Discount function
In economics, 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 f(t) having a negative first derivative and with ct (or c(t) in continuous time) defined as consumption at time t, total utility from an infinite stream of consumption is given by: U ( { c t } t = 0 ∞ ) = ∑ t = 0 ∞ f ( t ) u ( c t ) {\displaystyle U{\Bigl (}\{c_{t}\}_{t=0}^{\infty }{\Bigr )}=\sum _{t=0}^{\infty }{f(t)u(c_{t})}} Total utility in the continuous-time case is given by: U ( { c ( t ) } t = 0 ∞ ) = ∫ 0 ∞ f ( t ) u ( c ( t ) ) d t {\displaystyle U{\Bigl (}\{c(t)\}_{t=0}^{\infty }{\Bigr )}=\int _{0}^{\infty }{f(t)u(c(t))dt}} provided that this integral exists.