Editing Time preference (section)

=== Hyperbolic discounting ===
Although the exponential equation provides a nice rationale for discounting in accordance with utility theory, the apparent rate, when measured in the lab, is not constant. It actually declines over time. This means that the difference between receiving $10 tomorrow and $11 in two days is different from receiving $10 in 100 days and $11 in 101 days. Although the difference between the values and the times is the same, people ''value'' the two options at a different discount rate. The $1 is more heavily discounted between tomorrow and two days than it is between 100 and 101 days, meaning that people prefer the $10 option more in the two day case than in the 100 day case.

Such preferences fit a hyperbolic curve. The first hyperbolic delay function was of the form<ref name=":0" />

<math>D(k) = 
\begin{cases} 
1 & \text{if } k = 0 \\ 
\beta \delta^k & \text{if } k > 0 
\end{cases}</math>

This function describes a difference between the discount rate today and the next period, and then constant discounting after. It is commonly called the <math>(\beta, \delta)</math> model.

A simple hyperbolic delay discounting equation is that of

<math>\frac{v}{V} = \frac{1}{1 + kD}</math>

Where <math>v</math> is the discounted value, <math>V</math> is the non-discounted value, <math>k</math> is the discount rate, and <math>D</math> is the delay.<ref>{{Cite journal |last=Rachlin |first=Howard |date=May 2006 |title=Notes on Discounting |journal=Journal of the Experimental Analysis of Behavior |language=en |volume=85 |issue=3 |pages=425–435 |doi=10.1901/jeab.2006.85-05 |issn=0022-5002 |pmc=1459845 |pmid=16776060}}</ref> This is one of the most common hyperbolic discounting functions used today, and is especially useful in comparing two discounting scenarios, as the <math>k</math> parameter can be easily interpreted.