scholarly journals Delay Effect and Subadditivity. Proposal of a New Discount Function: The Asymmetric Exponential Discounting

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 367 ◽  
Author(s):  
Salvador Cruz Rambaud ◽  
Piedad Ortiz Fernández
Keyword(s):  
Intertemporal Choice ◽  
Delay Effect ◽  
Discount Function ◽  
Preference Relations ◽  
Monetary Rewards ◽  
Exponential Discount ◽  
Exponential Discounting ◽  
Decreasing Impatience

The framework of this paper is intertemporal choice and, more specifically, the so-called delay effect. Traditionally, this anomaly, also known as decreasing impatience, has been revealed when individuals reverse their preferences over monetary or non-monetary rewards. In this manuscript, we will analyze the delay effect by using preference relations and discount functions. The treatment of the delay effect with discount functions exhibits several scenarios for this paradox. Thus, the objective of this paper is to deduce the different expressions of the delay effect and their mathematical characterizations by using discount functions in stationary and dynamic settings. In this context, subadditivity will be derived as a particular case of decreasing impatience. Finally, we will introduce a new discount function, the so-called asymmetric exponential discount function, able to describe decreasing impatience.

scholarly journals A New Approach to Intertemporal Choice: The Delay Function

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 807
Author(s):  
Salvador Cruz Rambaud ◽  
Isabel González Fernández
Keyword(s):  
Behavioral Finance ◽  
Intertemporal Choice ◽  
Discount Function ◽  
New Approach ◽  
Preference Relations ◽  
Delay Function ◽  
Different Types ◽  
Variation Rate ◽  
The Way

The framework of this paper is intertemporal choice, which traditionally has been studied with preference relations and discount functions. However, the interest of econophysics in this topic makes time become a central magnitude. Therefore, the aim of this paper is to introduce the concept of delay function and, by using this tool, to analyze the concept of impatience and the different types of inconsistency. In behavioral finance, consistency is correlated with the concept of symmetry because, in this case, the indifference between two rewards does not change when the same delay is added to their respective availability dates. Moreover, we have shown the way to derive a discount (respectively, delay) function starting from the expression of its corresponding delay (respectively, discount) function by requiring some suitable conditions for this construction. Finally, we have deduced the concept of instantaneous variation rate and Prelec’s measure of inconsistency in terms of the delay function.

Discount models in intertemporal choice: an empirical analysis

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Isabel María Parra Oller ◽  
Salvador Cruz Rambaud ◽  
María del Carmen Valls Martínez
Keyword(s):  
Empirical Analysis ◽  
Intertemporal Choice ◽  
Hyperbolic Discounting ◽  
Least Squares Regression ◽  
Discount Function ◽  
Content Type ◽  
The Individual ◽  
Gains And Losses ◽  
First Time ◽  
The Empirical Analysis

PurposeThe main purpose of this paper is to determine the discount function which better fits the individuals' preferences through the empirical analysis of the different functions used in the field of intertemporal choice.Design/methodology/approachAfter an in-depth revision of the existing literature and unlike most studies which only focus on exponential and hyperbolic discounting, this manuscript compares the adjustment of data to six different discount functions. To do this, the analysis is based on the usual statistical methods, and the non-linear least squares regression, through the algorithm of Gauss-Newton, in order to estimate the models' parameters; finally, the AICc method is used to compare the significance of the six proposed models.FindingsThis paper shows that the so-called q-exponential function deformed by the amount is the model which better explains the individuals' preferences on both delayed gains and losses. To the extent of the authors' knowledge, this is the first time that a function different from the general hyperbola fits better to the individuals' preferences.Originality/valueThis paper contributes to the search of an alternative model able to explain the individual behavior in a more realistic way.

scholarly journals Correction to: Modeling the inconsistency in intertemporal choice: the generalized Weibull discount function and its extension

2018 ◽  
Vol 14 (3) ◽  
pp. 427-427
Author(s):  
Salvador Cruz Rambaud ◽  
Isabel González Fernández ◽  
Viviana Ventre
Keyword(s):  
Intertemporal Choice ◽  
Discount Function

scholarly journals Temporal delay discounting in acutely ill and weight-recovered patients with anorexia nervosa

2015 ◽  
Vol 45 (6) ◽  
pp. 1229-1239 ◽  
Author(s):  
F. Ritschel ◽  
J. A. King ◽  
D. Geisler ◽  
L. Flohr ◽  
F. Neidel ◽  
...  
Keyword(s):  
Anorexia Nervosa ◽  
Delay Discounting ◽  
Intertemporal Choice ◽  
Temporal Discounting ◽  
Self Control ◽  
Hyperbolic Model ◽  
Cross Sectional ◽  
Monetary Rewards ◽  
Cross Sectional Design ◽  
Age Range

Background.Patients with anorexia nervosa (AN) are characterized by a very low body weight but readily give up immediate rewards (food) for long-term goals (slim figure), which might indicate an unusual level of self-control. This everyday clinical observation may be quantifiable in the framework of the anticipation-discounting dilemma.Method.Using a cross-sectional design, this study compared the capacity to delay reward in 34 patients suffering from acute AN (acAN), 33 weight-recovered AN patients (recAN) and 54 healthy controls. We also used a longitudinal study to reassess 21 acAN patients after short-term weight restoration. A validated intertemporal choice task and a hyperbolic model were used to estimate temporal discounting rates.Results.Confirming the validity of the task used, decreased delay discounting was associated with age and low self-reported impulsivity. However, no group differences in key measures of temporal discounting of monetary rewards were found.Conclusions.Increased cognitive control, which has been suggested as a key characteristic of AN, does not seem to extend the capacity to wait for delayed monetary rewards. Differences between our study and the only previous study reporting decreased delay discounting in adult AN patients may be explained by the different age range and chronicity of acute patients, but the fact that weight recovery was not associated with changes in discount rates suggests that discounting behavior is not a trait marker in AN. Future studies using paradigms with disorder-specific stimuli may help to clarify the role of delay discounting in AN.

scholarly journals Are Delay and Interval Effects the Same Anomaly in the Context of Intertemporal Choice in Finance?

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 41
Author(s):  
Salvador Cruz Rambaud ◽  
Piedad Ortiz Fernández
Keyword(s):  
Reference Point ◽  
Intertemporal Choice ◽  
Hyperbolic Discounting ◽  
The Other ◽  
Delay Effect ◽  
Interval Effect ◽  
Or Groups ◽  
Delay Effects ◽  
The One ◽  
The Relationship

Traditionally, the interval and delay effects have been identified and considered as the same anomaly in the context of intertemporal choice, when individuals or groups of individuals make their decisions about reward preferences. This has supposed that most studies on this topic have been focused on the delay effect and, consequently, that the discount functions provided by the existing literature have considered only this effect. This is the case of hyperbolic discounting, which has been used to describe the delay, but not the interval effect. Therefore, the main objective of this paper is to carry out a detailed analysis of both anomalies, which will allow us to mathematically relate them, thus finding their analogies and differences. To do this, we will first analyze the concept of delay effect and later the different definitions of the interval effect. The main conclusion of this paper is twofold. On the one hand, if the benchmark for valuation is fixed, the delay effect coincides with the so-called decreasing interval effect. On the other hand, if the assessment reference point is the beginning of each interval, both anomalies are different. These findings make necessary to redefine the concept of interval effect. Finally, we will analyze the relationship between the interval effect, the delay effect and the subadditivity

scholarly journals Time-Consistent Investment and Consumption Strategies under a General Discount Function

2021 ◽  
Vol 14 (2) ◽  
pp. 86
Author(s):  
Ishak Alia ◽  
Farid Chighoub ◽  
Nabil Khelfallah ◽  
Josep Vives
Keyword(s):  
Variational Method ◽  
Portfolio Management ◽  
Open Loop ◽  
Explicit Representation ◽  
Time Inconsistency ◽  
Management Problem ◽  
Discount Function ◽  
Special Cases ◽  
Consumption Strategies ◽  
Exponential Discounting

In the present paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that gives rise to time-inconsistency of the decision-maker. We consider equilibrium policies within the class of open-loop controls that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

scholarly journals Testable Implications of Models of Intertemporal Choice: Exponential Discounting and Its Generalizations

2020 ◽  
Vol 12 (4) ◽  
pp. 114-143
Author(s):  
Federico Echenique ◽  
Taisuke Imai ◽  
Kota Saito
Keyword(s):  
Intertemporal Choice ◽  
Revealed Preference ◽  
Parametric Methods ◽  
Consumption Data ◽  
Exponential Discounting ◽  
Preference Tests

We present revealed-preference characterizations of the most common models of intertemporal choice: the model of exponentially discounted concave utility, and some of its generalizations. Our characterizations take consumption data as primitives, and provide nonparametric revealed-preference tests. We apply our tests to data from two recent experiments and find that our axiomatization delivers new insights and perspectives on datasets that had been analyzed by traditional parametric methods. (JEL C91, D11, D15)

scholarly journals Exponential discounting in security games of timing

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Jonathan Merlevede ◽  
Benjamin Johnson ◽  
Jens Grossklags ◽  
Tom Holvoet
Keyword(s):  
Intertemporal Choice ◽  
Time Discounting ◽  
Distant Future ◽  
Common Assumption ◽  
Games Of Timing ◽  
Advanced Persistent Threats ◽  
Security Games ◽  
Attack And Defense ◽  
Best Responses ◽  
Exponential Discounting

Abstract Strategic game models of defense against stealthy, targeted attacks that cannot be prevented but only mitigated are the subject of a significant body of recent research, often in the context of advanced persistent threats (APTs). In these game models, the timing of attack and defense moves plays a central role. A common assumption, in this literature, is that players are indifferent between costs and gains now and those in the distant future, which conflicts with the widely accepted treatment of intertemporal choice across economic contexts. This article investigates the significance of this assumption by studying changes in optimal player behavior when introducing time discounting. Specifically, we adapt a popular model in the games of timing literature, the FlipIt model, by allowing for exponential discounting of gains and costs over time. We investigate changes of best responses and the location of Nash equilibria through analysis of two well-known classes of player strategies: those where the time between players’ moves is constant, and a second class where the time between players’ moves is stochastic and exponentially distributed. By introducing time discounting in the framework of games of timing, we increase its level of realism as well as applicability to organizational security management, which is in dire need of sound theoretic work to respond to sophisticated, stealthy attack vectors.

scholarly journals An Extension of the Concept of Derivative: Its Application to Intertemporal Choice

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 696
Author(s):  
Salvador Cruz Rambaud ◽  
Blas Torrecillas Jover
Keyword(s):  
Behavioral Finance ◽  
Intertemporal Choice ◽  
Point Of View ◽  
Financial Mathematics ◽  
Economic Problems ◽  
Scientific Disciplines ◽  
The One ◽  
Novel Concept ◽  
Decreasing Impatience

The framework of this paper is the concept of derivative from the point of view of abstract algebra and differential calculus. The objective of this paper is to introduce a novel concept of derivative which arises in certain economic problems, specifically in intertemporal choice when trying to characterize moderately and strongly decreasing impatience. To do this, we have employed the usual tools and magnitudes of financial mathematics with an algebraic nomenclature. The main contribution of this paper is twofold. On the one hand, we have proposed a novel framework and a different approach to the concept of relative derivation which satisfies the so-called generalized Leibniz’s rule. On the other hand, in spite of the fact that this peculiar approach can be applied to other disciplines, we have presented the mathematical characterization of the two main types of decreasing impatience in the ambit of behavioral finance, based on a previous characterization involving the proportional increasing of the variable “time”. Finally, this paper points out other patterns of variation which could be applied in economics and other scientific disciplines.

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