Multiproduct pricing in major league baseball: a principal components analysis.
Depken, Craig A., II ; Grant, Darren
I. INTRODUCTION
How do firms set their prices? The question is easily answered if
the firm sells a single good at a single price: There should be an
inverse relationship between the percentage markup and the elasticity of
product demand q, as prescribed in the famous Lerner Index relation (P -
MC)/P = 1/[eta].
One could easily overlook the dual nature of this relation--it is
both structural and heuristic. It is structural because it is formally
derived from fundamentals; it is heuristic because it can be articulated
in simple, intuitive terms. Thus this one equation gives, in fact, two
overlapping descriptions of how firms set their prices. Both have value:
textbooks in economics and marketing focus on heuristics, whereas
structural models are preferred for policy analysis.
Of course most firms do not sell one good, but many related goods,
at nonuniform prices. Then the concordance between structural models and
heuristics cannot be sustained. Structural models of nonlinear multiproduct pricing, elaborate equations derived from first principles,
are precise but difficult to simplify, interpret, or apply. In
comparison, heuristics such as tying, bundling, or two-part pricing are
crude but intuitive and easy to use. There are still two overlapping
descriptions of how firms set their prices, but they are no longer
conjoined.
This divergence is recognized in theory, but not in application.
Empirical studies of multiproduct pricing have uniformly adopted a
structural approach, despite many obstacles to its use: a large number
of relevant variables, both dependent and independent; scant data on
some of these variables; and estimation difficulties. These formidable
obstacles have sharply limited the number of empirical analyses of
multiproduct pricing and have compromised the ease and rigor with which
they are conducted. As a result, despite the ubiquity of this problem in
practical business decisions, the literature does not contain a set of
stylized facts that tell us how multiproduct firms set their prices.
Our solution--inductive, not deductive; practical, not perfect;
heuristic, not structural--relies upon a factor-analytic technique,
principal components analysis. This reveals, rather than imposes,
structure in the data, breaking down price co-movements into a few
independent patterns that can be easily described, naturally
interpreted, and rigorously tested. In this paper we explicate this
method and apply it to pricing in Major League Baseball (MLB), a topic
of long-standing interest in sports economics and the quintessential multiproduct pricing problem.
Pricing in MLB occurs in geographically isolated markets in which
most teams are local monopolists; all sell multiple products including
tickets, parking, and concessions, at prices that vary substantially and
nonuniformly across teams and across time. Several factors emphasized in
the general theory of multiproduct pricing are potentially relevant: The
general demand for any team's "product bundle" fluctuates
substantially over time, whereas the products sold by the team are
related in demand and potentially subject to nonlinear pricing, such as
second-degree price discrimination, in order to maximize the capture of
consumer surplus.
Yet a structural analysis of multiproduct pricing in MLB is
impractical for all the reasons listed above. The required concession
quantity or revenue data are simply not available; nor are good
instruments for prices. And the formal theory of multiproduct pricing is
not well developed for this straightforward yet nontrivial case, which
combines an obligatory entry fee (the ticket price) with complementary,
discretionary, multiple-purchase concessions. Acquiring a basic
understanding of pricing in this market requires a methodology that
needs little a priori theoretical structure while accommodating many
prices but limited data on quantities, costs, and demand--precisely the
province of principal components.
Our analysis of the pricing decisions of all MLB teams from 1991 to
2003 indicates that the factors stressed by theory are indeed relevant.
A general demand effect explains about half of the joint variation in
prices charged by teams, while changes in price differentials across
products associated with second-degree price discrimination and demand
complementarities explain another 20%, with price discrimination
seemingly the more important of the two.
Recognizing the novelty of using principal components in this
context, we begin by describing how this method is used to analyze
multiproduct pricing, and how it compares to structural modeling.
Section III discusses the theory and evidence on the pricing decisions
of MLB teams and describes the data, whereas Section IV presents the
empirical results. The final section provides conclusions.
II. MULTIPRODUCT PRICING: THEORY AND PRACTICE
A. Structural Models
A structural model expresses prices in terms of demand parameters
and costs, specifying the functional form using economic theory. These
models can be used to test theory, to infer the type of competition
present in the market, or for policy analysis. For example, Guilietti
and Waterson (1997) use structural estimation to determine whether
competitive retailers price their products as predicted by Bliss (1988),
whereas Dub6 (2005) uses it to infer the effects of mergers in the soft
drink industry on prices and welfare.
Each price markup in a structural model depends, in general, on the
firm-level own- and cross-elasticities of demand for all products in the
market. This focus on behavioral fundamentals makes these models well
suited to policy analysis, but also demands a lot of data, because the
number of parameters is sizeable even when there are just a few products
and increases rapidly when there are more products. Estimation thus
requires extensive price and quantity data, along with a mechanism to
account for price endogeneity. This can be difficult to execute.
Guilietti and Waterson, for example, were forced to aggregate 31
products into seven categories to permit estimation, and utilized
industry, rather than firm, elasticities because of data limitations.
According to Dub6 et al. (2005), the problem is magnified when analyzing
price dynamics, which has not yet been done using the structural
approach.
Structural estimation also requires a lot from theory. When the
appropriate theoretical relationships can be imposed on the analysis,
the inferences one can draw from the data are increased and
strengthened. But this is often hard to do for multiproduct pricing,
where these complex relationships are sensitive to model assumptions,
according to Spence (1980); "rather opaque as to intuitive
content," according to Sibley and Srinagesh (1997); and
"difficult to apply empirically," according to Bliss (1988).
(1) Even if a structural model is properly specified, estimation is not
simple. Dubr, for example, relies on numerical techniques (Monte Carlo
integration and the method of simulated moments) to obtain parameter
estimates of his model.
Thus the power of structural modeling comes at a cost, which limits
both the type and number of empirical studies that can be executed. This
motivates the introduction of an alternative method that can uncover
relationships and answer questions structural modeling cannot, while
requiring less data, theoretical structure, and computational effort.
B. Principal Components
Our factor-analytic perspective treats prices as governed by a few
uncorrelated, unobserved, underlying latent variables. Principal
components analysis can be employed to recover these latent variables
and their relation to prices. A common technique for "untangling
complex patterns of association in multivariate data" (Green,
1978), principal components have been used to analyze market prices, in
Doll and Chin (1970); asset prices, in Roll and Ross (1980) and others;
business cycles, in Forni and Reichlin (1998); industry profitability,
in Slade (2004); and government regulation in the U.S. economy, in Goff
(1996). Here, the patterns obtained provide a heuristic description of
the primary determinants of prices that can be linked to theory via
empirical tests that can be feasibly conducted in a wide range of
applications.
One observes a sequence of prices set by a multiproduct firm on
each of N goods. The analysis decomposes each price vector, [P.sub.j],
into a linear function of N independent latent variables, or principal
components, [Z.sub.k], weighted by scalar coefficients [A.sub.k]:
[P.sub.j.t] = [summation][A.sub.k,j] [Z.sub.t,k], k = 1, ..., N. These
terms are determined by the way the principal components are calculated;
none are prespecified. Analytical structure is imposed on the data, as
in any parametric analysis, but not theoretical structure.
Each latent variable [Z.sub.k] is determined, up to an arbitrary
scale factor, by demeaning the price matrix P and calculating the
eigenvectors and eigenvalues of [P.sup.T]P, placing the latter in a
diagonal matrix [lambda] and the associated orthonormal eigenvectors in
matrix A. (2) These matrices satisfy [A.sup.T][P.sup.T]PA = [lambda].
The matrix of principal components Z is set equal to the matrix product
PA; its covariance matrix is then [lambda]. Prices are then
reconstructed by P = [ZA.sup.T], as above. Although the number of
components equals the number of prices, typically most variation is
explained by a few components with meaningful interpretations; the
others are essentially noise, or "scree." The sum of the
eigenvalues equals the sum of the variances of all prices, so the
fraction of joint price variation attributable to component k equals
[[lambda].sub.kk]/tr([lambda]).
The eigenvalues and associated eigenvectots are conventionally put
in descending order. Then the first eigenvector contains weights, or
factor loadings, that yield the linear combination of the prices in P
with the largest variance, restricting the sum of the squared weights to
equal one. The second eigenvector yields that linear combination
(independent of the first eigenvector) with the largest
"remaining" variance, and so on. If the main factors
underlying price variation are those suggested by economic theory, the
first few eigenvectors should be interpretable as such, while the
remaining eigenvectors should be uninterpretable and should explain
little of the variation in P. Interpretations of the eigenvectors can be
framed as hypotheses about the associated principal component and tested
in the usual way using observed independent variables X. The
interpretation of component k should imply certain coefficient signs on
the vector g in the regression [Z.sub.k] = [gamma]X + v.
Thus principal components analysis is wholly complementary to
structural estimation. The inferences yielded are general, expressed in
terms of the latent variables, not detailed. The data requirements are
modest and computation is trivial. These features are particularly
valuable given that the basic mechanics of multiproduct pricing have not
been comprehensively documented because of computational complexities
and data limitations.
C. Comparison and Illustration
We illustrate these points with a two-good, stylized quasi-structural model that closely echoes our analysis of MLB pricing.
The first good has only fixed costs, so a monopolist sets the prices of
these goods based on general demand, measured by a cardinal index G, and
the variable cost of producing the second good, C. The goods are
complements, so increases in C raise the price of good 2, [P.sub.2], and
lower the price of good 1, [P.sub.1] (Forbes, 1988):
[P.sub.1] = [[alpha].sub.1] + [[beta].sub.1]G - [[gamma].sub.1]C
[P.sub.2] = [[alpha].sub.2] + [[beta].sub.2]G + [[gamma].sub.2]C
where all parameters are positive. If G and C are measured, these
parameters can be estimated directly, using regression, and the
monopolist's behavior is fully explained. In many instances,
however, including our own, only imperfect proxies are available. Then
one cannot break down prices, or price variation, into components
associated with demand or costs using structural methods. But
appropriately weighted linear combinations of these prices reflect
demand and costs perfectly:
[P.sub.1] + w [P.sub.2] = [alpha] + [beta]G
[P.sub.1] + [w.sup.*] [P.sub.2] = [[alpha].sup.*] +
[[gamma].sup.*]C
If these weights (w = [[gamma].sub.1]/[[gamma].sub.2], [w.sup.*] =
-[[beta].sub.1]/[[beta].sub.2]) could be ascertained, perfect correlates
of G and C can be created and then used to reconstruct prices: the
monopolist's behavior is, again, fully explained.
Principal components analysis approximates these weights when the
effects of G and C on prices are roughly independent, as illustrated in
Figure 1. The ellipse delineates, or circumscribes, the bivariate distribution of realized prices. The effect of general demand shifts on
prices is illustrated by the line GG, with slope
[[beta].sub.2]/[[beta].sub.1], and that of cost changes is illustrated
by the line cc, with slope -[[gamma].sub.2]/[[gamma].sub.1]. Principal
components will extract the exact weights w, w* if
[[beta].sub.2][[beta].sub.2]/[[beta].sub.1][[gamma].sub.1] = 1, and will
approximate them otherwise, as the major axis, MM, containing the price
combination with the greatest variance differs somewhat from the line
GG, and similarly for the minor axis, mm. (3)
Crucially, one need not merely speculate whether the approximation
is good. Instead, the estimated weights--along with the interpretations
of the principal components that they generate--can be subjected to
formal testing. Even more crucially, one need not have perfect data to
do this. Simply conduct auxiliary regressions relating each component to
observed demand and cost proxies [??] and [??]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [??], [[??].sup.*] are the weights yielded by the principal
components analysis and [epsilon], [[epsilon].sup.*] are the error
terms. If these components accurately reflect the contributions of
demand and costs, four testable hypotheses should be satisfied: 13 >
0, [gamma] = 0, [[beta].sup.*] = 0, and [[gamma].sup.*] < 0. If so,
theoretical content has been successfully extracted from the
interactions between prices.
The technique is even more valuable when one is analyzing multiple
prices instead of just two, so that [P.sub.1] and [P.sub.2] represent
matrices, not vectors. One set of goods is complementary to the other
set, so the same model applies to the prices appropriately grouped. In
addition to their previous functions, the vectors of weights [??] and
[[??].sup.*] yielded by the analysis now also suggest the appropriate
groupings--prices that have similar weights across all key eigenvectors.
If general demand is the strongest influence on prices, the first
eigenvector should have positive weights on all prices, and the next,
reflecting the influence of costs, should have negative weights on the
prices in [P.sub.1] and positive weights on those in [P.sub.2]. The
remaining components represent noise. The validity of these weights can
be tested using the above regressions, just as before.
[FIGURE 1 OMITTED]
A final advantage of the principal components methodology is that
it can handle pricing influences other than costs or demand,
"institutional factors" such as price restrictions or the
ability to price discriminate. These may not be easily quantified or
formally modeled, and so can be hard to analyze or identify using
structural methods. This is not so with principal components, in
contrast, because these factors often can he treated, or thought of, as
latent variables. This property turns out to be important for our
analysis, as both institutional factors listed above influence pricing
in MLB.
D. Implementation
A major concern with this analytical approach is the potential for
unjustified interpretations. This is addressed in two ways. First, we
ensure the application being considered is appropriate for the
technique. This is not automatically true. Much economic data represent
an aggregate of multiparty, decentralized decision-making, where many
factors generate correlations between the dependent variables,
complicating and weakening the interpretation of the factor loadings.
But the monopoly pricing problem is, instead, austere: a single entity
controls all pricing decisions, and there are strong priors about the
main variables that drive these decisions (demand parameters and costs)
and their general influence on prices. Clearly identifiable, plausible
pricing patterns in accord with these priors and supported by auxiliary
regressions like those depicted above can reasonably be interpreted as
such. Furthermore, in our specific application, we conduct a preliminary
theoretical analysis that confirms that prices in this market are
plausibly determined by a few underlying, independent factors.
Second, the heuristics produced by the principal component analysis
are not considered definitive, but rather hypotheses to be tested.
Initially this is done informally, by conducting principal component
analyses on the same prices in other markets. Our interpretations are
suspect if the factor loadings are similar in markets in which the
factors underlying pricing are believed to be different, or vice versa.
Finally, formal hypothesis tests are conducted by regressing Z on X, as
outlined above.
To summarize, we adopt the following general three-step procedure:
(1) Conduct a preliminary theoretical analysis to determine whether
the dependent variables are plausibly determined by a small number of
independent latent variables.
(2) Conduct the principal component analysis; identify and
interpret the major principal components.
(3) Conduct formal and informal tests of the interpretations given
to those principal components.
III. MLB: AN INDUSTRY CASE STUDY OF MULTIPRODUCT PRICING
A. Studies of Pricing in MLB
MLB presents a classic case of multiproduct pricing. The game
"package" purchased by most MLB fans includes a combination of
tickets, parking, food concessions, and other concessions such as
programs. These products are relatively homogeneous across firms within
the industry, and most ball clubs are isolated local monopolies. (The
eight teams that play in the same metropolitan area, such as the New
York Yankees and the New York Mets, have distinctly different fan bases
[see Depken, 2000] and significant monopoly power.)
For years the trade publication Team Marketing Report (TMR) has
reported the posted prices of tickets, parking, and several concessions
for the four major professional sports leagues in the United States, and
from these calculated a Fan Cost Index (FCI) reflecting the expenses
incurred by a hypothetical family of four that attends a game, parks at
the stadium, and consumes a typical mix of concessions. For 2004, the
MLB average FCI was $155.52, of which ticket costs, at $78.98, were
barely half, suggesting that expenditures on parking and concessions are
quantitatively important. Revenue figures further support this
contention. (4)
Nonetheless, a comprehensive analysis of the relationship between
the various prices in MLB (or any other sporting league) has yet to be
undertaken--again, because data limitations do not allow it to be done
using traditional methods. Instead, Depken (2001), Marburger (1997),
Scully (1989), and Zimbalist (1992) focus on the relationship between
ticket prices and attendance, whereas Ferguson et al. (1991), Scully
(1989, 111-113), and Zimbalist (1992, 214) enquire whether ticket prices
are set optimally given the near-zero cost of seating additional fans.
For single-good pricing, this would be where demand is unit elastic
and revenue is maximized. Most studies, such as Krautmann and Berri
(2007), instead find that prices are set where ticket demand is
inelastic. This vexing outcome may be explained by accounting for other
costs of attendance, which are omitted from most of these studies, and
their relation to ticket prices. (5) One potential explanation,
suggested by Marburger (1997) and Krautmann and Berri (2007), is that
the revenue lost from ticket price reductions can be recaptured through
increased concession demand. But our empirical findings also suggest a
second possibility: For a given level of general demand, ticket prices
and concession prices move in opposite directions, biasing downward
elasticities that are estimated using ticket prices alone. Thus our
analysis can advance these strands of the sports economics literature
while also contributing to industrial organization.
B. Multiproduct Pricing in MLB
Although the above-mentioned features of MLB make it suitable for
analysis from an empirical perspective, other features of the market
make it suitable from a theoretical perspective. In particular, economic
theory has identified three general reasons that the prices of goods
sold by a multiproduct monopolist would be related, and each is well
represented in MLB.
The first possible source of price correlations is a change in
general demand for the game "package," stemming perhaps from a
surge in team popularity or greater income in the team's market
area. An increase in general demand should exert upward price pressure
for all goods in the game package, because each good is quite distinct
and has a positive income elasticity. Similarly, a decrease in demand
should exert downward price pressure on all goods. Thus the
"pricing signature" of general demand shifts is positive
co-movement of all prices. This signature is unique, and will not be
generated by a general increase in costs, as shown below.
Demand interrelations between goods also affect price setting.
Here, the most important relationship is between tickets and
concessions, which enhance the game-viewing experience. Tickets and
concessions can therefore be considered two composite, complementary
goods. Forbes (1988) shows how the prices of two complements respond to
changes in cost or demand. Increases in both products' demand, or
both products' costs, should increase both products' prices.
But an increase in the cost of, or demand for, just one of the products
will increase its price and decrease the price of its complement.
Because the marginal cost of tickets--additional game
attendance--is virtually zero, increases in factor prices should affect
the cost of concessions only. Thus cost increases will raise concession
prices and, through Forbes' logic, decrease ticket prices.
Similarly, idiosyncratic demand shifts for one of these composite goods
will increase its price and lower the price of the other good. The
pricing signature of cost or idiosyncratic demand shifts is a negative
relation between ticket prices and concession prices. This signature is
clearly distinct from that of general demand shifts.
Finally, product prices can be related because of nonlinear pricing
that attempts to maximize the capture of consumer surplus in the face of
heterogeneous consumer demand (product demand that differs across
consumer types). Common forms of nonlinear pricing include second-degree
price discrimination, in which consumers can pay a fixed "entry
fee" in order to purchase some range of quantities at a price below
"list," and the selling of bundled products at a discount. In
MLB, nonlinear pricing can generate price interactions across tickets
and concessions because some of the surplus generated by lowering
concession prices can be extracted in ticket prices, as exactly one
ticket is required of each patron. The degree to which this is done
depends on the extent of consumer heterogeneity and on teams'
ability to extract surplus through ticket prices. Although all stadiums
offer a range of seating options and ticket prices, some have a greater
range than others. Those teams are probably the most able to extract
consumer surplus in this way, and should choose to have higher average
ticket prices and lower concession prices in consequence, as shown in
Rosen and Rosenfield's (1995, 373) extensive theoretical analysis
of ticket pricing.
The interpretation of principal components in terms of the economic
forces just discussed is now reasonably clear. The eigenvector
associated with the general demand component should have positive factor
loadings on all prices, and should be correlated with demand shifters
such as income and team winning percentage. The eigenvector associated
with cost or idiosyncratic demand shifts should have oppositely signed
factor loadings on tickets and concessions, as should the eigenvector
associated with price discrimination. Clearly, the pricing signatures of
these last two economic forces need not be distinguishable; one
principal component may contain the effects of both. If so, evidence on
their relative importance may be gleaned from auxiliary regressions that
relate this component to costs, idiosyncratic demand shifters, and
stadium characteristics that influence teams' abilities to extract
consumer surplus through ticket pricing.
C. Data and Descriptive Statistics
The principal components analysis is conducted on the prices of
seven goods sold by all MLB teams from 1991 to 2003, as reported by TMR:
tickets (average per-game season ticket prices), official stadium
parking, beer, soda, hot dogs, ball caps, and programs. Beer and soda
prices are reported for different size drinks and so are normalized to
20 oz. All prices are converted to year 2000 dollars using the Consumer
Price Index. All prices are reported at the beginning of the season;
promotional price changes are not included in the data.
Over this period, MLB added four teams, dramatically realigned the
divisions within the American and National Leagues, introduced
inter-league play, and expanded the postseason play-offs to include wild
card teams. Therefore, the price data describe relatively homogeneous
products across all firms within an industry that has continued to
evolve even while the individual firms have remained relatively isolated
local monopolies. Each of these properties is conducive to testing
hypotheses about multiproduct pricing.
Table 1 presents the descriptive statistics for all prices, in the
upper panel, and for those variables (described further below) used in
the auxiliary regressions, in the lower panel. The real price of tickets
averaged $13.95 over the sample period, whereas the average real price
of parking was $7.30. Prices within the stadium averaged $5.18 for a 20
oz beer, $2.58 for a 20 oz soda, $2.27 for a hot dog, $3.56 for a
program, and $11.79 for a ball cap. The greatest variance was displayed
in ticket prices, which is not surprising given the different local
market and stadium characteristics across teams, and the smallest
variance was in the prices of hot dogs and soda.
Table 2 reports the correlation matrix of real prices. As can be
seen, the correlation between the prices of any two goods in the sample
is generally positive, but never greater than .60. Prices are neither so
uncorrelated that the goods can be viewed as having independent demands,
nor so correlated that they can be treated as a single "composite
good." The positive correlations suggest that the dominant
influence on price is the general demand for baseball, but their modest
magnitudes suggest that the multiproduct pricing considerations
discussed above, which induce negative relations between prices, are
also possible.
IV. PRICING IN MLB: EMPIRICAL RESULTS
A. Principal Component Analysis for MLB
Table 3 presents the basic principal component analysis: the seven
eigenvalues that solve the characteristic root, as well as the
eigenvector and proportion of overall variation associated with each
eigenvalue. The first component accounts for 40% of the total variation
in real prices, with the subsequent three components accounting for
approximately 15% each. The others, with very small eigenvalues and
incomprehensible eigenvectors, appear to be irrelevant scree.
By carefully examining the factor loadings in these first four
eigenvectors, the seven products represented can be usefully grouped
into three categories: "obligatory purchases" (tickets and
parking), food concessions, and nonfood concessions. (6) Both goods in
the first category have same-signed factor loadings in every
eigenvector. Similarly, factor loadings on the three goods in the second
category are consistently similar in sign and magnitude, with one small
exception. In contrast, for the two nonfood concessions, the signs and
magnitudes of the various factor loadings are haphazard with respect to
the other prices and each other.
The heuristics uncovered by the analysis can be expressed in terms
of these categories. Three patterns are apparent upon examination of the
factor loadings. First, the largest component has positive loadings on
all prices, interpretable as a general demand effect, as before. This
can include the secular trend in MLB attendance overall, interrupted by
the 1994 players' strike, and intertemporal demand differences for
individual teams as their performance varies over time. This variation,
and the change in demand that results, is known to be substantial. This
component suggests that its effect on prices is substantial as well.
Second, for each of the next three components, the factor loadings
of obligatory purchases and food concessions take opposite signs but
have similar sums. We can consider this the difference between the
prices of these two composite goods, consistent with the other pricing
influences discussed previously. This includes the interplay between
complementary goods, in which idiosyncratic demand shifts or changes in
concession costs increase the price of one good and decrease the price
of the other, and second-degree price discrimination, in which a higher
"entry fee" on obligatory purchases is coupled with lower
prices for repeat-purchase food concessions.
Third, the haphazard factor loadings for the nonfood concessions,
along with the weak correlations between these prices and the other
prices in Table 2, suggest that the prices of these concessions are not
integrated into pricing decisions for the other products. This
conclusion was buttressed, on further investigation, by the discovery
that the prices of caps are regulated by MLB and by informal discussions
with a team official indicating that programs and food concessions are
considered "separate markets," with prices presumably determined independently for each.
Table 4 presents results when nonfood concessions are excluded from
the analysis. This distills out the interplay between the five remaining
prices and checks the separateness of nonfood concessions: if these are
excluded, the first two patterns should appear as before. They do.
General demand explains half of the combined variation in the remaining
prices, and the trade-off between obligatory purchases and food
concessions about 20%. The literature suggests, variously, retaining for
further analysis those components that are easily interpretable, that
are suggested by theory, or that meet the "Kaiser criterion"
or the "scree test." The first component meets all of these;
the second fails only the Kaiser criterion, marginally; the others fail
all convincingly. We thus retain the first two components for further
analysis, and treat the rest as noise.
B. Principal Component Analyses for Related Markets
One simple way to check our interpretations is to conduct analogous
principal component analyses of the same set of prices in alternative
markets. The first, the National Football League (NFL), is similar to
MLB, with local monopolies that experience varying product demand
selling tickets, parking, and concessions. The forces that govern
pricing in MLB should also appear here; thus we should expect to uncover
similar factor loadings. The second market is the national market for
the five most closely aligned goods for which the Bureau of Labor
Statistics creates price indices: entertainment, parking, beer consumed
outside of the home, soda, and hot dogs. Each is converted to a
"real price index" by deflating with the overall Consumer
Price Index. These aggregate prices, not specific to major league
sports, should be set by competitive forces, not multiproduct pricing.
There is no reason to expect similar loadings here; indeed, there is no
reason to expect any meaningful patterns at all.
The results for these two markets, for the same 1991-2003 period,
are presented in Tables 5 and 6. For the NFL, in Table 5, both the
eigenvalues and eigenvectors are very similar to those uncovered for
MLB. The first component, representing general demand, has positive,
similar factor loadings across all prices, and explains 50% of overall
price variation. The second component has oppositely signed factor
loadings on obligatory purchases and food concessions, and explains
about 20% of overall price variation. The other components are small and
uninterpretable.
In contrast, the eigenvalues and eigenvectors for the national
market, in Table 6, are quite different. The first component, which
explains 73% of price variation, includes large positive factor loadings
on three prices, a small positive loading on a fourth, and a large
negative loading on a fifth. This has no obvious interpretation and no
relationship to the economic forces outlined previously. The second
component does include positive factor loadings on tickets and parking
and negative loadings on food, as before, but these are dominated by an
overwhelming factor loading on hot dog prices, whereas the others are
near zero. This component essentially extracts the price of hot dogs,
which does not move in concert with the other prices. This, too, does
not relate to the economic forces outlined previously. In contrast, the
analogous components for MLB and the NFL have substantial loadings on
all prices, which sum to 1 for food concessions and -I for obligatory
purchases. This is easily interpretable as the difference in the prices
of these two composite goods.
The analysis thus far has focused on the unexpurgated variation of
real prices charged by professional sports teams, including both
cross-team and cross-time variation. We believe this is appropriate;
there is no reason to exclude either source of variation in advance.
However, one may legitimately wonder whether price changes within teams
across time exhibit similar patterns. To examine this question, we
replicated the principal components analysis on MLB and NFL prices that
were purged of team and year effects, by regressing each price on a full
set of team and year fixed effects and using the residuals in the
analysis in the place of the original price data. By culling all
nationwide and fixed-team influences from prices, we focus the analysis
on local price dynamics.
In this analysis, available from the authors upon request, the
principal components reveal the same qualitative relationships as those
shown in Tables 4 and 5. The largest influence on prices remained a
general demand effect, which explained about 40% of the variance, and
the second largest remained a price trade-off between obligatory
purchases and food concessions, which explained about 20%. Of the 20
total factor loadings (2 components x 5 prices x 2 markets), 19 were
similar in sign and magnitude, with one being zero instead of the
expected negative sign. These results show that our original conclusions
are reasonably robust and suggest that the factors generating price
variation across teams are similar to those governing local price
dynamics.
Finding that the principal components are similar in a similar
market, different in a different market, and robust to the purging of
team and year effects lends additional credibility to our
interpretations, as these outcomes would be very unlikely to occur by
happenstance.
C. Auxiliary Regressions
Finally, in Table 7, we test our interpretations of the two (MLB)
components with economic content by relating them to a common set of
city-, team-, and stadium-specific variables that should affect prices
or price interactions. Several of these are commonly included in other
economic studies of professional sports: season attendance, city
per-capita income, city population, once-lagged team win percentage, the
age of the stadium and its square, and a dummy for whether the stadium
is single-purpose. We also construct a proxy for the local wages of
amusement workers. (7) The descriptive statistics for these variables,
reported in Table l, are similar to those reported in other studies.
Lagged win percentage and the wage variable are not available for
first-year expansion teams and Canadian teams, so these observations are
dropped.
Two models are used. Model I is sparse and direct. Attendance is
used as a measure of general demand, the real wage of amusement workers
as a proxy for variable costs, and a dummy for a single-purpose stadium
as an indicator of the ability to price discriminate. Second-degree
price discrimination should be more feasible for teams whose stadiums
better permit a range of seating options and ticket prices, to better
extract surplus from consumers. Single-purpose stadiums, such as Oriole
Park at Camden Yards (Baltimore), Safeco Field (Seattle), PETCO Park
(San Diego), and Rangers Ballpark (Arlington, Texas), provide a wider
variety of sight lines as reflected in their greater number of ticket
options, and are thus more conducive to this pricing strategy.
Attendance, which is possibly endogenous, is instrumented by the
following close correlates: population, lagged income, lagged winning
percentage, stadium age, and its square--see Coates and Humphreys (2005)
and Depken (2004). Model II simply replaces attendance with these
instruments. All first and second stage models also include a time
trend, and both Model I and Model II are estimated using the random
effects estimator, deemed appropriate vis-a-vis the fixed effects
estimator or pooled ordinary least squares using Hausman specification
tests.
The first component was associated with a general demand effect.
The auxiliary regressions presented in the first two columns of Table 7
support this interpretation in two ways: by rejecting the null on
variables that are associated with demand shifts, and by accepting the
null on variables that are not. Attendance is highly significant in
Model I, while four of five instruments for attendance are significant
with the expected signs in Model II. In contrast, the single-purpose
stadium dummy and real amusement wage are insignificant. Note,
incidentally, that observed general demand shifters explain only about
60% of this component's variation, so reduced form regressions
alone would vastly understate the price variation attributable to
general demand shifts.
We have interpreted the second component as the difference between
the prices of food concessions and obligatory purchases. This could be
associated with the extent of second-degree price discrimination,
negatively, as greater ability to extract surplus through ticket pricing
should push down concession prices. Both Model I and Model II support
this supposition, as the coefficient on the single-purpose stadium dummy
is significant with the expected sign. It could also be associated with
variable cost increases, positively, as these should raise concession
prices and lower ticket prices. This hypothesis is not supported; the
coefficient on the real wage variable takes the "wrong" sign
and is insignificant. Thus, the price discrimination explanation for
this component is preferred over the demand complementarities
explanation. However, given the modest fit of these regressions and the
absence of idiosyncratic demand shifters in our regressions, it is
possible that demand complementarities still influence price
interactions. Finally, general demand shifters should be unrelated to
the second component. This is confirmed, as the attendance variable is
insignificant in Model I, while only one of five Model II demand proxies
is significant with the expected sign.
Our last set of regressions used the same models to predict the
prices of programs and hats. Based on the factor loadings and price
correlations, we concluded that these prices were set separately from
the other prices, as if programs and hats belonged to a separate market.
If so, these prices should not be closely related to our key explanatory
variables. The regression results support this expectation. Other than
the time trend, no coefficient is significant in any regression.
From these findings we draw three main conclusions. First, general
demand shifts, only partly traceable to observables, generate roughly
half of all price variation in the products sold at MLB games. This
conclusion holds whether one pools all teams for all years or focuses on
within-team price dynamics. Second, teams engage in price discrimination
that involves a trade-off between ticket and concession prices. In this
market (and in the NFL), multiproduct pricing considerations contribute
meaningfully to price variation, but are less important than general
demand shifts. Third, program and cap pricing are not integrated with
the setting of other prices, partly because of price constraints imposed
by MLB.
V. CONCLUSIONS
Structural analysis of multiproduct pricing is complicated by the
challenge of linking prices to a large number of own- and cross-price
elasticities and costs using a theoretical relationship that can be
difficult to specify a priori. The principal components technique,
instead, extracts information directly from the observed interactions
among prices, simultaneously reducing the complexity of the analysis and
broadening the scope of the conclusions that can be drawn from it.
In our examination of pricing in MLB, these conclusions are
fundamental and yet new--because here, as in many other markets, the
questions we address have not been previously explored, for lack of a
practical way of doing so. Our analysis clusters seven goods into three
categories, natural yet not obvious ex ante, whose behavior is distinct.
Price variation within and across categories can be explained by
elementary theory and by institutional factors. Although multiproduct
pricing considerations explain a nontrivial amount of price variation in
this market, general shifts in product demand, driven substantially but
not exclusively by variation in team success, remain the dominant
influence on price. Overall, the economic forces stressed by theory do
appear to be the most important influences on pricing in MLB.
These conclusions are of intrinsic interest, but they also inform
the relevant sports economics literature. We have shown that, by
accounting for demand interrelationships and engaging in price
discrimination, teams' pricing methods are more sophisticated than
previously modeled. Although changes in team performance shift the
demand both for tickets and concessions, for a given level of general
demand there is a negative interplay between ticket and concession
prices. This suggests that the price elasticity of the full cost of game
attendance is larger in magnitude than that estimated using ticket
prices alone, and supports previous claims that optimal ticket pricing
need not require unitary elasticity of ticket demand.
ABBREVIATIONS
FCI: Fan Cost Index
MLB: Major League Baseball
NFL: National Football League
TMR: Team Marketing Report
doi: 10.1111/j.1465-7295.2010.00263.x
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(1.) Mirrlees (1976) set out the complete solution for nonlinear
monopoly multiproduct pricing, a complex function of elasticities,
costs, and endogenous Lagrange multipliers. Simple relations appear only
under restrictive conditions that exclude many realistic pricing
situations such as that analyzed here. Armstrong (1999) shows that
cost-based two-part tariffs are nearly optimal when goods are neither
substitutes nor complements and the number of goods is large. Two-part
tariffs are also obtained when duopolies compete sufficiently and
certain other conditions apply; see Armstrong and Vickers (2001) and
Rochet and Stole (2002). Sibley and Srinagesh (1997) show that optimal
nonlinear prices can be determined separately for each market when
preferences satisfy a strong condition called the "uniform ordering
of demand curves," whereas Bliss (1988) shows that competitive
retailing margins are a constant percentage across all goods when
consumers are "fixed budget shoppers."
(2.) Following custom, each variable is normalized by its standard
deviation; principal components analysis can be sensitive to the scale
of the variables. Doing this also highlights the interplay of prices,
the focus of the analysis. See Green (1978) and Johnson and Wichern
(1982) for thorough explications of the principal components technique.
(3.) Throughout this example, the weights are applied just to
[P.sub.2], but this is only to simplify the exposition. Principal
components actually weight both prices, with the sum of the squared
weights equal to one. The first component could thus be expressed as
sin([theta])[P.sub.1] + cos([theta])[P.sub.2], where [theta] =
arctan([[beta].sub.2]/[[beta].sub.1]) if the "exact weights"
are extracted.
(4.) Team-specific revenue figures are difficult to obtain. In
2004, the average MLB team bad $51.7 million in gate revenue and $90.6
million in "'other revenue," which includes media,
stadium advertising, and concession revenues (see
http://www.rodneyfort.com/SportsData/BizFrame.htm). Silver (2006)
reports that the 1997 Cleveland Indians received a revenue of $49
million from tickets, $17 million from merchandise, and $14 million from
concessions and catering.
(5.) Notable exceptions are Welki and Zlatoper (1994), who include
the cost of parking in a study of NFL demand, Depken (2000) who includes
average concession expenditures in a study of baseball demand, and
Winfree et al. (2004), who proxy for the total cost of attending a game
using a travel time measure. None of these measures, however, completely
reflects the full costs of attending a sporting event.
(6.) Because one need not purchase stadium parking to gain access
to the event, some stadiums have limited parking, and others have
generous private parking, the "obligatory" nature of stadium
parking is, of course, only approximate. This grouping is indicated by
the analysis. however, whatever its name.
(7.) Attendance and team quality data were obtained from MLB, city
income from the Bureau of Economic Analysis, city population from the
Census Bureau, and stadium characteristics from Munsey and Suppes at
www.ballparks.com. The real wage measure was determined using the method
pioneered by Coates and Humphreys (2002), utilizing the Regional
Economic Information System generated by the Bureau of Economic
Analysis. These data provide the full number of employees in the service
sector (all subsectors) and the full compensation of the amusement and
entertainment sector. RWAMUSE, a proxy for the real wage rate, is
measured as the total real compensation in the amusement sector divided
by total service sector employment. There are some gaps in the city
data, and the data series ends in 2001 because of data limitations.
Missing observations are imputed using city-specific
interpolation/extrapolation. As the employment sector is broader than
the compensation sector, the estimates of RWAMUSE are not directly
interpretable, but proxy for differences in variable costs across cities
and time.
CRAIG A. DEPKEN, II and DARREN GRANT *
* The authors acknowledge the helpful comments of anonymous
referees, Bill Crowder, Brian Goff, Jahn Hakes, Courtney LaFountain.
Steve Shmanske, Mike Ward, Ron Warren, and seminar participants at the
Southern Economic Conference, the Western Economic Association Meetings,
the Academy of Economics and Finance Meetings, Southern Methodist
University, and the University of Texas at Arlington. This paper is
complementary to a contemporaneous study by Stewart and Jones (2010),
who investigate whether professional baseball teams are multiproduct
firms in a much different way. Stewart and Jones test whether baseball
teams can be treated as providing one product (an "event") or
two products ("entertainment" and "performance") in
a production framework, using a generalized cost function approach. They
find that professional baseball teams are multiproduct firms, but that
the products are weakly separable in production. They do not investigate
price setting in their study.
Depken: Associate Professor, Department of Economics, University of
North Carolina--Charlotte, 220 Friday Building, Charlotte, NC 28223.
Phone 1-704-687-7484, Fax 1-704-687-4014, E-mail depken@uncc.edu
Grant: Assistant Professor, Department of Economics and
International Business, Sam Houston State University, Huntsville, TX
77341. Phone 1-936-294-4324, Fax 1-936-294-3488, E-mail dgrant@shsu.edu
TABLE 1
Descriptive Statistics
Variable Description
Tickets Average per-game season ticket price
Parking Price of parking
Beer Price of 20 oz beer
Soda Price of 20 oz soda
Hot dogs Price of hot dog
Programs Price of program
Hats Price of ball cap
Attendance Total season home attendance (hundreds
of thousands)
Lagged income Previous year's MSA per-capita income
($ thousands)
Population MSA population (millions)
Lagged win percentage Previous season's wins (fraction)
Stadium age Age of team's stadium (years)
Real amusement wage Real wage of amusement workers
(arbitrary units; see text)
Single-purpose stadium Team's stadium is single-purpose (0/1)
Time trend Time trend (1 = 1991)
Variable Mean SD Minimum Maximum
Tickets 13.95 4.51 8.29 39.83
Parking 7.30 2.85 2.93 19.19
Beer 5.18 0.92 3.19 10.57
Soda 2.58 0.53 1.41 4.70
Hot dogs 2.27 0.55 0.78 4.23
Programs 3.56 1.06 0.69 7.35
Hats 11.79 2.15 4.73 20.00
Attendance 22.16 7.22 9.05 44.83
Lagged income 30.49 4.49 22.47 47.14
Population 6.27 5.49 1.60 21.31
Lagged win percentage 0.50 0.06 0.32 0.70
Stadium age 30.11 24.37 0.00 89.00
Real amusement wage 1.20 0.38 0.58 2.34
Single-purpose stadium 0.61 0.48 0.00 1.00
Time trend 6.16 3.16 1.00 11.00
Notes: Price data (reported in upper panel) describe all MLB
teams from 1991 through 2003 and were obtained from various
issues of TMR. All prices, incomes, and wages converted to 2000
dollars using the Consumer Price Index from the Bureau of Labor
Statistics. Attendance and team win percentage obtained from MLB.
Population and income obtained from Census Bureau. Stadium
characteristics obtained from Munsey and Suppes at
www.ballparks.com. Wage data obtained from the Regional Economic
Information System of the Bureau of Economic Analysis, as
described in the text. The price data comprise a sample of 372
observations used in the principal component analysis. Stadium,
income, wage, and population data are 342 observations for U.S.
baseball teams (two Canadian teams not included).
TABLE 2
Correlation Matrix of Real Ticket, Parking, and Concession Prices
Tickets Parking Beer Soda Hot Dogs Programs Hats
Tickets 1.00
Parking 0.59 1.00
Beer 0.45 0.33 1.00
Soda 0.53 0.27 0.50 1.00
Hot dogs 0.45 0.17 0.48 0.52 1.00
Programs 0.14 0.12 0.02 0.10 0.18 1.00
Hats 0.05 -0.02 0.06 0.13 0.14 0.15 1.00
Notes: Price data describe all MLB teams from 1991 through 2003.
obtained from various issues of TMR, and were converted to 2000
dollars using the Consumer Price Index from the Bureau of Labor
Statistics.
TABLE 3
Principal Components of All MLB Prices: Eigenvalues and Eigenvectors
Eigenvector Eigenvector Eigenvector Eigenvector
Variable One Two Three Four
Real prices of:
Tickets 0.49 -0.18 0.21 0.15
Parking 0.36 -0.33 0.53 0.41
Beer 0.44 -0.12 -0.29 -0.04
Soda 0.46 0.04 -0.26 -0.11
Hot dogs 0.43 0.20 -0.30 -0.36
Programs 0.15 0.54 0.64 -0.46
Hats 0.10 0.72 -0.14 0.67
Eigenvalue 2.79 1.14 0.98 0.80
Proportion of 0.40 0.16 0.14 0.11
variance
explained
Eigenvector Eigenvector Eigenvector
Variable Five Six Seven
Real prices of:
Tickets -0.33 -0.12 -0.73
Parking 0.10 -0.14 0.53
Beer 0.80 0.21 -0.15
Soda -0.46 0.63 0.31
Hot dogs -0.10 -0.70 0.24
Programs 0.15 0.17 -0.04
Hats 0.04 -0.03 -0.02
Eigenvalue 0.52 0.46 0.31
Proportion of 0.07 0.07 0.04
variance
explained
TABLE 4
Principal Components of Five MLB Prices: Eigenvalues and Eigenvectors
Variable Eigenvector One Eigenvector Two Eigenvector Three
Real prices of:
Tickets 0.50 -0.30 -0.32
Parking 0.37 -0.74 0.07
Beer 0.45 0.19 0.86
Soda 0.47 0.28 -0.32
Hot dogs 0.43 0.49 -0.25
Eigenvalue 2.73 0.95 0.54
Proportion of 0.55 0.19 0.11
variance
explained
Variable Eigenvector Four Eigenvector Five
Real prices of:
Tickets 0.03 -0.74
Parking 0.15 0.53
Beer -0.07 -0.14
Soda -0.71 0.31
Hot dogs 0.68 0.22
Eigenvalue 0.47 0.31
Proportion of 0.09 0.06
variance
explained
TABLE 5
Principal Components of NFL Prices
Variable Eigenvector One Eigenvector Two Eigenvector Three
Real prices of:
Tickets 0.47 -0.43 -0.18
Parking 0.44 -0.55 -0.02
Beer 0.38 0.63 -0.47
Soda 0.41 0.27 0.85
Hot dogs 0.52 0.18 -0.15
Eigenvalue 2.50 0.98 0.67
Proportion of 0.50 0.20 0.13
variance
explained
Variable Eigenvector Four Eigenvector Five
Real prices of:
Tickets 0.68 -0.31
Parking -0.39 0.59
Beer 0.21 0.43
Soda 0.15 0.08
Hot dogs -0.56 -0.60
Eigenvalue 0.44 0.41
Proportion of 0.09 0.08
variance
explained
TABLE 6
Principal Components of Real Price Indices
Variable
(price index) Eigenvector One Eigenvector Two Eigenvector Three
Tickets 0.50 -0.09 0.03
Parking 0.50 -0.13 0.22
Beer 0.49 0.10 0.56
Soda -0.49 0.16 0.79
Hot dogs 0.14 0.97 -0.15
Eigenvalue 3.65 0.98 0.16
Proportion of 0.73 0.20 0.03
variance
explained
Variable
(price index) Eigenvector Four Eigenvector Five
Tickets 0.72 -0.48
Parking 0.16 0.81
Beer -0.57 -0.33
Soda 0.35 0.04
Hot dogs 0.09 0.09
Eigenvalue 0.13 0.08
Proportion of 0.03 0.02
variance
explained
Note: Variables are real price indices,
constructed as described in the text.
TABLE 7
Auxiliary Regressions (Coefficient Estimates,
with Standard Errors in Parentheses)
Dependent
Variable [right arrow] Component One Component Two
Independent
Variable [down arrow] Model I Model II Model I Model II
Attendance 0.99 * - -0.25 --
(0.18) (0.13)
Population -- 0.04 -- 0.06 *
(0.03) (0.02)
Lagged -- 0.09 * -- -0.03
income (0.02) (0.02)
Lagged win -- 2.64 * -- -0.47
percentage (0.82) (0.53)
Stadium age -- -0.04 * -- 0.03 *
(0.02) (0.01)
Stadium age -- 0.05 * -- -0.04 *
squared (0.01) (0.02)
(divided
by 100)
Single- 0.43 0.34 -0.65 * -0.46 *
purpose (0.22) (0.24) (0.15) (0.17)
stadium
Real 0.01 0.23 -0.08 -0.33
amusement (0.26) (0.24) (0.17) (0.18)
wage
Time trend 0.25 * 0.20 * 0.03 * 0.05 *
(0.02) (0.02) (0.01) (0.02)
[R.sup.2] 0.48 0.68 0.06 0.26
Wald statistic 655.9 * 775.93 * 39.65 * 63.8 *
([chi square])
Dependent
Variable [right arrow] Program Price Cap Price
Independent
Variable [down arrow] Model I Model II Model I Model II
Attendance 0.30 -- -0.39 --
(0.16) (0.36)
Population -- 0.03 -- 0.05
(0.02) (0.05)
Lagged -- -0.02 -- -0.04
income (0.02) (0.05)
Lagged win -- 0.75 -- 0.23
percentage (0.53) (1.64)
Stadium age -- -0.01 -- 0.03
(0.01) (0.02)
Stadium age -- 0.00 -- -0.03
squared (0.01) (0.03)
(divided
by 100)
Single- -0.37 -0.47 0.20 -0.25
purpose (0.19) (0.25) (0.38) (0.44)
stadium
Real -0.03 -0.17 0.02 -0.38
amusement (0.22) (0.25) (0.44) (0.50)
wage
Time trend 0.12 * 0.11 * 0.08 * 0.12 *
(0.01) (0.02) (0.03) (0.05)
[R.sup.2] 0.19 0.22 0.01 0.02
Wald statistic 120.1 * 128.1 * 8.3 12.2
([chi square])
Notes: N = 342. Canadian teams and first-year expansion teams are
excluded because the lagged winning percentage or real wage
cannot be measured. A random effects estimator was applied after
Hausman specification tests. The first two principal components
from Table 4 are the dependent variables, along with the real
prices of programs and hats.
* Significant at the 5% level in a two-tailed test.