Exponential Discounting
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In
economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
, exponential discounting is a specific form of the
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 having a n ...
, used in the analysis of choice over time (with or without
uncertainty Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision ...
). Formally, exponential discounting occurs when total
utility In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish ...
is given by U \Bigl(\_^ \Bigr) = \sum_^ \delta^(u(c_t)) where is consumption at time , is the exponential
discount factor In finance, discounting is a mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Effic ...
, and is the instantaneous utility function. In
continuous time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
, exponential discounting is given by U \Bigl(\_^ \Bigl) = \int_^ e^u(c(t))\,dt Exponential discounting implies that the
marginal rate of substitution In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no ext ...
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 at date to earn a utility benefit of at time . At date , this investment opportunity is considered favorable; hence, this function is: . Now consider from the perspective of date , this investment opportunity is still viewed as favorable given . To view this mathematically, observe that the new expression is the old expression multiplied by . Therefore, the preferences at is preserved at ; 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 In economics, hyperbolic discounting is a time inconsistency, time-''inconsistent'' model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers. Acc ...
have more empirical support.


See also

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Temporal discounting In behavioral economics, time preference (or time discounting,. delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good at an earlier date compared with receiving it at a later ...
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Hyperbolic discounting In economics, hyperbolic discounting is a time inconsistency, time-''inconsistent'' model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers. Acc ...
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Intertemporal choice In economics, intertemporal choice is the study of the relative value people assign to two or more payoffs at different points in time. This relationship is usually simplified to today and some future date. Intertemporal choice was introduced by ...
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Keynes–Ramsey rule In macroeconomics, the Keynes–Ramsey rule is a necessary condition for the optimality of intertemporal consumption choice. Usually it is expressed as a differential equation relating the rate of change of consumption with interest rates, time ...


References

*{{cite book , first1=Andreu , last1=Mas-Colell , author1-link=Andreu Mas-Colell, author2-link=Michael Whinston , first2=Michael D. , last2=Whinston, author3-link=Jerry Green (economist) , first3=Jerry R. , last3=Green , title=Microeconomic Theory , publisher=Oxford University Press , year=1995 , pages=733–736 , isbn=0-19-507340-1 , url=https://books.google.com/books?id=KGtegVXqD8wC&pg=PA733 Intertemporal economics