Discounted utility

In economics, discounted utility is the utility (desirability) of some future event, such as consuming a certain amount of a good, as perceived at the present time as opposed to at the time of its occurrence. It is calculated as the present discounted value of future utility, and for people with time preference for sooner rather than later gratification, it is less than the future utility. The utility of an event x occurring at future time t under utility function u, discounted back to the present (time 0) using discount factor ${\displaystyle \beta ,}$ Is

Since more distant events are less liked, ${\displaystyle 0<\beta <1.}$

Discounted utility calculations made for events at various points in the future as well as at the present take the form

where ${\displaystyle u(x_{t})}$ is the utility of some choice ${\displaystyle x}$ at time ${\displaystyle t}$ and T is the time of the most distant future satisfaction event. Here, since utility comparisons are being made across time when the utilities are combined in a single evaluation, the utility function is necessarily cardinal in nature.

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