# Multiproduct Pricing in Major League Baseball: A Principal Components Analysis

## Article excerpt

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. …

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