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  $$f(t)$$ having a negative first derivative and with $$c_t$$ (or $$c(t)$$ in continuous time) defined as consumption at time t, total utility from an infinite stream of consumption is given by


 * $$U(\{c_t\}_{t=0}^\infty)=\sum_{t=0}^\infty {f(t)u(c_t)}$$.

Total utility in the continuous-time case is given by


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

provided that this integral exists.

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