Economists routinely model how people choose between different options. For example, a common issue is time delay: would you choose to receive £100 today, or prefer £200 next month? The classical normative model – purporting to describe the optimal way of behaving – claims that a decision maker should confront such problems with exponential discounting. In judging the value of a later payment relative to an earlier one, the later payment should be de-valued or discounted by an exponential function of the time between them. For money payments, this perspective can be derived through a no-arbitrage argument, assuming both payments are guaranteed and the earlier payment can be invested during the delay to earn compound interest.
Real people often don’t behave this way. One striking observation, for example, is that people often reverse their preferences, switching as time passes from preferring the later to the earlier payment, something never predicted by standard exponential discounting. To account for empirical data, researchers have proposed a variety of alternative and more descriptive discounting models which use non-exponential functions to better fit observed behaviour. Meanwhile, many economic theorists have laboured to save the classic model through various connivances – assuming some uncertainty in the potential payments, for example, or the possibility of systematic irrational cognitive biases.
In a new paper, LML Fellows Alex Adamou, Yonatan Berman, and Ole Peters, working with Diomides Mavroyiannis at Université Paris-Dauphine, take a different approach, proposing an alternative model of optimal discounting in which decision makers always follow a simple rule – they consistently try to maximise the growth rate of their wealth. Their analysis shows that this simple perspective offers a natural explanation for a wide range of empirically observed discounting behaviour, and elucidates how the optimal strategy often depends on variables such as a person’s existing wealth.
Adamou and colleagues consider a decision maker who has to choose between two specified payments to be made at distinct future times, and does so by comparing the growth rates of wealth associated with each option. This is not yet enough to fully define the problem, which requires further assumptions about the manner in which an individual’s wealth grows (through additive or multiplicative dynamics), as well as for the period over which the growth rate is computed. As the researchers demonstrate, the optimal method of discounting depends strongly on these assumptions, and in different cases yields four different forms of discounting: no discounting, exponential, hyperbolic and a hybrid of exponential and hyperbolic.
Importantly, two of the discount functions arising in the model, hyperbolic and hybrid, predict preference reversals as observed in the behaviour of real people. In addition, the hybrid discount function depends on the decision maker’s current wealth and underlying growth rate. Overall, the model thereby produces a richer set of predicted behaviours than other specifications. One further prediction of particular interest is that decision makers may switch from preferring the earlier to the later payment as their wealth increases. In other words, richer people discount less steeply. This is again consistent with empirical findings.
This is the first analysis proposing a normative model which predicts both exponential and non-exponential discounting without violating standard axioms of choice – the logical bedrock of decision theory. Moreover, the analysis has no need to assume any bias nor dynamic inconsistency on the part of the deciding agent, who at all times simply prefers the option with the highest growth rate. From this perspective, changes in discounting behaviour arise from actual changes of circumstance, not of mind. The authors believe that the predictions of this model can be tested in experiments, and expect to see such tests in the near future.
The paper is available at https://www.researchers.one/article/2019-10-7