Multiproduct pricing in major league baseball: a principal components analysis.

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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Marburger, D. R. "Optimal Ticket Pricing for Performance Goods." Managerial and Decision Economies, 18(5), 1997, 375-81.

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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 [phone], Fax [phone], E-mail [e-mail]

Grant: Assistant Professor, Department of Economics and International Business, Sam Houston State University, Huntsville, TX 77341. Phone [phone], Fax [phone], E-mail [e-mail]

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.