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WP-2017-005
Center for Advanced Economic Study
Fukuoka University
(CAES)
8-19-1 Nanakuma, Jonan-ku, Fukuoka,
JAPAN 814-0180
Faculty of Economics
Fukuoka University
Bandwagon Effects and Local Monopoly Pricing
In Professional Team Sports Market.
J M Kang, Y. J. Ryu, Peng Liu
CAES Working Paper Series
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Bandwagon Effects and Local Monopoly Pricing
in Professional Team Sports Market.
J. Moonwon Kang
Professor, Faculty of Economics, Fukuoka University, Japan.
Young Jin Ryu
Assistant Professor, Kitakyusyu City University, Japan
Peng Liu
Assistant Professor, Guangdong Ocean University, China.
Abstract:
This study establishes a case that bandwagon effects impart upward sloping segments to
market demand curves. As an example of such cases, we analyze a professional team
sports market characterized by local monopoly supply and demand induced by
bandwagon effects. Assuming consumers’ choice of attending the game is indivisible
and there exist systematic differences in loyalty to the local team among potential
attendants, we build a positive relation between ticket price and the number of entrants
to stadiums. We derive theoretical results that size of stadium does not affect the
optimum ticket pricing of monopoly team, and the strong of bandwagon effects is
associated with lower ticket prices. Further, we show that enlargements of stadium size
do not increase teams’ investments to team quality. A look at the evidence suggests our
theoretical results are largely consistent with real world observation in Japan and
Europe.
* This study is supported by the program for scientific research start-up funds of
Guangdong Ocean University.
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Introduction
In his 1991 paper, Gary Becker suggested that, when individual demand is positively
related to the market demand, i.e., when there exist bandwagon effects in market
demand, the demand curve for the good is positively sloped for a wide range, and this
upward sloping demand curve could explain the well-observed and perpetuating
phenomenon of excess demand(queues to attend the successful sporting events) or
excess supply(vacancies in the stadium) in sports product markets. Becker suggested
that, if the pleasure from attending the game is greater when many people want to attend
the same event, the demand curve first rises and then falls, and combined with the fixed
supply(given number of seats in the stadium), the upward sloping demand curve would
be a source of excess demand or supply. This view, however, has been criticized by
Gisser et. al.(2009) who assert that Becker`s hypothesis of upward rising demand curve
is valid only when bandwagon effect is preposterously large, and is inconsistent with
standard empirical observations.
Both Becker’s hypothesis and criticism by Gisser et. al. and others(1) remain
unsatisfactory because these studies do not make clear the corresponding
utility-maximization and profit-maximization problems while discussing the possible
relations among bandwagon effect, demand curve and market outcome. The purpose of
the present study is to establish a sensible case wherein Becker’s hypothesis is valid,
and to examine its implications in sports product market.
The model we present here is based on the following basic assumptions: one, we
describe individual’s choice of attending a professional sport game and assume this
choice is indivisible(attend or not), two, loyalties to local sport team differ among
potential attendants to the game, and strong loyalty to the team yields higher utility from
the game, three, there exists bandwagon effect, which produces positive relation
between the number of attendants and pleasures obtained from attending the game, four,
the sport team under our investigation is a local monopoly and, in the short-run, the cost
of providing the game is constant regardless of the number of attendants.
This study is organized as follows: Section 2 describes the basic model and establishes
a case in which the demand for the sports game is positively related to ticket prices for a
wide range. In Section 3, we explain how local monopoly pricing of match day ticket is
affected by bandwagon effects in demand sides. We will derive a theoretical view that
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the stadium size would have no influence to the optimum ticket pricing, which seems
largely consistent with real world observations in England Premier League and Japan
J-1 League. Section 4 extends the basic analysis to examine how changes in the strength
of bandwagon effect affect the ticket pricing and derives a result that strong bandwagon
effects will lower the monopoly price. We suggest a view that regional differences in
bandwagon effects could explain the wide discrepancies in match day ticket prices
observed among top leagues in European football. In Section 4, we further analyze the
possible relations among bandwagon effects, stadium sizes and investments in team’s
quality. We found that investments in team quality are not strongly related with stadium
sizes. Concluding remarks is given in Section 5.
Basic Framework
We assume the net surplus of the ith attendant to the game is(see L. Pepall and J.
Reiff(2016) among others for similar specification of utility function)
(1) {Ui = θiυ(nd) − λp
= 0
if attend the gameif not
where p is the ticket price, nd is the number of fans who want to attend the game with
the given ticket price p, and λ is the marginal utility of income which is assumed to be
identical to every individual. The parameter θi ∈ [0 1] ranks orders of population by
the intensity of their preference of the game(which would be related to their loyalty to
the team). We assume population is uniformly distributed on the line θ ∈ [0 1]. Thus,
normalizing total population n0 = 1 , the number of population whose preference
parameter θi is larger than θ̂ will be simply (1 − θ̂) (see Figure 1). It is important to
note that the utility associated with the game is defines as a function of nd, which is
not necessarily equal to actual number of attendants in our model as we will show
below(see footnote). We assume υ is a twice differentiable function and
υ′ =∂υ(nd)
∂nd> 0, ∙ υ′′ =
∂2υ(nd)
∂(nd)2< 0
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FIGURE 1
Bandwagon effect, originally defined by Leibenstein(1950) as “the desire to join the
crowd, be ‘one of the boys`, etc. – phenomena of mob motivations and mass psychology
either in their grosser or more delicate aspects”, is captured in this model by the
assumption of υ′ > 0.
Define θi∗ as in (2),
(2) θi∗υ(nd) − λp = 0
then, it would be apparent that, for given ticket price p, those fans whose preference for
the game θi is larger than θi∗ will want to attend the game, and the following relation
holds,
(3) nd = 1 − θi∗.
From (2) and (3), we obtain
(4) nd = 1 −λP
υ(nd) or P =
1
λ(1 − nd)υ(nd)
The equation (4) represents the demand schedule in the proposed model showing the
relation between P and nd(2). The demand curve implied in (4) can be upward sloping
for a wide range. This possibility can be shown as follows: Total differentiating (4)
yields the equation,
dnd = −λ
υ(nd)dP + P
υ′
[υ(nd)]2dnd
Substituting (4) into the above equation and rearranging, we have
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(5) ∂P
∂nd= β[(1 − nd)υ′ − υ] , β =
1
λ
Which is positive when nd is sufficiently small and the bandwagon effect υ′ is
sufficiently large (υ′ >υ
1−nd). Put differently, differentiating (2) with respect to nd, it
follows that,
∂P
∂nd= β {
∂θi
∂nd υ(nd) + θiυ
′(nd)}
The first term ∂θi
∂nd υ(nd) is negative, which implies that the marginal attendant’s
enthusiasm to the game is falling as the number of attendants increases(which we want
to call loyalty weakening effect), and the second term is positive when there exist
bandwagon effects. When the number of attendants, nd, is small, it is probable that the
bandwagon effect dominates loyalty weakening effect, and ∂P
∂nd > 0. In general case of
consumer demands, an individual adjusts quantity consumed to equate the marginal
utility of consumption and the utility evaluation of the given price. In the model we
investigate here, however, an individual can’t adjust her level of consumption because
the relevant choice is either 1(to attend the game) or 0(or not). In the specific model this
study presents here, we have positive bandwagon effects and the negative loyalty
weakening effect instead of the usual income and substitution effects. Both Bandwagon
effect and loyalty weakening effect changes systematically according to the changes in
the number of possible attendants as our analysis below will show.
To make our argument simple and clear, we will denote nd by n, and we will specify
the utility function υ as
(6) υ(n) = Anα, A > 0, 0 < α < 1
We could call α as the strength of bandwagon effect. We will think in a way that A
means team quality which is assumed to be fixed here. We will discuss the problems
related with monopoly investments in team quality in Section 4. The demand relation
(4), then, will be rewrote as,
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(7) n = 1 −λP
Anα
Or P = βAnα(1 − n). From (7), we can calculate,
(8) ∂n
∂P=
1
βAnα−1[α − (1 + α)n]
Which is positive when n <α
1+α. The demand curve is upward sloping when n <
α
1+α
reaches the maximum at n̂ =α
1+α , then decreasing thereafter as Figure 2 shows. The
maximum reservation price denoted by Pmax in Figure 2 can be calculated from (7) as,
(9) Pmax = βAαα
(1 + α)(1+α)
FIGURE 2
If we assume the marginal utility of income λ decreases as the income level rises, then,
other things being equal, an increase in income level of potential attendants shifts the
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demand curve upwards and Pmax becomes higher as the above Figure shows. As we
will show in the next section, Pmax is the profit-maximizing ticket prices in most cases,
and Figure 2 can explain historical trends of rising ticket prices associated with rising
household income.
Local Monopoly Pricing in Sports Product Market
We assume the team under our investigation is local monopoly and there is only one
team in the region. Further, we assume the local monopoly team maximizes the
short-run profit. Ignoring revenues from the media, sponsors, and other sources, for
simplicity of exposition, we define the profit obtained by the local monopoly team from
the game as in (10) for the case that the given size of stadium(denoted by n̅ ) is smaller
than n̂ .
(10) π = Pn − C
Where C represents the total cost which is assumed to be fixed and not varied by the
changes in the number of attendants. Here, firstly we explain the local monopoly team’s
profit maximization problem using the specific utility function (6), and then generalize
the results obtained. Note that, by the chain rule,
∂π
∂P= (
∂π
∂n) ∙ (
∂n
∂P)
If utility function is as given in (6), it follows from (7) and (8),
(11) ∂π
∂P= βAnα[(1 + α) − (2 + α)n] (
∂n
∂P)
When n < n̂(i.e., when ∂π
∂P> 0), if (1 + α) − (2 + α)n > 0(i.e., if
1+α
2+α> 𝑛), then
∂π
∂P> 0. Both
α
1+αand
1+α
2+αare monotonic increasing in α (0 < 𝛼 < 1), and 0 <
α
1+α<
1
2,
1
2<
1+α
2+α<
2
3, and thus,
α
1+α<
1+α
2+α. Therefore, in the case of n < n̂ (<
1+α
2+α),
∂π
∂P> 0
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without further conditions. Accordingly, if n̅ <1+α
2+α, then the profit maximizing pricing
will be Pmax as ∂π
∂P> 0 is positive within the relevant region.
This result can be generalized easily as follows. From (4), (10),
∂π
∂n= β[n(1 − n)υ′ + (1 − 2n) υ]
And
∂π
∂P= [n(1 − n)υ′ + (1 − 2n) υ] {
1
(1 − n)υ′ − υ}
In the case of ∂n
∂P=
1
(1−n)υ′−υ> 0, if n(1 − n)υ′ + (1 − 2n)υ > 0 then
∂π
∂P> 0.
Note that, when n <1
2, n(1 − n)υ′ + (1 − 2n) υ is positive. Thus, if we assume n̅ <
n̂ <1
2 which is reasonably the case, Pmax is a unique profit maximizing price.
Figure 3 explains the above arguments. When the given stadium size is n̅, the local
monopoly pricing will be Pmax and there exists an excess demands for the game of
n̂ − n̅(3).
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FIGURE 3
When the given stadium size n̅ is larger than n̂, we redefine the expected profit from
the game as in (12). Suppose that profit from the game is maximized at the point “a” in
Figure 3 where the number of attendants is n0 and the corresponding ticket price is P0
if there is a way to implement the point “a” , excluding the other possible outcome of
“b”. In the present model, however, there is no way to implement “a” and there arises a
problem of indeterminacy if n̅ > n̂. So, if the team is risk neutral, expected profit from
the game will be defined as follows,
(12) π =1
2(Pn′ + Pn) − C
where n̅ > n̂ and n′ is defined as in Figure 3. From (12),
(13) ∂π
∂P=
1
2[(n′ + n) + P (
∂n′
∂P+
∂n
∂P)]
Thus, we know that ∂π
∂P> 0 when n̅ > n̂ if
∂n
∂P is sufficiently small, i.e., if the price
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elasticity of demand is sufficiently small. From (7), the price elasticity of demand(εp)
can be calculated as,
(14) εp =1 − n
α − (1 − α)n
which is decreasing in α. Accordingly, if bandwagon effect is sufficiently strong(which
we assume as the case under our investigation), ∂π
∂P is positive when n̅ > n̂.
FIGURE 4(a)
FIGURE 4(b)
Even if the stadium size becomes larger to n̅′, the local monopoly pricing remains
unchanged and we can observe the vacancies as many as n̅′ − n̂. An important and
interesting observation can be made here that the level of monopoly pricing is not
affected by the stadium size. It seems that this theoretical result is largely consistent
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with some empirical observations. Figure 4(a) shows the relations between the stadium
sizes and match day highest and lowest ticket prices in England Premier League,
2016-17, and Figure 4(b) is the relation between the average ticket price and stadium
sizes in J1 League(1st division of professional football league in Japan) in 2016 season.
Figure 4 seem to support the above theoretical result(4).
Bandwagon Effects, Match Day Ticket Price, and Investments in Team Quality.
We will examine the relation between the strength of bandwagon effects and the
market demands. In below, we will prove that, for the given ticket price p, an increase in
α expands the demands for the game when n < n̂.
Firstly note that ∂Pmax
∂α> 0 and for given n,
(15) ∂Pmax
∂α= βA(1 − n)nd ln n < 0
Further, from the demands curve (7), P = Anα(1 − n), if we write P(α) = βAnα(1 −
n) and P(εα) = βAnεα(1 − n) where ε > 1 is an arbitrary number. Then, it is
evident that P(α) − P(εα) = βA(1 − n)nα(1 − nε) > 0 . When n > 1, 𝜀 > 1 ,
therefore P(α) > 𝑃(εα) for ∀n ∈ (0 1) . Therefore, an increase in α shifts the
demands curve downward as described in Figure 5, and Pmax is decreasing as the
bandwagon effect becomes stronger.
Combined with the results obtained in Section 3, The above result implies that the team
in a region of stronger bandwagon effect would charge cheaper ticket price. We want to
suggest a possibility that, though it would be a difficult job to measure the strength of
bandwagon effect α, regional differences in bandwagon effects can explain wide
discrepancies in match day and season ticket prices among top leagues in European
football(see, for example, G. Nufer and J. Fischer, 2013).
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FIGURE 5
Next, we will investigate the relation between the bandwagon effect and monopoly’s
investment to team quality.
When n̅ < n̂, we redefine the profit from the game as
(16) π = Pmax ∙ n̅ − (c + ωq)
= hA(q)n̅ − (c + ωq)
Where h = β (αα
(1+α)1+α) , and q means the total number of talents that the team
possesses, ω is the unit cost of talents, c is the other fixed costs(see, for example, S.
Kesenne, 2007). From above, it follows that,
∂π
∂q= hA′(q)n̅ − ω
∂2π
∂q2 = hA′′(q)n̅ < 0.
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If we define q∗ by A′(q∗) =ω
hn̅, it follows
(17) ∂q∗
∂ω=
1
A′′hn̅< 0
(18) ∂q∗
∂n̅= −
A′
A′′n̅> 0
(15) shows that the optimum investments to team quality increases as the stadium size
becomes larger when n̅ < n̂.
When n̅ > n̂, the profit is defined as
(19) π = Pmax ∙ n̂ − (c + ωq)
= hA(q)n̂ − (c + ωq)
And, following the same procedure as above, we can calculate that A′(q∗) =ω
hn̂ and
(20) ∂q∗
∂ω=
1
A′′hn̂< 0,
∂q∗
∂n̅= 0.
The relations (14) ~(17) reveals that, firstly, an increase in the price of talents decreases
investments to team quality, secondly, when the stadium size is smaller than n̂ so that
when there is no vacancy in the stadium, enlargements of the stadium size increases
investments to team quality, thirdly, when there are vacancies in the stadium,
enlargements of the stadium size does not affect investments to team quality.
Figure 6 shows the relation between stadium sizes and teams’ investments to players in
England Premier League(Fig. 6(a)) and Italian Serie A(Fig. 6(b))(5). These relation show
that, tough we can of course find differences between big market clubs and small
market clubs, it can be safely argued that teams’ investments to team quality are not
strongly related with stadium sizes.
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FIGURE 6(a)
FIGURE 6(b)
Concluding Remarks
Recent years have witnessed a boom of researches exploring the possible relations
between social relations and market demands, and this study can be viewed as one of
these researches. We elaborate Becker’s inferences(1991) that bandwagon effects could
bring an upward sloping demand curves, and examine how bandwagon effects would
affects local monopoly pricing in sports product market. We establish a case wherein
Becker’s inferences are valid and make explicit what has been implicit in Becker’s
arguments. We can derive further implications of the upward demanding curve
combined with local monopoly by specifying the strength of bandwagon effects. We
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explain the excess demand and supply, often observed in professional team sports
worldwide, as equilibrium phenomena wherein both consumers’ utilities and producer’s
profit are maximized. We provide a view that this seemingly awkward “equilibrium” is
ascribed to bandwagon effects associated with upward demand curves and market
structure of local monopoly.
Upward sloping demand curves would yield several new insights in studies about
competitive balance, revenue sharing, players’ labor markets and other important
research areas of professional team sports markets. We think the model we propose in
this study could be extended and applied in these directions. However, an important
question remains unanswered in this study, that is, the very basic question how we can
measure the bandwagon effect from the empirical data. We think this measurement
problem is related with our understanding or interpretation of social and psychological
forces underlying bandwagon-like behaviors. For example, it might be that bandwagon
effects are the results of identity confirming behaviors of consumers. If identity
confirming behavior enhances consumers’ utilities and identity confirming is made by
mimicking the behaviors of others(of same social identity group) as postulated by
Akerlof and Kranton(2000), then bandwagon effects in this study could be explained as
behaviors of regional identity confirmation. Under this situation, the strength of
bandwagon effects would be partly determined by the historical process under which
local professional sports club has been related to local identity of the region.
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Footnotes.
(1) Becker’s postulation is analyzed in a game theory framework by Karni and
Levin(1994), for which Plott and Smith(1999) indicate behavioral inconsistencies of
basic assumptions they adopted.
(2) The demand curve in Figure 2 shows the relation between the market demand and
the reservation price of the marginal attendant, which is somewhat different from the
ordinary demand curves representing the sum of individual demands for given
prices. We think this raises some confusion among the researchers. For example,
Gisser et.al.(2009) wrote, “ if the zigzagged curve that Becker(1991) labeled “d”
shown in Figure 1 is not really a demand curve then the question becomes: What is
it?”.
(3) In this study, we explain the excess demand and excess supply as phenomena
observed under “market equilibrium” – equilibrium in the sense that producer’s
profit and consumers’ utilities are maximized.
(4) The data for England Premier League is from BBC SPORT football(2016) and the
data for J1 League is from Football GEIST(2016).
(5) The data for Italian Serie A is from “Serie A Transfer and Wage Budgets”,
REALSPORT(Nov. 11, 2016) and the for stadium sizes are from BBC SPORT
football(2016) and “List of English Football Stadiums by Capacity(Wikipedia)”.
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Karni, Edi and Dan Levin(1994). Social Attitudes and Strategic Equlibrium:A
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Kesenne, Stefan(2007). The Economic Theory of Professional Team Sports, Edward
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Leibenstein, H(1950). Bandwagon, Snob and Veblen Effects in the Theory of
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