Research Note
Inelastic Sports Pricing and Risk
Per Andersen, University of Southern Denmark
Department of Business and Economics
Campusvej 55 ∙ DK-5230 Odense M ∙ Denmark
Tel.: +45 6550 3211
Fax: +45 6550 3237
E-mail: [email protected]
Martin Nielsen, University of Southern Denmark
Department of Business and Economics
Campusvej 55 ∙ DK-5230 Odense M ∙ Denmark
Tel.: +45 6550 3420
E-mail: [email protected]
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Research Note
Inelastic Sports Pricing and Risk
Abstract
Seemingly inconsistent with rational behavior, the empirical sports economics literature finds that
sports teams price in the inelastic part of the demand curve. This paper shows that this pricing
strategy may in fact be consistent with rational behavior of a risk averse sports team. To show
that, we develop a static model where a risk averse sports team faces a stochastic demand and
maximizes expected utility of profits. The result of the model is pricing in the inelastic range of
the demand curve. Furthermore, using comparative statics we show that increased fixed costs
lowers the price charged by the sports team.
Keywords: Inelastic pricing, risk aversion and pricing, sports pricing.
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I. Introduction
Classical microeconomic theory suggests that monopolies maximizing profits should set price in
the elastic range of the demand curve. Yet the sports economics literature has on regular basis
found empirical evidence that sports teams in fact price tickets in the elastic range of demand,
seemingly inconsistent with profit maximization. For recent references, see Krautman and Berri
(2007) and Fort (2004) for discussions of empirical studies. Several explanations have been
suggested to explain this phenomenon, which is often referred to as the paradox of inelastic
sports pricing. For example, Kesenne (1996, 2000) argue that the paradox is caused by the simple
fact that sports teams do not maximize profits, but something else for example the win
probabilities subject to a profit constraint. Berri and Krautmann (2007) argues that the observed
pricing behavior is consistent with profit maximization by considering the complementary
between tickets sold and concessions, especially if the cost of admitting another person into the
stadium is close to zero. Yet Bird (1982) maintains that the paradox could be ascribed to bad data
quality and inappropriate choice of methodology. The discussion indicates that consensus is
lacking in explaining what is called the paradox of inelastic sports pricing. While there may be
some explanatory power in each argument, we find that these explanations are insufficient
because they leave out an important aspect, namely, uncertainty and risk aversion.
While sports teams are normally free to set ticket prices, the prices are usually set weeks before
the match – and often even before the start of the season. Consequently, sports teams face
uncertainty since they set prices before knowing many of the factors determining the demand for
tickets, from weather conditions to performance of the teams and to the importance of the games.
Specifically, sports teams make their pricing decision before they know the optimal price for a
given match. Therefore, the presumption characterizing the literature in Sports Economics until
now, that sports teams act as if the environment were nonstochastic, fails. Instead, sports teams
must form expectations regarding the ticket demand and how they set prices will depend crucially
on their attitude towards risk.
In this paper, drawing heavily on methodologies introduced by Sandmo (1971) and Leland
(1972), we develop a model where a sports team faces a stochastic ticket demand for a particular
match. Knowing only the probability distribution of the demand function, the sports team must
set the ticket price before the ticket demand is realized. As a benchmark for comparison, we will
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first analyze and solve the maximization problem of a sports team maximizing expected profits.
Subsequently, we will consider the same sports team in the case where it is risk averse and thus
maximizes expected utility of profits. The model result is very strong in the sense that it predicts
pricing in the inelastic range of the demand curve. Besides resolving the paradox of inelastic
pricing, the model holds several interesting comparative statics. As an example, we show that
increased fixed costs will induce the risk averse sports team to lower their prices ceteris paribus.
II. A formal model of profit maximization under risk
Demand uncertainty is parameterized by a random variable, ε, drawn from a continuously
differentiable cumulative probability distribution F(ε), with strictly positive probability density,
dF(ε). We will assume that a sports team faces a demand which solely depends on a ticket price p
and this random variable, such that
denotes the demand function in state ε. The random
variable is correlated with all possible phenomena that could influence the demand on the match
day but are unknown to the sports team at the time of the pricing decision. For example, if the
sports team is performing better in the season that expected, ε is larger than unity and it will be
assumed to tilt the demand curve outwards. We will model this by letting uncertainty enter the
demand function through a multiplicative and additive shift in the demand curve. A demand
function that satisfies the stated assumptions is
,
(1)
where α determines the strength of the additive shift relative to the multiplicative shift. Following
the existing literature in Sports Economics, we will assume that marginal costs of the sports team
are zero, but that fixed costs F are positive (salaries etc. are assumed constant and completely
independent of price and output when the pricing decision is made). Further, assuming the
stadium is without capacity problems, ex post profits become
(2)
As a benchmark for comparison, we will first analyze and solve the maximization problem of a
sports team maximizing expected profits. Such a sports team would select the price maximizing
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(3)
The first order condition is
(4)
That is expected marginal revenues equal to zero, that is, pricing where the elasticity of demand
equals minus one. If a sports team do not care about risk, low demand states are equally as bad as
high demand states are good, and consequently it maximizes profits for the average demand.
We will now assume that that the sports team’s attitude towards risk can be summarized by a von
Neumann-Morgenstern utility function. This may be a strong assumption since pricing decisions
in sports teams are typically taken by a group of individuals, and group preferences may not
always satisfy the transitivity axiom required for the existence of a utility function. It is therefore
possible that our approach implicitly assumes that the sports team’s reactions to changes in its
environment are more predictable and stable than they really are. However, there are presumably
sports teams in which preferences are sufficiently similar within the group of decision makers to
guarantee the existence of a group reference function. This provides justification for the approach
taken in this paper. The utility function of the sports team is a concave, continuous, and
differentiable function of profits, so that
,
(5)
Hence, the sports team is assumed to be risk averse. By the introduction of risk aversion the
sports team selects the price that maximizes
(6)
The first order condition is
(7)
Evaluating the first order condition for the price maximizing expected profits yields
(8)
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Now, let
correspond to the profit when the random variable is unity. Then for
,
since
, we have
(9)
by assumption (5). The same inequality holds for
. Therefore,
deriving expectations on both sides of (9) yields
,
(10)
That is, the price maximizing expected profits generates an expected marginal utility of profits
less than zero which is inconsistent with the first order condition (7). Hence, the optimal price is
smaller than the price solving equation (4) which implies expected marginal revenues larger than
zero. Specifically, the risk averse sports team will price in the inelastic range of the demand
curve. Thus, the pricing pattern observed empirically may be explained simply by risk aversion.
The result is intuitively plausible. The variance of profits is an increasing function of price and in
response to that a risk averse firm charges a lower price than a firm maximizing expected profits.
III. The optimal price and fixed costs
In the literature, it is often mentioned that team owners argue for higher ticket prices as a
response to higher costs due to higher salaries (in the present model considered fixed costs).
Under certainty this argument is considered invalid since the marginal decision setting is
unaffected by the level of fixed costs. In the present model things are more complicated since the
level of fixed costs has an effect on the marginal utility of profits.
Doing comparative statics on the first order condition (7) yields
,
(11)
where D is negative from the second order condition for the maximization problem. Hence, to
know the sign of the effect, we must determine the sign of the numerator of (11). In order to do
that we will use the Arrow-Pratt measure of absolute risk aversion
,
(12)
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and we will assume that it is a decreasing function of profits; that is, compensation required for
accepting a given risk is falling with the profit level.
Now, let
correspond to the profit when the random variable satisfies
.
Then, using the assumption of decreasing absolute risk aversion, we have
for
(13)
Substituting from the definition of
yields
for
,
(14)
Clearly, we know that
for
since
,
(15)
by assumption. Multiplying (14) with the left-hand side of (15), we obtain
(16)
The same inequality holds for all . For if
, the inequality in (14) is
reversed, but so is that in (15). Consequently, deriving the expected values reveals the following
relationship
,
because
(17)
is a constant. But by the first order condition (7), the right hand side is zero, and
hence the left hand side is positive. Therefore, the numerator of equation (11) is positive and
since D is negative from the second order condition for the maximization problem, we conclude
that higher fixed costs imply a lower price. This result is contrary to the above mentioned
statements from team owners. The interaction is however fairly obvious. Higher fixed costs
reduce profits and therefore the willingness to take risks and again since variance is an increasing
function of price, the response is to decrease the price.
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IV. Conclusion
The paradox of inelastic sports pricing is perhaps a minor paradox than originally thought. It is
probably true that many sports teams engage in win maximization rather than profit
maximization, but this does not necessarily change first order conditions for ticket sales, see
Késenne and Pauwels (2006). It is probably true that a part of the explanation for pricing in the
inelastic range is the relation between sales of tickets and concessions, see Krautmann and Berri
(2007). It is probably also true that better methodologies tend to increase estimated elasticities,
see Forrest et al. (2002). Combining these approaches to explaining the paradox with integration
of effects of risk related behavior introduced in the present paper, the paradox is perhaps not a
paradox at all given the low marginal costs of admitting another fan is close to zero.
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References
Forrest, D., Simmons, R., and Feehan P. (2002). A Spatial Cross-sectional Analysis of the
Elasticity of Demand for Soccer. Scottish Journal of Political Economy 49, 336-355.
Fort, R. (2004). Inelastic Sports Pricing. Managerial and Decision Economics 25, 87-94.
Fort, R. (2007). Reply to “The Paradox of Inelastic Sports Pricing.” Managerial and Decision
Economics 28, 159-160.
Kahneman, D., and Tversky, A. (1979). “Prospect Theory: An Analysis of Decision under Risk.”
Econometrica 47, 263-291.
Késenne, S. and Pauwels, W. (2006). Club Objectives and Ticket Pricing in Professional Team
Sports. Eastern Economic Journal 32, 549-560.
Krautman, A. C., and Berri, D. J. (2007). Can We Find It at the Concessions? Understanding
Price Elasticity in Professional Sports. Journal of Sports Economics 8, 183-191.
Porter, P. K. (2007). The Paradox of Inelastic Sports Pricing. Managerial and Decision
Economics 28, 157-158.
Sandmo, A. (1971). On the Theory of the Competitive Firm under Price Uncertainty. American
Economic Review 61, 65-73.
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