Hyperbolic Discounting Emerges from the Scalar Property of Interval Timing

Ten dollars today is more attractive than the same amount of money tomorrow and is consistent with the well known proverb, “a bird in the hand is worth two in the bush.” We all know that the value of a reward is discounted over time. How the value discounts over time and what is the rationale for such discounting, however, is less clear. Classical economic theory has assumed an exponential form of discounting which has been repeatedly shown insufficient for explaining the data(Frederick and Loewenstein, 2002). Hyperbolic discounting, with a faster discount rate in the beginning than in the distant future, is found to fit the data better empirically. Consequently, it is of great interest to understand why people and other animals discount future rewards hyperbolically. In the past, a number of explanations have been offered. Not surprisingly these explanations can be categorized into either value-based models or time-based models. In value-based models, hyperbolic discounting emerges when discount rates are assumed to be stochastic (Farmer and Geanakoplos, 2009), or there is uncertainty of the risk that the future reward will not be realized (Sozou, 1998), or there is uncertainty of the timing of the realization (Dasgupta and Maskin, 2005). In time-based models, hyperbolic discounting can be shown to result from the perceptual distortion of time (Takahashi, 2005; Kim and Zauberman, 2009). Recently, Ray and Bossaerts (2011) offered another time-based explanation beginning with the crucial observation that subjective or psychological time is different from objective or physical time. They argued that the discounting function is indeed exponential with respect to the subject’s internal clock, but it appears to be hyperbolic to observers who use objective time – the measurement of which isn’t dependent upon biological systems and their inherent variability. Specifically, they made three assumptions: 1 Increment of psychological time is stochastic and auto-correlated (i.e., the internal clock is positive quadrant dependent). 2 Increment of psychological time is stationary. 3 Value discounting over psychological time is exponential.