Abstract
The aim of this paper is to obtain the family of the so-called generalized Weibull discount functions, introduced by Takeuchi (Game Econ Behav 71:456–478, 2011), by deforming the q-exponential discount function by means of the Stevens’ “power” law. The obtained discount functions exhibit different degrees of inconsistency and so they can be classified according to the value of their characteristic deforming parameters. Moreover, we extend the construction of the generalized Weibull discount function starting from any discount function instead of the q-exponential discounting. In any case, the value of the parameter \(\theta \) of these new discount functions is extended from (0, 1] to the union of the intervals \((-\,\infty ,0) \cup (0,+\,\infty )\).
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23 May 2018
In the original publication, the second author’s name was incorrect. The correct name should be Isabel González Fernández.
Notes
Here, the discount fraction is denoted by \(F_{t_1 \rightarrow t_2}\).
It is possible to find a discount function in which \(\displaystyle \lim _{t \rightarrow t_0} F(t) >0\) but, taking into account that F(t) is continuous, there will exist a \(t_1 > t_0\) (possibly \(t_1 = +\,\infty \)) such that \(\displaystyle \lim _{t \rightarrow t_1} F(t) \ge 0\), which describes the three considered cases.
There is another noteworthy deformation of time given by \(D(t) = \alpha \ln (1+\beta t)\), where \(\alpha > 0\) and \(\beta > 0\) (Takahashi 2005). Nevertheless, this deformation will not be considered in this paper.
The original name of this function is q-exponential, where \(q = 1 - \theta \). However, in order to avoid some confusion, we will adapt the notation to the parameter \(\theta \) introduced by Takeuchi (2011) and hereinafter it will be named the \(\theta \)-exponential discount function.
In Cruz Rambaud and Ventre (2015), this condition is equivalent to satisfy the so-defined Q-property.
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Acknowledgements
This paper has been partially supported by the project “La sostenibilidad del sistema nacional de salud: reformas, estrategias y propuestas”, reference: DER2016-76053-R, Ministerio de Economía y Competitividad.
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The original version of this article was revised: The second author’s name was incorrect and it has been corrected in this version.
We are very grateful for the comments and suggestions offered by two anonymous referees.
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Cruz Rambaud, S., González Fernández, I. & Ventre, V. Modeling the inconsistency in intertemporal choice: the generalized Weibull discount function and its extension. Ann Finance 14, 415–426 (2018). https://doi.org/10.1007/s10436-018-0318-3
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DOI: https://doi.org/10.1007/s10436-018-0318-3