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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.

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

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