General time consistent discounting

Modeling inter-temporal choice is a key problem in both computer science and economic theory. The discounted utility model of Samuelson is currently the most popular model for measuring the global utility of a time-series of local utilities. The model is limited by not allowing the discount function to change with the age of the agent. This is despite the fact that many agents, in particular humans, are best modelled with age-dependent discount functions. It is well known that discounting can lead to time-inconsistent behaviour where agents change their preferences over time. In this paper we generalise the discounted utility model to allow age-dependent discount functions. We then extend previous work in time-inconsistency to our new setting, including a complete characterisation of time-(in)consistent discount functions, the existence of sub-game perfect equilibrium policies where the discount function is time-inconsistent and a continuity result showing that “nearly” time-consistent discount rates lead to “nearly” time-consistent behaviour

Time consistent discounting

A possibly immortal agent tries to maximise its summed discounted rewards over time, where discounting is used to avoid infinite utilities and encourage the agent to value current rewards more than future ones. Some commonly used discount functions lead to time-inconsistent behavior where the agent changes its plan over time. These inconsistencies can lead to very poor behavior. We generalise the usual discounted utility model to one where the discount function changes with the age of the agent. We then give a simple characterisation of time-(in)consistent discount functions and show the existence of a rational policy for an agent that knows its discount function is time-inconsistent

Hyperbolic discounting may be time consistent

Using dynamic programming methodology, the paper analyzes the most general conditions for an additive utility functional to represent time consistent preferences. It challenges the conventional wisdom of the domain, which, following Strotz(1956), assume that only exponential discounting is compatible with time consistent behavior. The paper gives some examples of special time consistent hyperbolic discount functions and also discuss the relation between time consistency and stationarity.intertemporal choice, consumption and saving, time consistency, time discounting, exponential discounting, hyperbolic discounting

Staggered wages, inflation, and discounting

In the literature of staggered wages (Taylor, 1979, 1980; Blanchard, 1986; Ball and Cecchetti, 1991) the discount factor is neglected in the workers’ loss function. Yet discounting is to be viewed as an extra piece of micro-foundation with implications for discretionary monetary policy. We revisit the issue and show that discounting in the model of staggered wages actually lowers the time consistent steady inflation.Staggered wage model; Time consistent steady inflation; Discounting

Hyperbolic Discounting can represent Consistent Preferences

Among non Exponential Discounting (ED) models, introduced to capture time inconsistent choices, Hyperbolic Discounting (HD) recently gained particular relevance. This paper points out that, for some particular payoff structures, HD can also represent consistent preferencesTime Consistency, Hyperbolic Discounting

Heterogeneous discounting in consumption-investment problems. Time consistent solutions

In this paper we analyze a stochastic continuous time model in finite horizon in which agents discount the instantaneous utility function and the final function at constant but different instantaneous discount rates of time preference. Within this context we can model problems in which, when the time t approaches to the final time, the valuation of the final function increases compared with previous valuations in a way that cannot be explained by using a unique constant or a variable discount rate. We derive a dynamic programming equation whose solutions are time-consistent Markov equilibria. For this class of time preferences, we study the classical consumption and portfolio rules model (Merton, 1971) for CRRA and CARA utility functions for time- consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of stochastic terminal time is also discussed.dynamic programming, consumption and portfolio rules, heterogeneous discounting, time consistency

Hyperbolic Discounting Is Rational: Valuing the Far Future with Uncertain Discount Rates

Conventional economics supposes that agents value the present vs. the future using an exponential discounting function. In contrast, experiments with animals and humans suggest that agents are better described as hyperbolic discounters, whose discount function decays much more slowly at large times, as a power law. This is generally regarded as being time inconsistent or irrational. We show that when agents cannot be sure of their own future one-period discount rates, then hyperbolic discounting can become rational and exponential discounting irrational. This has important implications for environmental economics, as it implies a much larger weight for the far future.Hyperbolic discounting, Environment, Time consistent, Exponential discounting, Geometric random walk, Term structure of interest rates

Evolution as Learning Yields Hyperbolic Discounting

Learning is modeled as an infection, which jumps from person to person. The rate of infection mimics individual discount rates and induces savings behavior on its own. It is shown that the apparent discount rate, the combination of the agents' true discount rate and the infection rate, decreases over time and approaches the agents' true discount rate. This decrease, known as hyperbolic discounting, is consistent with what is observed in psychology studies, while the limiting case, exponential discounting, is consistent with market level observations. This model closes the gap between individual and market level observations of discounting behavior without explicitly assuming the two kinds of discounting nor relying on commitment mechanisms.discounting, genetic algorithms, learning

Interpreting Time Horizon Effects in Inter-Temporal Choice

We compare different designs that have been used to test for an impact of time horizon on discounting, using real incentives and two representative data sets. With the most commonly used type of design we replicate the typical finding of declining (hyperbolic) discounting, but with other designs find constant or increasing discounting. As a whole, the data are not consistent with any of these usual candidate discounting assumptions, and they also imply a violation of transitivity. The results have implications for interpreting previous evidence, and pose an important puzzle for understanding inter-temporal choice.time preference, hyperbolic discounting, self-control, dynamic inconsistency, intransitivity

Testable Implications of Quasi-Hyperbolic and Exponential Time Discounting

We present the first revealed-preference characterizations of the models of exponential time discounting, quasi-hyperbolic time discounting, and other time-separable models of consumers’ intertemporal decisions. The characterizations provide non-parametric revealed-preference tests, which we take to data using the results of a recent experiment conducted by Andreoni and Sprenger (2012). For such data, we find that less than half the subjects are consistent with exponential discounting, and only a few more are consistent with quasi-hyperbolic discounting