airports as an other
assume that µrt is random (and thus uncorrelated with the Xirt) to allow identification of the
effect of GINI as this variable varies only with r and t. Following Green (1994), we use GLS after
transforming the data with the Cochrane-Orcutt transformation to estimate the model. We use the 1987
data when lags are needed.
19We
20
The lagrange multiplier test for
σ µ2 = 0 gives χ12 = 1135.9. Brueckner et al. (1992) also find that
taking into account these type correlations in the error terms does not change dramatically the
inference.
21 The test consists in regressing the absolute value of the residual (here the GLS residual) on the
variable that is suspected to be the source of the heteroskedasticity (here the variable GINI). We
obtain a positive and significant coefficient for GINI (Wald test statistic=5.07).
15
explanatory variable (DIST). We find that the U-shaped pattern price-FSI is more
pronounced using this specification (model 7, table 3).
Next, we estimate the price equation on a sample that only include markets
that can be characterized as duopoly among the initial 400 routes. 22 In this case, we
measure FSI by the ratio of the two firms market shares.23 We also obtained a nonmonotonic relationship between prices and FSI (model 8, table 3).
The pattern price-FSI could also be the results of the combination of
economies of traffic density and a dominance effect. The price would first decline as
FSI increases not because collusion becomes more difficult but because there are
efficiency gains for the firms that increase their market share.24 Suppose there are
two firms sharing a market 50-50 (GINI=0), an increase in the market share of a firm
may mean that this firm is better able to exploit economies of traffic density and this
may lead to a price decrease that has to be matched by the competitors.25 Prices
would then rise as FSI further increases by the same dominance effect as described
before. There are several indications that seem to suggest that this may not be the
best explanation consistent with the data. First, the evidence on price variability
provide support for the explanation based on price coordination. Indeed, it is not
clear how traffic density could explain that result. Second, if efficiency gains explain
the price decrease as FSI increases, this effect should be less important in large
markets where economies of traffic densities may be exhausted even by airlines that
have a small market share. We estimate the price equation separately on two subsample (the 200 largest markets and 200 smaller markets), we find that the pattern
price-FSI is in fact more pronounced in the sample that include only the 200 largest
markets.26 Second, if it is the efficiency gains of the firms with a large market share
22
A dupoly market is defined as a market where the two leading firms control at least 90% of the
traffic on the route.
23 The inequality is measured by MSR2/MSR1 (where MSR1 is the market share of the largest firm
and MSR2 is the market share of the smaller firm).
24 It has been clearly established that the increase in traffic density on a route lower the cost-perpassenger-mile (see Caves et al. 1984).
25 The effect of GINI is indeed a market-wide effect affecting all the firms on a route.
26 For the 200 smallest markets in the sample, the estimated coefficients on GINI and GINI2 are
respectively (standard error in parenthesis) -0.1559 (0.1047) and 0.3328 (0.0913). For the 200 largest
markets, they are -0.3265 (0.1199) and 0.5547 (0.1052).
16
that explains the price decrease as FSI increases, this effect should be better captured
by the variable market share of the dominant airline on the route. We reestimated the
model with this variable as an explanatory variable. The effect of GINI remained
identical.
V . Conclusions and Policy Implications
In this paper, we reexamine the link between market power and concentration.
We show that recent theoretical advances have implications for this relationship that
have not yet been tested. We focus on the link between firm size inequality and
market power. We show on a model with capacity constraints and endogenous
conduct that this relationship may be more complicated that the positive link usually
expected by the traditional literature. In fact, we show that the combination of two
effects may lead to a U-shaped relationship between FSI and prices. Indeed, we show
that tacit price coordination may be easier when firms are symmetric in size. This
means that as FSI increases, prices may decrease as collusion becomes more difficult
to maintain.
As FSI further increases, a dominance effect resulting from the
existence of capacity constraints appears leading to a positive link between FSI and
prices. We also show that as collusion is easier in markets where firms are symmetric
in size, prices should be more stable in those markets. We test these predictions for
the U.S. domestic airline industry. We find results that are consistent with both
predictions.
FSI measured by a Gini coefficient of market shares have a non-
monotonic U-shaped effect on the average price and price variability appears lower in
markets where FSI is low.
A possible extension of this paper would be to let the choice of capacity be
endogenous and try to understand why firms in the same market choose different
level of capacity. This extension may however be quite difficult as it would certainly
suppose introducing some asymmetry between the firms.
One obvious policy
implication of our results concerns conditions to apply for allowing firms to merge.
For example, in a market with three firms, if two firms with 35% and 30% of market
share respectively merge, this will lead to an increase in firm inequality that may
upset part of the effect of the decrease in the number of firms. Using our empirical
17
results, prices would actually decrease following this merger by about 2.58%
(assuming the market share of the new firm is just the sum of the market shares of the
two merging firms). On the other hand, markets where the merger would lead to
more symmetric firm size would lead to higher prices. Testing this on actual merger
events could be a way to provide further evidences for our hypothesis.
18
FIGURE 1. The S-curve
FIGURE 2. Relationship between the expected price of firm 1 and the Gini in the
static Nash Equilibrium
19
FIGURE 3. Type of equilibrium
FIGURE 4. Maximum sustainable price
FIGURE 5. Relationship between firm 1 price varianceand the Gini.
20
Table 1: Results (endogenous variable, Log(Pirt))
Variables
Model 1
Model 2
Model 3
Model 4
L[COST]
0.3297***
(0.0354)
0.3286***
(0.0353)
0.3187***
(0.0352)
0.3275***
(0.0352)
L[N-DEST]
0.0599***
(0.0126)
0.0605***
(0.0126)
0.0574***
(0.0125)
0.0588***
(0.0125)
L[AIRPORT-MS]
0.0814***
(0.0039)
0.0809***
(0.0039)
0.0806***
(0.0038)
0.0809***
(0.0039)
L[AIRPORT-HERF] -0.0463***
(0.0159)
-0.0484***
(0.0158)
-0.0632***
(0.0158)
-0.0504***
(0.0158)
MSR
0.1016***
(0.0107)
0.1010***
(0.0107)
0.1016***
(0.0107)
0.1010***
(0.0108)
HERF
0.3621***
(0.0250)
--
--
--
(1/N)
--
0.7337***
(0.0210)
2.6980***
(0.2048)
1.5172***
(0.1593)
(1/ N) 2
--
--
-3.2090***
(0.3513)
-1.8410***
(0.2502)
CV 2 / N
--
0.2640***
(0.0210)
--
--
GINI
--
--
-0.3142**
(0.1261)
-0.2498***
(0.0790)
GINI 2
--
--
0.5208***
(0.0956)
0.4485***
(0.0691)
# obs.
11 915
11 915
11 915
11 915
0.8213
0.8219
0.8232
R2
***:significant at 0.01%; **:significant at 0.05%; *: significant at 0.1%.
Fixed effects not reported.
0.8221
21
Table 2. GLS results (endogenous variable Log(Pirt)).
Variables
model 5-OLS
model 6-GLS (1)
L[COST]
0.3405***
(0.0373)
0.3151***
(0.0304)
L[DEST]
0.0341**
(0.0134)
0.0283**
(0.0157)
L[AIRPORT-MS]
0.0845***
(0.0110)
0.0824***
(0.0046)
L[AIRPORT-HERF]
-0.0333**
(0.0164)
-0.0935***
(0.0226)
MSR
0.0845***
(0.0110)
0.0581***
(0.0117)
(1/N)
1.3800***
(0.1657)
1.3504***
(0.2700)
(1 / N) 2
-1.5797***
(0.2626)
-1.5567***
(0.4022)
GINI
-0.3063***
(0.0822)
-0.3406***
(0.1162)
GINI 2
0.5059***
(0.0721)
0.5646***
(0.1079)
# of obs.
10 312
10 312
R2
0.8380
--
(1) std in parentheses corrected for heteroskedasticity (see White 1980)
***:significant at 0.01%; **:significant at 0.05%; *: significant at 0.1%.
Fixed effects not reported.
22
Table 3. Alternative specifications results
Variables
model 7
model 8
L[COST]
0.2298***
(0.0528)
0.3151***
(0.0967)
L[N-DEST]
0.1728***
(0.0163)
0.0749*
(0.0403)
L[AIRPORT-MS]
0.0587***
(0.0054)
0.1270***
(0.0117)
L[AIPORT-HERF]
0.0071***
(0.0081)
0.0117
(0.0406)
ROUTE-MS
0.1660***
(0.0156)
--
(1/N)
1.4596***
(0.1443)
--
(1/ N) 2
-2.020***
(0.2313)
--
MSR2/MSR1
--
-0.3523***
(0.0864)
(MSR2 / MSR1) 2
--
0.2831***
(0.0870)
GINI
-0.8621***
(0.0830)
--
2
GINI
0.9502***
(0.0699)
--
L(DIST)
0.4578***
(0.0055)
R2
0.5821
0.8973
# obs.
11 915
1641
***:significant at 0.01%; **:significant at 0.05%; *: significant at 0.1%.
Fixed effects not reported.
23
Appendix 1. Average Price and Market Share Under A Static Nash Equilibrium
With Capacity Constraint
Using Levitan and Shubik (1972) and Krep and Scheikman (1983) results we can
derive the support for the prices distribution p ∈ p L , p H with
[
pL =
(
1
1 − k 2 (2 − k 2 )
2
]
)
 1 − k2 
pH = 

 2 
For firm 1: the distribution function over this support is
F1 ( p) = 1 −
pL
p
for
[
p ∈ p L , pH
]
1 − k 2 (2 − k 2 )
and a mass point at pH given by 1 - F1 (pH) =
1 − k2
For firm 2:

(1 − k 2 )2  1
F2 ( p) =  (1 − p) −
4 p  k 2

[
p ∈ pL , pH
]
Given this, we can easily compute the average price and market share for the two
firms:
E ( p1 )
=∫
pH
pL
 1 − k (2 − k ) 
2
2 
F1′ ( p1 ) p1 dp1 + p H 


1 − k2


1 − k (2 − k ) 
2
2

= p L ln( p H ) − ln( p L ) + p H 

1 − k2



[
E ( p2 )
]
(1 − k )
=
2
4k 2
2
[ln( p ) − ln( p )] − 21k ( p
H
L
2
For a given price p1 , firm 1 market share is:
24
2
H
− p L2 )
s1 ( p1 ) = F2 ( p1 )
max{0, 1 − k 2 − p1 }
min{k 2 , 1 − p2 } + max{0, 1 − k 2 − p1
[
In the price range p ∈ p L , p H
]
+ 1 − F ( p )) . 1
} (
2
1
we have:
1 - k2 - p ≥ 0
so that k2 ≤ 1 - p
So we get
s1 ( p1 ) = F2 ( p1 )
1 − k 2 − p1
+ 1 − F2 ( p1 ) . 1
k 2 + (1 − k 2 − p1 )
(
)
 k 
= 1 − F2 ( p1 )  2 
 1 − p1 
So that the average market shares are:
E ( s1 ) =
∫
pH
pL
(
)
F1′ ( p) s1 ( p) . dp + 1 − F1 ( p H ) . s1 ( p H )
p (1 − k 2 ) 2
= L
4
p
H

p 1 − k 2 − p H 
1
1 
p
p
ln
ln
1
−
−
−
−
+ L 
(
)


2 
p 2p p
pH  1 − pH 

L
E ( s2 ) = 1 − E ( s1 )
The expected market shares of both firms depends positively of its capacity share,
graphically we have:
25
We can also compute the variance of the prices:
(1 −
k 2 (2 − k 2 )
) − [ E ( p )]
Var ( p1 )
= p L [ p H − p L ] + p H2
Var ( p2 )
2

1  (1 − k 2 )
1
p H − p L ] − [ p H3 − p L3 ] − E ( p2 )
=

[
k 2 
4
3

1 − k2
[
The expected static Nash equilibrium profits are:
 1 − k2 
E (π 1 ) = 

 2 
E (π 2 ) =
2
pL k2
26
2
1
]
2
Appendix 2
Collusion at p can be sustained if the following incentives constraints are respected:
2
1
 1 − k2  
IC1 = p (1 − p) − p s1 (1 − p) −  p s1 (1 − p) − 
 ≤0
 2  
r 
[
]
(
[
]
])
[
IC2 = p Min k 2 , (1 − p) − p Min k 2 , s2 (1 − p) −
(
)
1
p Min [k 2 , s2 (1 − p)] − p L k 2 ≤ 0
r
Proposition:
For all p such that
 1 − k2 
p (1 − p) ≥ 

 2 
2
if IC2 ≤ 0 then IC1 ≤ 0 .
Proof:
Note that since k2 > s2 (1 - p) for all p ≤ 1, we only have two cases to consider.
a) k2 > (1 - p)
In this case, we have (1 - s1) p (1 - p) < (1 - s2) p (1 - p) as s2 < s1 for
k2 < k1 = 1. So we need to show that:
 1 − k2 
p s2 (1 − p) − p L k 2 ≤ p s1 (1 − p) − 

 2 
2
Using Davidson and Denecher (1990) theorem 1 we know that:
pL k2
 1 − k2 
≥ k2 

 2 
2
So that we only need to show that:
(s
1
− s2 ) p (1 − p) ≥ (1 − k 2 )
 1 − k2 


 2 
2
If (s1 - s2) ≥ (1 - k2), the inequality above will hold. This is the case since:
(s1 - s2) =
1 − k 22
1 + k 22
≥ 1 − k 2 which gives
27
1+ k2
1 + k 22
≥ 1 which is true for k 2 ≤ 1 þ
b) k2 < (1 - p)
In this case, we have to prove that:
[
2
1 
 1 − k2  
p k 2 − p s2 (1 − p) −  p s2 (1 − p) − k 2 
  ≥
 2  
r 
[
p (1 − p) − p s1 (1 − p) −
]
]
1
r
2

 1 − k2  
p
s
1
−
p
−
)  2  
 1(


From a, we know that:
 1 − k2 
 1 − k2 
p s2 (1 − p) − k 2 
 ≤ p s1 (1 − p) − 

 2 
 2 
2
2
we need to show that:
p k2 - p s2 (1 - p) ≥ p (1 - p) - p s1 (1 - p) or
(k2 - s2 (1 - p)) ≥ (1 - p) (1 - s1) or
k2 ≥ 2 (1 - p) s2
Since (1 - p) is maximum at p = 0, we need to show that:
k2
1 ≥
≥ 2s2 or
2k 2
1 + k 22
leads to (k 2 − 1)
2
≥ 0 which is true. þ
28
Appendix 3. Determination of the Maximum Collusive Price
We have proven in appendix 2 that IC2 ≤ 0 implies IC1 ≤ 0 for all p such that
 1 − k2 
p (1 − p) ≥ 

 2 
2
(1)
We can easily show that s2 (1 - p) ≤ k2 for all k2 ≤ 1. This means that IC2 becomes:
[
[
]
]
IC2 = p Min k 2 , (1 − p) − p s2 (1 − p) −
[
]
1
p s2 (1 − p) − p L k 2 ≤ 0
r
Let's define rL such that for r ≤ rL the monopoly price pm = 0.5 is sustainable.
Since IC2 is strictly increasing in r, rL is defined by:
rL =
[
0.25 s2 − p L k 2
[
]
0.5 Min k 2 , 0.5 − 0.25 s2
]
(2)
Note that for pm (1) is respected for any k2 .
Let's define rH such that for r > rH no collusion (p > 0) is possible. As IC2 is
strictly increasing in r, rH will be defined by
r =
Max
p
p s2 (1 − p) − p L k 2
[ pMin[k , (1 − p)] − p s (1 − p)]
2
(3)
2
For k2 > (1 - p), pm = 0.5 solve (3). For k2 < (1 - p), p* solve (3) with p* given
by:


k 
2 p L +  4 p L2 − 4 p L  1 − 2  
s2  


p* =
2
0.5
1− k2 
Note that (1-p*) ≥ k2, p* (1 - p*) ≥ 
 and r(p*) > rL , for all relevant k2
 2 
(see figure 3) so that rH = r(p*). For a specific r, we can obtain (by numerical
resolution) k2L and k2H . For k2 ∈ [ k2L, k2H [, the maximum sustainable price is the
largest price that solve IC2 = 0 (as IC2 is convex in p). This price is given by:
2
29
2
~
p =
1 
1  1


 1 
−  k 2 − s2 − s2  +  k 2 − s2  1 +   − 4  1 +  s2 p L k 2

 r
r 
r  r

1

2 s2  1 + 

r
30
Appendix 4. Data and Variables
I. Data Source
(1) Origin & Destination (O & D) Survey Data Bank 1A, 1987 - 1993, second quarter.
(2) The Air Carrier Financial Statistics Quarterly and Air Carrier Traffic Statistics Monthly from
U.S. Department of Transportation, 1987 - 1993. The cost variables are computed using these data.
II. Record and Route Selection Criteria
Some records were eliminated in aggregating the data by airline, route, and time level. A record
is an observation for a specific airline, time, origin, destination, itinerary and price. The following
records were eliminted.
(1) Records with non-credible fares. There are a variety of reporting errors in the O&D data
(fares are occasionally misreport or miscode).
(2) Records that correspond to unusual itineraries like open jaw tickets (example: BostonSeattle-Tampa)
(3) Records with itineraries that involve more than one connection each way.
(4) Interline tickets. This is when a passenger uses more than one airline to complete a trip.
(5) Record that includes at least on segment in a non-coach class.
From the original pool of over 11,000 markets, we select the 400 largest domestic routes based
on 1986 traffic.
III. Variables
Here are the detailed description of the different variables used in the analysis. The level of an
observation is route-airline-time. To reach this level, different itineraries and price levels for the
same route on an airline need to be aggregated. (for example, an airline that flies the route New York
- Los Angeles, may have a direct flight or a indirect flight going through Chicago.) The different
variables were aggregated using the number of passengers on the different itineraries as a weighting
system.
Pirt : Pirt is the average coach fare paid by the passengers for a one-way trip on a particular
airline, route and quarter. A round trip is considered as two one-ways and thus the price is divided by
two. Both restricted (Y) and unrestricted economy fares (YD) are included in our sample.
COST : is the ratio of the firm total operating cost divided by the total number of passengermile.
N-DEST : N-DEST is the weighted sum of number of destination offered by an airline at both
endpoints (the weight is the percentage of the airline travelers on the route originating from each
endpoint).
AIRPORT-MS: AIRPORT-MS is the weighted average of the airline market share of originating
passengers at the two endpoint airports (the weight is the percentage of the airline passengers on the
route that originates at each endpoint). For each endpoint airport, the airline market share of
originating passengers is the ratio of the number of the airline passengers that originate from the
airport (whatever their destination), divided by the total number of passengers that originate from the
airport.
AIRPORT-HERF : For each endpoint, the Herfindahl is computed the market shares computed
on all the passengers originating at that airport. On a particular route, AIRPORT-HERF is the
31
weighted average of Herfindahl at each endpoint of the route (the weight is the percentage of
passengers on the route that originate at each endpoint).
MSR : MSR is an airline’s share of the market defined as the number of passengers traveling on
the airline divided by the total number of passengers on the route.
The other variables are defined in the text.
Mean and standard deviation for the sample
Variables
Mean
(std)
Pirt
152.0
(52.29)
COST
0.1377
(0.0185)
N-DEST
107.4
(45.60)
AIRPORT-MS
0.1550
(0.1224)
AIPORT-HERF
0.2339
(0.0896)
MSR
0.1957
(0.2350)
HERF
0.4179
(0.2350)
N
7.9045
(2.1443)
CV 2 / N
0.2931
(0.1657)
GINI
0.7218
(0.1342)
MSR2/MSR1 (1)
0.3568
(0.2894)
DIST
1007
(628)
(1) computed on duopoly markets only.
32
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