## Hyperbolic and exponential discounting

In his post on Hyperbolic and exponential discounting, Murray Bourne objects to the comment by 'kissmetrics' in Six Advantages of Hyperbolic Discounting that "the phrase *hyperbolic discounting* is despicable jargon".

But actually I tend to agree with that judgement - though not to the credit of the 'kissmetrics' author.

One problem is that the term is often used in contexts where the specifically "hyperbolic" model is not what is relevant; in fact most of the "applications" in the 'kissmetrics' article fall into that category. It is also, in my opinion, inappropriate to name a model on the basis of a (relatively) esoteric aspect compared to the simplicity of its underlying assumption. Let me elaborate.

The basic idea of the 'kissmetrics' piece is that people often favour an immediate reward over a later reward which is worth more than the accepted reward would have been with accumulated interest at the prevailing rate, or equivalently that they prefer to postpone a payment now in favour of making a later payment which includes more than the interest they might earn in the meantime. There are actually lots of rational explanations for both of these (based in the first case on things like at least some of the cost of handling a deferred payment being independent of the length of the deferment and in the second on the fact that the onus is on the creditor to extract the funds if the buyer finds the product unsatisfactory and wants to break the deal), but that is not my main concern here. What I am interested in first is that the mathematical model needed to describe this behaviour is not necessarily "hyperbolic"; it could just be a "spread" between the interest rate that the individual applies to transactions and what could be earned in the bank, or any of dozens of other equally likely models. Indeed the distinction between exponential and "hyperbolic" discounting only really shows up in the nature of its "tail" behaviour for long periods and the fact that in the "hyperbolic" model the effective discount rate f(t)/f(t+1) is not constant (and in fact, though it may start high, it approaches 1 as t goes to infinity). Of course many other models also have this property in a way that is just as good at matching the data and to call all of them "hyperbolic" is incorrect.

But even when the hyperbolic model is being used, I do object a bit to the label because it plays on an aspect of the model which is much less "natural" than the fact that it is based on *linear* growth. Surely it would be more generally understandable and informative to call it the linear growth discounting model.

Some terminological mind games:

Although the function f(t)=1/(1+rt) has a hyperbolic graph it is not a hyperbolic function (because there are other functions which got that name first). But it is a rational function although many people think that using it for discounting is not rational!

Linear growth leads to hyperbolic discounting, but what leads to linear discounting? and is anything wrong with it? (Hint: Vernor Vinge and Ray Kurzweil may have the answer.)