Optimal pricing strategies for a cluster of goods: own- and cross-price effects with correlated tastes

ABSTRACT

Contingent valuation methods are used to identify observed and unobserved preferences of goods and services. We apply these methods, in the context of multivariate probit analysis, to compute willingness to pay for each product of a cluster of goods conditional on having purchased another offered good of the cluster. We also provide a derivation of compensated cross-price elasticities based on unobservable factors, proving to be convenient in situations where cross-prices are not part of the demand equations. As goods belonging to a cluster typically embed correlated taste, their pricing strategy should consider all offered goods simultaneously rather than individually. Therefore, we solve for the set of optimal prices of a social planner whose objective function weights both the producer’s revenues and the consumer’s joint latent utility. We show an application to collegiate sports events, but these methods can be extended in a straightforward fashion to other goods and services. Supplementary materials for this article are available online.

Abbreviations: C30, Multiple or Simultaneous Equation Models, General; C35, Discrete Regression and Qualitative Choice Models; C40, Special Topics of Econometric and Statistical Methods; D12, Empirical Analysis of Consumer Economics; D40, Market Structure, Pricing, and Design; D60, Welfare Economics; Z20, Sport Economics

KEYWORDS:

• Cluster of goods
• bundling
• unobserved taste
• optimal pricing policy
• compensated cross price elasticities
• sport tickets

JEL CLASSIFICATION:

• C30
• C35
• C40
• D12
• D40
• D60
• Z20

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary material

Supplemental data for this article can be accessed here.

Notes

¹ See Online Appendix 1.A for more details on willingness to pay and a complete list of applications in the sports economics literature.

² Guiltinan (1987), Rao et al. (1987), Crawford (2008), Stremersch and Tellis (2002), Venkatesh and Mahajan (2009), Prasad, Venkatesh and Mahajan (2017), provide literature reviews and analyse price bundling of goods and services, and DeGraba and Mohammed (1999) and Chalip and McGuirty (2004) study product bundling for sports events (or concerts).

³ They state that this model allows the estimation of the cross-correlation coefficients separately from the joint purchase outcomes. Additionally, they state that the cross-correlation estimates are smaller in magnitude but equal in sign to those of multivariate probit.

⁴ In this two-stages approach, first a probit model describes the participation decision (i.e. a non-zero consumption decision) and then a Poisson regression truncated at zero identifies the consumption decision.

⁵ Thibaut et al. (2014, 2017, 2018) estimate income effects for a variety of sport and provide a list of studies on this topic.

⁶ Further insights on sport pricing can be found in Drayer and Rascher (2013).

⁷ For example, in sport events, loyalty to a team, often crosses sports because of an individual’s residence, circle of friends, or school affiliation.

⁸ We arbitrarily set the increment to 1%, but any other increment that generates sufficient variation in prices would suffice.

⁹ Figure 1 assumes that the two goods are substitutes, but the same exercise can be conducted for complementary goods.

¹⁰ Revenue maximization is useful in case of goods and services characterized by a cost structure dominated by fixed costs, as in our case with sports events.

¹¹ We do not have access to systematized data on attendance for the rest of the mentioned sports.

¹² Simulated maximum likelihood is more accurate than conventional or exact maximum likelihood in estimating multivariate models, probits in particular, due to the difficulty in finding closed forms of the probabilities maximized as the number of equations increases (Cappellari and Jenkins 2006).

¹³ See Table A.1 in Online Appendix for detailed results of the unconditional reservation prices for each sport.

¹⁴ Tables A.6 and A.7 in the Online Appendix 6 show, respectively, the median $P^{jk}$and median $P^j$ used to construct each cell of Table 1.

¹⁵ Detailed results for each sports pair are not shown due to space, but are available from the authors upon request.

¹⁶ As we already noticed, our setup allows pursuing a pricing strategy for each individual in the sample which is consistent with goods or services when a first-degree price discrimination is possible.

¹⁷ A male, who played varsity sports in high school, did not play intramurals in high school but did in Iowa State, and neither has relatives graduated from Iowa State nor was a Cyclone fan before coming to school.

¹⁸ The tradeoff between revenues and consumer’s utility indicates that if consumers have zero joint utility, it requires the planner to have revenues higher than 1 (the maximum attainable) in order to have the same utility as that of the optimum.

¹⁹ It is likely that for most goods and services this is the only scenario of interest.

²⁰ Recall that price discrimination is not possible in this case, so only one price must be available for each sport at the tickets office.