Editing Time preference (section)

=== Quasi-hyperbolic discounting ===
The last major model is that of quasi-hyperbolic discounting. Researchers found that there is a first day effect, meaning that people greatly value immediate rewards over those in the future. Like the previous example, imagine now that you are offered $10 today or $11 tomorrow. You are also offered $10 tomorrow or $11 in two days. The preference for the $10 in the today case is typically greater than the preference for the $10 tomorrow case.

This can be captured by a quasi-hyperbolic curve, wherein there is a fitted parameter for the magnitude of the first day effect. This is commonly called the beta-delta model, wherein there is a beta parameter that accounts for the present bias. The equation for utility over time looks like<ref>{{Cite journal |last1=O'Donoghue |first1=Ted |last2=Rabin |first2=Matthew |date=1999 |title=Doing It Now or Later |url=https://www.jstor.org/stable/116981 |journal=The American Economic Review |volume=89 |issue=1 |pages=103–124 |doi=10.1257/aer.89.1.103 |jstor=116981 |issn=0002-8282|url-access=subscription }}</ref>

<math>U_t(u_t, u_{t+1}, \dots, u_T) = \delta^t u_t + \beta \sum_{s=t+1}^T \delta^{s-t} u_s
</math>

This explains that the sum of your current and all future utilities is equal to a delta parameter multiplied by your current utility plus all your future discounted utilities (scaled by beta).