Abstract
We consider a model, due to Andersen and Nielsen (Econ Lett 118(2):262–264, 2013), concerning the behavior of a risk-averse sports team under uncertainty in demand: the team chooses a value for the price of its ticket, but the ticket demand is stochastic at the moment of decision. For this model, we carry out a decision analysis by studying several comparative-static effects not considered by the authors in their paper. Specifically, we examine the effect of changes in the team’s risk aversion, and also the effect of a variation in the risk of the random demand. Furthermore, we enhance the model by considering a proportional profit tax, and we study the effect of a variation in the tax rate. We derive some conditions under which the sports team finds optimal to reduce the ticket price as a consequence of a rise in the tax rate.
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Notes
- 1.
The effect of fixed costs is examined in [2] only for the case of a team that exhibits decreasing absolute risk aversion.
- 2.
According to [2, p. 262], the number α “determines the strength of the additive shift relative to the multiplicative shift.” On the other hand, it is assumed that f is a function of class \(\mathcal{C}^{2}\) on \(\left [0, +\infty \right )\).
- 3.
- 4.
A sufficient condition for U ′ ′ < 0 is the following: f ′ < 0 and f ′ ′ ≤ 0.
- 5.
- 6.
Notice that the standard deviation of the random variable \(x(\,p,\varepsilon )\) (see (1)) is \({\bigl (\,f(\,p)+\alpha \bigr )}\sigma\). Thus a variation in σ is indeed equivalent to a variation in the standard deviation of the random demand x.
- 7.
Notice that M( p τ ∗) > 0, since Lemma 1 remains valid if we write π τ instead of π (the proof would be, mutatis mutandis, the same).
References
Álvarez-López, A.A., Rodríguez-Puerta, I.: Teoría de la empresa bajo incertidumbre con mercado de futuros: el papel de los costes fijos y de un impuesto sobre los beneficios. Rect@ 10(1), 253–265 (2009)
Andersen, P., Nielsen, M.: Inelastic sports pricing and risk. Econ. Lett. 118(2), 262–264 (2013)
Lippman, S.A., McCall, J.J.: The economics of uncertainty: selected topics and probabilistic methods. In: Arrow, K.J., Intriligator, M.J. (eds.) Handbook of Mathematical Economics, vol. 1, chap. 6. North-Holland, Amsterdam (1982)
Rodríguez-Puerta, I., Sebastiá-Costa, F., Álvarez-López, A.A., Buendía, M.: Una herramienta de análisis teórico en la teoría de la empresa bajo incertidumbre. Revista de Métodos Cuantitativos para la Economía y la Empresa 11, 33–40 (2011)
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Acknowledgements
We are grateful to two referees for their very helpful comments. Álvarez-López would also like to thank the financial support provided by the Spanish Interministerial Commission of Science and Technology (CICYT: Comisión Interministerial de Ciencia y Tecnología), under the Project with the reference number ECO2012-39553-C04-01.
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Appendix
Appendix
The following lemma slightly generalizes a result taken from [3]:
Lemma 2
Let ψ and ϕ be two real functions defined on \(\mathbb{R}\) such that ψ > 0 and ϕ is increasing. If ξ = ψ ⋅ϕ, and X is a real random variable such that the expectation \(E\!\left [X\,\psi (X)\right ]\) is finite, and such that the probability of the set \(\left \{X\neq 0\right \}\) is positive, then:
and the reverse inequality holds when ϕ is decreasing. In addition, if ϕ is strictly increasing or strictly decreasing, the corresponding inequalities also hold strictly.
Proof
See [4]: in Lemma 1, write \(F \equiv 1\) and Z = X. See also [1]. □
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Álvarez-López, A.A., Rodríguez-Puerta, I. (2015). Decision Analysis in a Model of Sports Pricing Under Uncertain Demand. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_3
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