Exponential discounting

In economics, exponential discounting is a specific form of the discount function, used in the analysis of choice over time (with or without uncertainty). Formally, exponential discounting occurs when total utility is given by

$$U(\{c_t{\}}_{t=t_1}^{t_2})=\sum _{t=t_1}^{t_2}{\delta }^{t-t_1}(u(c_t))$$

where Template:Mvar is consumption at time Template:Mvar, Template:Mvar is the exponential discount factor, and Template:Mvar is the instantaneous utility function.

In continuous time, exponential discounting is given by

$$U(\{c(t){\}}_{t=t_1}^{t_2})=\underset{t_1}{\overset{t_2}{\int }}{e}^{-\rho (t-t_1)}u(c(t))dt$$

Exponential discounting implies that the marginal rate of substitution between consumption at any pair of points in time depends only on how far apart those two points are. Exponential discounting is not dynamically inconsistent. A key aspect of the exponential discounting assumption is the property of dynamic consistency— preferences are constant over time. In other words, preferences do not change with the passage of time unless new information is presented. For example, consider an investment opportunity that has the following characteristics: pay a utility cost of Template:Mvar at date Template:Math to earn a utility benefit of Template:Mvar at time Template:Math. At date Template:Math, this investment opportunity is considered favorable; hence, this function is: Template:Math. Now consider from the perspective of date Template:Math, this investment opportunity is still viewed as favorable given Template:Math. To view this mathematically, observe that the new expression is the old expression multiplied by Template:Math. Therefore, the preferences at Template:Math is preserved at Template:Math; thus, the exponential discount function demonstrates dynamically consistent preferences over time.

For its simplicity, the exponential discounting assumption is the most commonly used in economics. However, alternatives like hyperbolic discounting have more empirical support.

See also

• Temporal discounting
• Hyperbolic discounting
• Intertemporal choice
• Keynes–Ramsey rule

References

• Template:Cite book