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 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:

{{mathimg|61a934cd75f1640703c9175647fe8439f183b3ac|width=27.382ex|height=6.843ex|va=-3.005ex|alt={\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:

{{mathimg|a77c8ed25041d33691734efdd05d9f38fecf7ec8|width=34.068ex|height=5.843ex|va=-2.338ex|alt={\displaystyle U{\Bigl (}\{c(t)\}_{t=0}^{\infty }{\Bigr )}=\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.