Positive Temporal Dependence of the Biological Clock Implies Hyperbolic Discounting

Temporal preferences of animals and humans often exhibit inconsistencies, whereby an earlier, smaller reward may be preferred when it occurs immediately but not when it is delayed. Such choices reflect hyperbolic discounting of future rewards, rather than the exponential discounting required for temporal consistency. Simultaneously, however, evidence has emerged that suggests that animals and humans have an internal representation of time that often differs from the calendar time used in detection of temporal inconsistencies. Here, we prove that temporal inconsistencies emerge if fixed durations in calendar time are experienced as positively related (positive quadrant dependent). Hence, what are time-consistent choices within the time framework of the decision maker appear as time-inconsistent to an outsider who analyzes choices in calendar time. As the biological clock becomes more variable, the fit of the hyperbolic discounting model improves. A recent alternative explanation for temporal choice inconsistencies builds on persistent under-estimation of the length of distant time intervals. By increasing the expected speed of our stochastic biological clock for time farther into the future, we can emulate this explanation. Ours is therefore an encompassing theoretical framework that predicts context-dependent degrees of intertemporal choice inconsistencies, to the extent that context can generate changes in autocorrelation, variability, and expected speed of the biological clock. Our finding should lead to novel experiments that will clarify the role of time perception in impulsivity, with critical implications for, among others, our understanding of aging, drug abuse, and pathological gambling.