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

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.

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

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.

The Brain as a Hierarchical Organization

Based on recent neuroscience evidence, we model the brain as a dual-system organization subject to three conflicts: asymmetric information, temporal horizon, and incentive salience. Under the first and second conflicts, we show that the uninformed system imposes a positive link between consumption and labor at every period. Furthermore, decreasing impatience endogenously emerges as a consequence of these two conflicts. Under the first and third conflicts, it becomes optimal to set a consumption cap. Finally, we discuss the behavioral implications of these rules for choice bracketing and expense tracking, and for consumption over the life cycle. (JEL D11, D74, D82, D87, D91)