Identifying own- and cross-price eects using contingent valuation of college sport tickets

Identifying own- and cross-price eﬀects using
contingent valuation of college sport tickets
XXXI Jornadas de Economia del BCU, Montevideo
Francisco Rosas
Universidad ORT Uruguay & cinve
Santiago Acerenza
Universidad ORT Uruguay
Peter F. Orazem
Iowa State University
August 19th 2016
Rosas (ORT & cinve) Own & Cross price eﬀects August 19th 2016 1 / 25

Background
Broad Motivation
Willingness to pay for diﬀerent goods is an important area of
study in economics and also an important input for the
productive sector.
Understanding willingness to pay if a consumer already bought
another good is important to make optimal bundling policies.
Revealing unobserved complementarities between goods is also
important to understand consumer behavior.

Background
Speciﬁc Motivation
National Collegiate Athletic Association (NCAA) is very popular
in the US, but only yields surpluses for a couple of the sports.
This implies a considerable burden to University administrators.
Spending on sports of Division I-A Universities amounted to
$7785 million, and generated revenues for $8220 million.
Very heterogeneous reality by sport, or by mens and womens
sports, or by University within Division.
An important factor for increasing revenues is attendance to
college sport games.
For that, understanding the demand for sport tickets, willingness
to pay for sports, and the relationship between sports is an
important input to generate maximizing revenue policies.

Background
Objectives
For that, we estimate an eight-variate probit model that allows
us to identify not only important own demand parameters but
also some unobserved correlations shared by all the sports.
This eight-variate framework will allow us estimate unconditional
and conditional willingness to pay for sports.
We aim also to generate information that if accounted for,
promoters can generate cross-marketing strategies to increase
attendance and revenues.
Finally, we aim to produce a framework that can be generally
applicable to other settings.

Background
Related literature
Contingent valuation methods to elicit preferences
Willingness to pay for basketball and baseball venues (Johnson
and Whitehead, 2000).
For attracting a professional hockey team (Johnson, Groothuis,
and Whitehead, 2001).
For retaining professional football and basketball teams
(Johnson, Mondello, and Whitehead, 2007).
For supporting amateur sports and recreation programs
(Johnson et al. 2007).

Background
Related literature
Previous applications of multi-variate Probits
Ashford and Swoden(1970) applied the model to a biological
system.
Contoyannis and Jones (2004) use it for studying health
production function and the parameters of lifestyle equations.
Chib and Greenberg (1998) applied it to commuting choice and
buying a car choice;.
Young et al. (2009) uses it in the modeling of the types of
claim in a portfolio of insurance policies.
Amemiya (1978) developed a variant of the multivariate probit
where one of the variables is partially observed.

Data
Sample
Generated a rich data through an artiﬁcial environment
developed in the undergraduate population at Iowa State
University.
In 2007, random sample of 2000 students, invited by Email to a
web-based survey.
Response rate 23.5% (470 individuals)

Data
Sample
Each individual was given the opportunity to purchase or refuse
a ticket to attend a college game.
1) women basketball, 2) men basketball, 3) football, 4)
volleyball, 5) wrestling, 6) gymnastics, 7) hockey, 8) women
soccer
Prices were randomly generated from a uniform distributions
with mean in the market price.
Asked demographic information and questions related to
participation and interest in sports
We require suﬃcient variation in prices to identify individual’s
behavior.
Because each respondent received a randomly drawn price,
there is no correlation with the control variables

Data
Descriptive Statistics
Success of the price randomization; a plot of the probability of a
positive purchase response by random price oﬀered: Law of demand

Data
Descriptive Statistics
Diﬀerences in demand between people who purchased football tickets
and those who did not. Possible complementarities between the two
sports; will be formally tested later

Model
Random Utility Model (RUM)
Suppose latent utility Bj∗
i of ticket purchase for sport
j = 1, 2, ..., J for individual i. It sets an J = 8 equations system
Bj∗
i = βj
0 + βj
1Pj
i + Z’iδδδj
− ηj
i
Where Pj
i is the exogenously oﬀered price to i for sport j, Z’i is a
vector of individual characteristics and ηj
i the unobserved factor.
As each ticket purchase decision is viewed as independent
(time), unobserved factors are correlated




η1
i
η2
i
...
ηJ
i



 ∼ N


0,



1 · · · ρ1j
...
...
...
ρ1j · · · 1







Model SUR estimation results
Random Utility Model (RUM)
The likelihood function of the model, arising from computing all
the combinations of purchase-not purchase, is:
n
i=1
log(ΦJ[q1
i (β1
0 +β1
1P1
i +Z’iδδδ1
), ..., qJ
i (βJ
0 +βJ
1 PJ
i +Z’iδδδJ
),QQQ·ρρρ])
where ΦJ is the J-variate Standard Normal Distribution fcn
qj
i = 2 × Bj
i − 1
Qjk = qj × qk , for j = k, and Qjk = 1 for j = k
ρρρ is J × J matrix of correlation coeﬃcients between equations

Results
RUM estimation results
Simulated maximum likelihood estimation of seemingly unrelated
model. Dependent variable: Binary choice of purchasing ticket of
sport j: Bj
i

Results
RUM estimation results
Estimation of the unobserved correlation coeﬃcients between
equations: ρρρ

Results Willingness to pay
WTP or Reservation prices
Parameter estimates of the SUR model are used to compute
Reservation prices
Deﬁned as the price at which individual is indiﬀerent between
purchasing or not sport j ticket
Pr[Bj
i = 1] = Φ[ˆβj
0 + ˆβj
1pj
i + Z’i
ˆδδδ
1
] = 0.50
for all j and for all i
where Φ is the uni-variate Standard Normal Distribution fcn
We compute a reservation price for each individual, and for each
sport: pj
i

Results Willingness to pay
WTP, results
Estimation of reservation prices (WTP): pj
i

Results Conditional willingness to pay
Conditional reservation prices
Dropping subscript i, we compute a reservation price for sport j
that equalizes to 0.5 the expected value of purchasing a ticket of
sport j (Bj
= 1), CONDITIONAL on purchasing a ticket of
sport k (Bk
= 1), and given the observed decision for (−k)
0.5 =
Pr[Bj
= 1, Bk
= 1, B−k
= b−k
]
Pr[Bk = 1, B−k = b−k ]
0.5 =
ΦJ [(ˆβj
0 + ˆβj
1pjk
+ Z’ˆδδδ
j
), (ˆβk
0 + ˆβk
1 Pk
+ Z’ˆδδδ
k
), . . . ,QQQ · ρρρ]
ΦJ−1[(ˆβk
0 + ˆβk
1 Pk + Z’ˆδδδ
k
), . . . ,QQQ · ρρρ]
pjk
implicitly solves this non-linear equality, using Bisection method
Compute a CONDITIONAL reservation price for each individual, and for
each pair of sports: pjk
i

Results Conditional willingness to pay
Conditional reservation prices, results
Percentage increment of Conditional reservation prices relative to
Unconditional reservation price:
pjk
i −pj
i
pj
i
Conditional reservation prices are higher than their unconditional
counterparts for each combination of sports
People who already have a ticket to one sport (j) have higher
willingness to pay for the other sport (k)

Results Own and Cross-price eﬀects
Own-price elasticities
We compute both own- and cross-price elasticites (and marginal
eﬀects), directly from the likelihood estimation
The probability of purchasing a ticket for sport j represents the
estimated demand of individual i for sport j.
The own-price elasticities of individual i for sport j is:
εjj
i =
1
n
n
i=1
∂Pr[Bj
i = 1]
∂Pj
i
Pj
i
Pr[Bj
i = 1]
This is straightforward to compute

Cross-price elasticities
Example of two sports: j, k

Cross-price elasticities
Given that there are no cross-prices in demand equations, we
derive an alternative way of estimating cross-price elasticities
(we exploit the unobserved correlation among sports)
The joint probability of purchase a combination of tickets is:
C = Pr[Bj
= 1, Bk
= 1, B−k
= b−k
]
C = ΦJ[(ˆβj
0 + ˆβj
1Pj
+ Z’ˆδδδ
j
), (ˆβk
0 + ˆβk
1 Pk
+ Z’ˆδδδ
k
), . . . ,QQQ · ρρρ]
We shock Pj
(1 + 1%) and implicitly solve for the Pj
consistent
with a probability = C, denoted as Pj∗
We use Bisection methods to solve this implicit function for each
individual and sport pair
Cross-price elasticities is obtained by plugging (Pj∗
Pj − 1) into
own-price elasticity equation

Cross-price elasticities, results
Median of each individual estimated own- and cross-price elasticities
A 1% change in womens basketball price induces a median
reduction of 0.46% in demand(in the probability of purchasing
wbb tickets)
A 1% change in womens basketball price induces a median
increase of 0.49% in demand for men bb

Conditional reservation prices, results
The fact that cross-price elasticity is positive, implies that sports
are substitutes.
The demand is more sensible to changes in own prices than in
prices of the other sports
The unobserved correlation between sports (ρjk) was estimated
as positive, interpreted as “pure taste for sports”
We complement this interpretation in that the observed factors
shows a substitutive behavior between sports.

Final Remarks
Conclusion
Using data from a contingent valuation survey we studied the
decision of purchasing or not 8 diﬀerent sports and their
relationships.
We estimated the reservation prices (or willingness to pay) of
each sport and compared them with the actual prices oﬀered.
We also estimated reservation prices of a sport conditional on
buying another sport.
In almost all sports, people that already desires one sport is
willing to pay more for the sport evaluated, than otherwise.
This result shows the possibility of making product bundles
between sports.

Final Remarks
Conclusion
Finally, we derive a method for estimating cross-price elasticities
for a sport when the prices of the other sport do not appear in
the decision equation
We exploit the unobserved correlation and the multivariate
probit framework.
Results are intuitive and make economic sense
This result can also be used for marketing purposes.

Identifying own- and cross-price eects using contingent valuation of college sport tickets

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