A note on the relevance of the q-exponential function in the context of intertemporal choices

doi:10.1016/j.physa.2005.08.056

Physica A: Statistical Mechanics and its Applications

Volume 364, 15 May 2006, Pages 385-388

https://doi.org/10.1016/j.physa.2005.08.056 Get rights and content

Abstract

This paper shows that the q-exponential function well known in the deformed algebra inspired in the Tsallis's nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience. Moreover, we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions.

Introduction

A large part of the behavioral economic literature has been paying attention to a deviation from the expected utility theory known as dynamic inconsistency. Dynamic consistency requires that the ranking of consumption streams be unchanged over time. But, empirical evidence has shown that this is not true in all situations [1], [2], [3], [4]. For example, if people are asked to choose between (a1) $$ 1000$ in 1 year and (a2) $$ 1050$ in 1 year and 1 week or (b1) $$ 1000$ today and (b2) $$ 1050$ in 1 week, then according to the expected utility theory someone who chooses (a2) in the first situation must choose (b2) in the second situation. However, greater impatience for immediate rewards can make one choose (a2) and (b1).

According to [5], the only way for agents to achieve intertemporal consistency is to discount the cash flows from all delayed events by a constant rate-per-time unit. Therefore, in the context of the classical expected theory, the exponential function has played a fundamental role.¹ The problem is that with the discounting supported by the exponential function, the phenomena presented in the example above cannot be modelled. For that reason, the hyperbolic discount was introduced [6] as an alternative to the exponential discount model in order to take the phenomena of increasing impatience described above into account.

In this paper, I introduce the definition of the q-exponential utility function based on the q-exponential function discount which is an extension of both previous ideas: (1) the concept of the classical utility function based on exponential discount and (2) the concept of the hyperbolic utility function based on the hyperbolic discount. Actually, the q-exponential is a very well-known function in the deformed algebra inspired in nonextensive thermodynamics [7] and was very well studied by [8].

This paper is organized as follows. In Section 2, the q-exponential function is introduced. In Section 3, an example of intertemporal choice is presented. Finally, Section 4 presents some conclusions of this work.

Section snippets

The q-exponential utility function

The q-exponential utility function can be defined as $U (x_{1}, x_{2}, \dots, x_{T}) = \sum_{t = 1}^{T} β_{q} (t) u (x_{t}),$ where u is an instantaneous utility function, T is a fixed time in the future and $β_{q} (t)$ , the q-exponential discount, is the inverse of the q-exponential function given by $β_{q} (t) = \frac{1}{e_{q} (α t)} = \frac{1}{[1 + (1 - q) α t]^{1 / (1 - q)}}$ with $q \in [0, 1]$ . One should note that when $q \to 1$ then $β_{q} (t)$ recovers the classical exponential discount. On the other hand, when $q = 0$ then $β_{q} (t)$ yields the hyperbolic discount [9]. Moreover, the definition of the q

The buffer-stock problem with q-exponential discount

In this paper, the buffer-stock problem² with q

Conclusions

This paper has introduced a very interesting connection between the Tsallis's thermodynamic statistics and behavioral economics⁶ by means of the q-exponential function.

Moreover, it has been shown that the q-exponential function can be used to take the phenomena known as increasing patience into account and the parameter q can be seen as a

Acknowledgements

The author would like to express his gratitude to one anonymous referee who made several important suggestions that helped the improvement of the paper.

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A note on the relevance of the q-exponential function in the context of intertemporal choices

Abstract

Introduction

Section snippets

The q-exponential utility function

The buffer-stock problem with q-exponential discount

Conclusions

Acknowledgements

Organ. Behav. Human Decision Process.

J. Econ. Behav. Organ.

Physica A

Chaos, Solitons and Fractals

Physica A

Physica A

Physica A

J. Econ. Psychology

Rev. Econ. Stud.