Discount function

A discount function is used in economic models to describe the weights placed on rewards received at different points in time. For example, if time is discrete and utility is time-separable, with the discount function ${\displaystyle f(t)}$ having a negative first derivative and with ${\displaystyle c_{t}}$ (or ${\displaystyle c(t)}$ in continuous time) defined as consumption at time t, total utility from an infinite stream of consumption is given by

${\displaystyle U(\{c_{t}\}_{t=0}^{\infty })=\sum _{t=0}^{\infty }{f(t)u(c_{t})}}$.

Total utility in the continuous-time case is given by

${\displaystyle U(\{c(t)\}_{t=0}^{\infty })=\int _{0}^{\infty }{f(t)u(c(t))dt}}$

provided that this integral exists.

Exponential discounting and hyperbolic discounting are the two most commonly used examples.