hyperbolically is where V is the present value, P is the annual cash flow, D is the number of annual payments and k is the factor governing the discounting. Criticism Several alternative explanations of non-exponential discounting have been proposed. An article from 2003 noted that this pattern might be better explained by a similarity heuristic than by hyperbolic discounting. Subjects have also reported changing relative preferences as they see more details of what they are choosing—a “temporal construal” effect. A study by Daniel Read introduces "subadditive discounting": the fact that discounting over a delay increases if the delay is divided into smaller intervals. This hypothesis may explain the main finding of many studies in support of hyperbolic discounting—the observation that impatience declines with time–while also accounting for observations not predicted by hyperbolic discounting. However, although these observations depart from exponential discounting, they do not entail preference reversal as time from the choice to the earlier reward increases. Arousal of appetite or emotion does sometimes lead to preference reversal, and this has been the most widely accepted alternative to a simply hyperbolic function: hyperboloid or quasi-hyperbolic discounting fuses exponential curves with an arousal bump as a visceral reward becomes imminent. Such cases are obviously important, but still do not account for cases where either both or neither choice is made during arousal. The most obvious objection to hyperbolic discounting is that many or most people learn to choose consistently over time in most situations. Similarly, a 2014 paper criticized the existing studies for mostly using data collected from university students and being too quick to conclude that the hyperbolic model of discounting is correct. Human experiments have frequently reported wide between-subject variations. If overcoming the tendency to temporary preference takes learning, the next obvious task for experimenters is to test theories of how and when this learning occurs (e.g. Ainslie, 2012). See also Akrasia Deferred gratification Intertemporal choice Temporal motivation theory Time preference Time value of money References Further reading zh-yue:雙曲線折扣 Cognitive biases Behavioral finance Intertemporal economics