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We study the intertemporal consumption and portfolio rules in the model with the general hyperbolic absolute risk aversion (HARA) utility. The equivalent approximation approach is employed to obtain the Hamilton-Jacobi-Bellman (HJB) equations, and a remarkable property is shown: portfolio rules are independent of the discount function. Moreover, both the consumption and portfolio rates are non-increasing functions of wealth. Particularly illustrative cases examined in detail are the models with the most adopted discount functions, including exponential discounting and hyperbolic discounting. Explicit solutions for intertemporal decisions are found for these special cases, revealing that individual’s time preferences affect the consumption rules only. Moreover, the time-consistent consumption rate under hyperbolic discounting is larger than its counterpart under exponential discounting.

This paper studies a continuous-time, nite-horizon contracting problem with renegotiation and dynamic inconsistency arising from non-exponential discounting. The problem is formulated as a dynamic game played among the agent, the principal and their respective future "selves", each with their own discount function. We identify the principal optimal renegotiation-proof contract as a Markov Perfect Equilibrium (MPE) of the game, prove such a MPE exists, and characterize the optimal contract via an extended Hamilton-Jacobi-Bellman system. We solve the optimal contract in closed form when the discount functions of the selves are related by time di erence, a property that is satis ed by common forms of non-exponential discounting such as quasi-hyperbolic discounting and anticipatory utility. In particular, quasi-hyperbolic discounting leads to a U-shaped action path and anticipatory utility leads to a humshaped path, both are qualitatively di erent from the monotonic action path that would arise under exponential discounting.

We propose a simple and robust two-step estimator for discount factors in a class of dynamic discrete choice models. The estimator follows from constructive identification results, including a new identification result for a general, time separable discount function. The estimator is derived as the solution to well-behaved sample moment conditions which are linear in the discount factors and are independent of the utility function. The estimator is therefore easy to implement, computationally light, and in contrast to existing estimators, robust to biases from finite sample approximations to the unknown utility function. We apply the estimator to data on mortgage defaults under an identifying assumption of time homogeneity of the utility function. We compare the performance of the proposed estimator to alternative two-step estimators that jointly estimate the discount factor and the utility function. The results show that our proposed estimator's robustness to finite sample approximation bias and its computational ease do not necessarily come at a material expense of precision.

A new notion of equilibrium, which we call strong equilibrium, is introduced for timeinconsistent stopping problems in continuous time. Compared to the existing notions introduced in Time-Consistent Stopping Under Decreasing Impatience and On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems, which in this paper are called mild equilibrium and weak equilibrium respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.

In the accumulation paradigm animals press one manipulandum to accumulate pellets or seconds of access to food, and then press another manipulandum, or run some distance, to collect it. The accumulation may be interpreted as delay discounting, with the animals adjusting the distal amount to compensate for its distance or delay. The amount accumulated before being collected is a linear function of the distance or time that the experimental paradigm stipulates for collecting it. That linear function follows from the Unit-Price/Unit-Amount axiom. The inverse of the linear compensation functions gives a delay-discount hyperbola. The advantages of the accumulation paradigm and analytic framework for delay discounting studies are noted. Compensation functions are then derived from a behavioral regulation model, which generalizes them to contexts where the individual's budget for response cost becomes over-taxed. In turn, such compensation/regulation models lead directly to representative demand functions. In sum, regulation models provide a theoretical grounding for demand functions, compensation functions, and delay discounting hyperbolas. The parameter that links them is the unit amount k, the slope of the compensation function and of the discount function, the setpoint for consumption in a regulation model, and the ideal quantity consumed at minimal price in a demand analysis.

In this paper, we formalize mathematically the concept of improving sequence effect which is one of the main anomalies of the Discounted Utility model (Samuelson, 1937). The improving sequence effect implies a preference for a given sequence of outcomes which increase over time and has been empirically demonstrated for both monetary and non-monetary results (hedonic experiences and health-related outputs). Nevertheless, to the date, there is not a mathematical treatment of this anomaly in the context of intertemporal choice which allows us to relate this paradox to other anomalies, such as the delay and the magnitude effects. In this way, the present manuscript has filled this gap. More specifically, we have proved that the improving sequence effect for monetary rewards cannot be rationalized by using a separable discount function, but only considering a non-separable discount function. Moreover, under certain conditions, we have proved that the delay and the magnitude effects are necessary conditions for the existence of the improving sequence effect.

Objectives This article explains how to optimize Bayesian D-efficient discrete choice experiment (DCE) designs for the estimation of quality-adjusted life year (QALY) tariffs that are unconfounded by respondents' time preferences. Methods The calculation of Bayesian D-errors is explained for DCE designs that allow for the disentanglement of respondents' time and health-state preferences. Time preferences are modelled via an exponential, hyperbolic, or power discount function and the performance of the proposed DCE designs is compared with that of several conventional DCE designs that do not take nonlinear time preferences into account. Results Based on the achieved D-error, asymptotic standard error, and estimated sample size to obtain statistically significant estimates of the discount rate parameters, the proposed designs outperform the conventional DCE designs. Conclusions We recommend that applied researchers use appropriately optimized DCE designs for the estimation of QALY tariffs that are corrected for time preferences. The TPC-QALY software package that accompanies this article makes the recommended designs easily accessible for health-state valuation researchers.

In this paper, we apply the LR-fuzzy random variables to estimate a discount function that associated with a generalized interest rate as well as fuzzy random variable future lifetimes, and establish some life annuity and endowments models. A novel fuzzy net premium model is obtained. Finally, a statistical simulation is given for illustrating the proposed models.

Background: The Mediterranean Diet (hereinafter MD) is considered a healthy dietary pattern. Adherence to this pattern can be assessed by means of the KIDMED test by which individuals are assigned an index and classified into three groups of adherence to MD: high, medium, and low. In addition, impulsivity or impatience in intertemporal choice has been defined as a strong preference for small immediate rewards over large delayed ones. Objective: This study examines the relationship between dietary habits, specifically Mediterranean dietary pattern, measured by the KIDMED index, and the exhibited impatience in intertemporal choices, by means of the parameter k (discount rate of the hyperbolic discount function). Methods: A sample of 207 university students answered a questionnaire based on two tests: the KIDMED test, to assess the degree of adherence to MD, and an intertemporal choice questionnaire, to assess impatience or impulsivity. Individuals were grouped depending on their KIDMED score and then the discount rate or impulsivity parameter was calculated for each group. Results: Discount rates were inversely related to the degree of adherence to MD. The values of overall k were 1.53, 1.91, and 3.71% for the groups exhibiting high, medium and low adherence to MD, respectively. We also found higher k-values for larger rewards (magnitude effect) in the three groups. Conclusion: High adherence to MD is related to less steep time discounting, which implies less impulsivity (more self-control) or lower discount rates. Conversely, low adherence to MD is related to steeper time discounting, which implies impulsivity or higher discount rates. These findings could be used to identify the target population where policy interventions are needed in order to promote healthier diet habits.

This paper makes a comprehensive empirical analysis on US Government bond (USGB) prices for a period of 60 months, including the financial crisis in 2008. The model is a cross-sectional model with stochastic discount function that takes into account bond attributes of coupon rate and maturity and that simultaneously values individual fixed-coupon bonds. First, we briefly clarify the theoretical relation between our stochastic discount function approach and the interest rate approach in mathematical finance. Then we propose two specific and effective models. Third, the derived yields are compared to swap rates to see the validity of the models.

This paper studies an optimal dividend problem for a company with non-exponential discounting. The surplus process is described by a dual model, and the target is to find a dividend strategy that maximizes the expected discounted value of dividends until ruin. The non-exponential discount function leads to a time-inconsistent problem. We aim at seeking the equilibrium strategy derived by taking our problem as a non-cooperate game, which is a time-consistent strategy. An extended Hamilton---Jacobi---Bellman equation system and a verification theorem are provided to derive the equilibrium strategy and the equilibrium value function. For the case of pseudo-exponential discount function, closed-form expressions for the equilibrium strategy and the equilibrium value function are derived. In addition, some numerical illustrations of our results are showed.

Analysis of an original Internet-based survey reveals that debt holding is related to time discounting through: (i) present bias, measured by the degree of declining impatience in the generalized hyperbolic discount function; (ii) borrowing aversion, captured by a sign effect in that future losses are discounted at lower rates than future gains; and (iii) impatience, measured by the overall discount rate. Hyperbolic respondents are classified naive if their answers reveal them to be time-inconsistent procrastinators, and otherwise sophisticated. Naive respondents with more steeply declining impatience are more likely to be debtors. The sign effect relates negatively to borrowing. Survey responses indicative of high or declining impatience are associated with credit card borrowing and other overborrowing indicators.

The importance of delay discounting to many socially important behavior problems has stimulated investigations of biological and environmental mechanisms responsible for variations in the form of the discount function. The extant experimental research, however, has yielded disparate results, raising important questions regarding Gene X Environment interactions. The present study determined the influence of stimuli that uniquely signal delays to reinforcement on delay discounting in two inbred mouse strains using a rapid-acquisition procedure. BALB/c and C57BL/6 mice responded under a six-component, concurrent-chained schedule in which the terminal-link delays preceding the larger-reinforcer were presented randomly across components of an individual session. Across conditions, components were presented either with or without delay-specific auditory stimuli, i.e., as multiple or mixed schedules. A generalized matching-based model was used to incorporate the impact of current and previous component reinforcer-delay ratios on current component response allocation. Sensitivity to reinforcer magnitude and delay were higher for BALB/c mice, but within-component preference reached final levels faster for C57Bl/6 mice. For BALB/c mice, acquisition of preference across blocks of a component was faster under the multiple than the mixed schedule, but final levels of sensitivity to reinforcement were unaffected by schedule. The speed of acquisition of preference was not different across schedules for C57Bl/6 mice, but sensitivity to reinforcement was higher under the multiple than the mixed schedule. Overall, differences in the acquisition and final form of the discount function were determined by a Gene X Environment interaction, but the presence of delay-specific stimuli attenuated genotype-dependent differences in magnitude and delay sensitivity.

In this paper, the dosage of the times in the repeated and fractionable actions is studied. The interventions in time are abstract in nature and deal also with the psychological aspect of the individual behavior, in a wider perspective of utility, or the management of procedures in output systems. Furthermore, decomposability of discount functions is explored, and time period splitting plans are considered. Such procedures are analyzed, for instance, in the treatment of drug addiction, and various applications in behavioral sciences have been found. We wonder whether it would be convenient an infinite subdivision, that is, a finest fragmentation of time period (for a medical treatment, or a financial process, or a man-machine interrelation); or, on the contrary, a possible efficient subdivision must be bounded. Specifically, we deal with the problem of finding the expression of a discount function able to distinguish between favorableness and unfavorableness of the refinements of splitting levels for general processes.

International audience; Under non-exponential discounting, we develop a dynamic theory for stopping problems in continuous time. Our framework covers discount functions that induce decreasing impatience. Due to the inherent time inconsistency, we look for equilibrium stopping policies, formulated as fixed points of an operator. Under appropriate conditions, fixed-point iterations converge to equilibrium stopping policies. This iterative approach corresponds to the hierarchy of strategic reasoning in game theory and provides “agent-specific” results: it assigns one specific equilibrium stopping policy to each agent according to her initial behavior. In particular, it leads to a precise mathematical connection between the naive behavior and the sophisticated one. Our theory is illustrated in a real options model.

The aim of this paper is to obtain the family of the so-called generalized Weibull discount functions, introduced by Takeuchi (Game Econ Behav 71:456–478, 2011), by deforming the q-exponential discount function by means of the Stevens’ “power” law. The obtained discount functions exhibit different degrees of inconsistency and so they can be classified according to the value of their characteristic deforming parameters. Moreover, we extend the construction of the generalized Weibull discount function starting from any discount function instead of the q-exponential discounting. In any case, the value of the parameter $$\theta $$ of these new discount functions is extended from (0, 1] to the union of the intervals $$(-\,\infty ,0) \cup (0,+\,\infty )$$ .

Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity-concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month).

For an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.

The aim of this paper is to find a suitable discount function able to describe the progression of a certain addiction or disease under treatment as a discounting process. In effect, a certain indicator related to a disease whose curve decays over time in a manner which is mathematically similar to the way in which discounting has been modelled. The discount functions observed are analyzed in experiments which study addictive and other problematic behaviors as well as some alternative hyperbola-like discount functions in order to fit the patience exhibited by the subject after receiving the treatment. Additionally, it has been experimentally found that people with addiction display high rates of discount (impatience) and preference reversals (dynamic inconsistency). This excessive discounting must be correctly modelled by a suitable discount function, otherwise, it can become a trans-disease process underlying addiction and other disorders. The (generalized) exponentiated hyperbolic discount function is proposed to describe the progression of a disease with respect to the treatment, since it maintains the property of inconsistency by exhibiting a decreasing discount rate after an initial period in which the opposite occurs.

The paper [12] examines a concept of equilibrium policies instead of optimal controls in stochastic optimization to analyze a mean-variance portfolio selection problem. We follow the same approach in order to investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. Equilibrium policies are characterized in this context by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions., 28 pages. arXiv admin note: text overlap with arXiv:0901.2484, arXiv:0708.0588 by other authors

Evaluation of medical devices via clinical trial is often a necessary step in the process of bringing a new product to market. In recent years, device manufacturers are increasingly using stochastic engineering models during the product development process. These models have the capability to simulate virtual patient outcomes. This article presents a novel method based on the power prior for augmenting a clinical trial using virtual patient data. To properly inform clinical evaluation, the virtual patient model must simulate the clinical outcome of interest, incorporating patient variability, as well as the uncertainty in the engineering model and in its input parameters. The number of virtual patients is controlled by a discount function which uses the similarity between modeled and observed data. This method is illustrated by a case study of cardiac lead fracture. Different discount functions are used to cover a wide range of scenarios in which the type I error rates and power vary for the same number of enrolled patients. Incorporation of engineering models as prior knowledge in a Bayesian clinical trial design can provide benefits of decreased sample size and trial length while still controlling type I error rate and power.

We assume that perfectly patient agents estimate the value of future events by generating noisy, unbiased simulations and combining those signals with priors to form posteriors. These posterior expectations exhibit as-if discounting: agents make choices as if they were maximizing a stream of known utils weighted by a discount function, D(t). This as-if discount function reflects the fact that estimated utils are a combination of signals and priors, so average expectations are optimally shaded toward the mean of the prior distribution, generating behavior that partially mimics the properties of classical time preferences. When the simulation noise has variance that is linear in the event's horizon, the as-if discount function is hyperbolic, D(t)=1/(1+at). Our agents exhibit systematic preference reversals, but have no taste for commitment because they suffer from imperfect foresight, which is not a self-control problem. In our framework, agents that are more skilled at forecasting (e.g., those with more intelligence) exhibit less discounting. Agents with more domain-relevant experience exhibit less discounting. Older agents exhibit less discounting (except those with cognitive decline). Agents who are encouraged to spend more time thinking about an intertemporal tradeoff exhibit less discounting. Agents who are unable to think carefully about an intertemporal tradeoff -- e.g., due to cognitive load -- exhibit more discounting. In our framework, patience is highly unstable, fluctuating with the accuracy of forecasting.

In this paper, we consider the consumption–investment problem in a regime-switching model with both the discount function and relative/absolute risk aversion depending on the exogenous environment. We obtain the solutions to this time-inconsistent optimal control problem for both the sophisticated agent and the naive agent for different utilities. We also give some comparisons between the strategies for these two kinds of agents. Our results show that at any time, given the same wealth, the naive agent with power (resp. exponential) utility consumes less (resp. more) than her/his sophisticated counterpart in general, and the sophisticated agent is more sensitive to the change of the exogenous environment.

In this paper, we revisit the consumption–investment problem with a general discount function and a logarithmic utility function in a non-Markovian framework. The coefficients in our model, including the interest rate, appreciation rate and volatility of the stock, are assumed to be adapted stochastic processes. Following Yong (2012a,b)’s method, we study an N -person differential game. We adopt a martingale method to solve an optimization problem of each player and characterize their optimal strategies and value functions in terms of the unique solutions of BSDEs. Then by taking limit, we show that a time-consistent equilibrium consumption–investment strategy of the original problem consists of a deterministic function and the ratio of the market price of risk to the volatility, and the corresponding equilibrium value function can be characterized by the unique solution of a family of BSDEs parameterized by a time variable.

Personality traits are stable predictors of many life outcomes that are associated with important decisions that involve tradeoffs over time. Therefore, a fundamental question is how tradeoffs over time vary from person to person in relation to stable personality traits. We investigated the influence of personality, as measured by the Five-Factor Model, on time preferences and on neural activity engaged by intertemporal choice. During functional magnetic resonance imaging (fMRI), participants made choices between smaller-sooner and larger-later monetary rewards. For each participant, we estimated a constant-sensitivity discount function that dissociates impatience (devaluation of future consequences) from time sensitivity (consistency with rational, exponential discounting). Overall, higher neuroticism was associated with a relatively greater preference for immediate rewards and higher conscientiousness with a relatively greater preference for delayed rewards. Specifically, higher conscientiousness correlated positively with lower short-term impatience and more exponential time preferences, whereas higher neuroticism (lower emotional stability) correlated positively with higher short-term impatience and less exponential time preferences. Cognitive-control and reward brain regions were more activated when higher conscientiousness participants selected a smaller-sooner reward and, conversely, when higher neuroticism participants selected a larger-later reward. The greater activations that occurred when choosing rewards that contradicted personality predispositions may reflect the greater recruitment of mental resources needed to override those predispositions. These findings reveal that stable personality traits fundamentally influence how rewards are chosen over time., National Institute on Aging (K01 AG040197)

In this paper, we study the dividend maximization problem with a non-constant discount rate in a diffusion risk model. We assume that the dividends can only be paid at a bounded rate and restrict ourselves to Markov strategies. This is a time inconsistent control problem. The equilibrium HJB-equation is given and the verification theorem is proven for a general discount function. Considering a mixture of exponential discount functions and a pseudo-exponential discount function, we get equilibrium dividend strategies and the corresponding equilibrium value functions by solving the equilibrium HJB-equations.

The aim of this paper is to derive an index able to indicate if a discount function exhibits increasing or decreasing impatience, and, even, in the last case, whether the decreasing impatience is moderate or strong. Moreover, it will be shown that the sign of this indicator coincides with the sign of the convexity index of the discount function when only considering the cases of increasing and decreasing impatience. Consequently, this parameter supposes an improvement of Prelec’s index of convexity. The main advantage of this novel measure is that, the same as Prelec’s index, it uses the differential calculus and, moreover, can be easily plotted by showing the changes from a type of impatience to another one according to time.

The present value of a forward contract for any asset that does not pay a dividend is calculated by discounting its forward price by the risk-free rate. We show that the discount function for assets that have a non-zero correlation with interest rates, has to be adjusted to account for the correlation between the asset and interest rates. Put-Call parity is also violated and needs to be adjusted as well for such assets. It is shown that the risk-free rate is asset dependent. The adjustment to the price is small for short dated forwards, but increases quadratically with time to maturity.

espanolLa mayor parte de la evidencia empirica sobre la eleccion de recompensas a lo largo del tiempo ha demostrado que la tasa de descuento utilizada no es constante, tal y como fue sugerido por Samuelson (1937). En efecto, la tasa de descuento puede depender del tiempo (pudiendo aparecer el efecto impaciencia decreciente, la estacionariedad o la impaciencia creciente) o de la cuantia (mostrando el efecto magnitud, la separabilidad o la reversion del efecto magnitud). Una funcion de descuento, capaz de explicar casi todos estos estados de impaciencia con respecto al tiempo y la cuantia, es la denominada funcion q-exponencial deformada por la cuantia, recientemente incorporada al ambito de la Econofisica. El objetivo principal de este articulo es la aplicacion del EM-index y de una nueva version del factor hiperbolico a esta novedosa funcion de descuento, y asi comprobar su validez. Estos indices constituyen nuestra propuesta para medir los efectos magnitud y plazo, y ambos estan inspirados en el factor hiperbolico introducido por Rohde (2010). EnglishMost empirical evidence on choices involving dated rewards has found that the discount rate used in intertemporal choices by individuals is not constant, as suggested by Samuelson (1937). In effect, the discount rate can depend on both time (showing decreasing impatience, stationarity or increasing impatience) and amount (showing magnitude effect, separability or magnitude effect reversal). A discount function, able to explain almost all the stages of impatience with respect to time and amount, is the so-called q-exponential discount function deformed by the amount which has been recently incorporated into the ambit of econophysics. The main objective of this manuscript is to apply the MEindex and a new version of the hyperbolic factor to this novel discount function, thus proving its validity. These indexes are our proposal to measure the magnitude and the delay effects, and both have been inspired in the hyperbolic factor introduced by Rohde (2010).

Discounting refers to the way in which the value of an outcome depends on the delay until it is obtained. If an organism’s discount function is known, then its rate of discounting at any delay can be found. If the function is not known, the normalised area under an estimate of the discount function has been used as a measure that summarises the strength of discounting over a range of delays. We propose a new measure of the strength of discounting: the overall discount rate W, which is the drop in value from the start to the end of the value curve divided by the area under this curve. We show that our measure has various advantages over the normalised area, namely it can be linked to the instantaneous rate of discounting and it respects the special nature of exponential discounting. It does not give a unique value to each curve, but we prove that this is incompatible with the requirement that similar discount functions be assigned similar values, and so this is not a defect of the measure.

If we assume that the purpose of a movement is to acquire a rewarding state, the duration of the movement carries a cost because it delays acquisition of reward. For some people, passage of time carries a greater cost, as evidenced by how long they are willing to wait for a rewarding outcome. These steep discounters are considered impulsive. Is there a relationship between cost of time in decision making and cost of time in control of movements? Our theory predicts that people who are more impulsive should in general move faster than subjects who are less impulsive. To test our idea, we considered elementary voluntary movements: saccades of the eye. We found that in humans, saccadic vigor, assessed using velocity as a function of amplitude, was as much as 50% greater in one subject than another; that is, some people consistently moved their eyes with high vigor. We measured the cost of time in a decision-making task in which the same subjects were given a choice between smaller odds of success immediately and better odds if they waited. We measured how long they were willing to wait to obtain the better odds and how much they increased their wait period after they failed. We found that people that exhibited greater vigor in their movements tended to have a steep temporal discount function, as evidenced by their waiting patterns in the decision-making task. The cost of time may be shared between decision making and motor control.

An optimal multiple-stopping problem with a random horizon and a general structure of rewards is considered. The problem can be formulated as follows. A decision maker has commodities of the same type for sale. Therefore, he can accept at most offers, which he observes sequentially. The decision on acceptance or rejection of the offer must be made on the basis of the past and current observations. Each accepted offer brings in a profit of if the offer is accepted before a random horizon and 0 otherwise. That means that after time the commodities cannot be sold. The aim of the decision-maker is to maximize the expected total profit. An optimal selection strategy is obtained by applying a general theorem which shows that a model with random horizon can be reduced to a model with a discount function. A case with random horizon independent of time and magnitude of offers is analysed in detail. As applications, two problems are solved: a multiple-stopping problem with a random horizon, a discount function and ...

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.

This According to discount asset function and double convex portfolio, the attainability of the asset optimization was discussed under the market models with transaction costs,. By virtue of stochastic analysis and the maximal value properties confined in some convex region, the attainability of asset optimization was given under discount function form and corresponding to utility function and expectation of utility function in this paper.

Most theoretical and empirical research on intertemporal choice assumes a deterministic and static perspective, leading to the widely adopted delay discounting models. As a form of preferential choice, however, intertemporal choice may be generated by a stochastic process that requires some deliberation time to reach a decision. We conducted three experiments to investigate how choice and decision time varied as a function of manipulations designed to examine the delay duration effect, the common difference effect, and the magnitude effect in intertemporal choice. The results, especially those associated with the delay duration effect, challenged the traditional deterministic and static view and called for alternative approaches. Consequently, various static or dynamic stochastic choice models were explored and fit to the choice data, including alternative-wise models derived from the traditional exponential or hyperbolic discount function and attribute-wise models built upon comparisons of direct or relative differences in money and delay. Furthermore, for the first time, dynamic diffusion models, such as those based on decision field theory, were also fit to the choice and response time data simultaneously. The results revealed that the attribute-wise diffusion model with direct differences, power transformations of objective value and time, and varied diffusion parameter performed the best and could account for all three intertemporal effects. In addition, the empirical relationship between choice proportions and response times was consistent with the prediction of diffusion models and thus favored a stochastic choice process for intertemporal choice that requires some deliberation time to make a decision.

People often consider how their actions influence others when making decisions. However, we are not equally generous to everyone alike. Our willingness to share resources declines as a function of social distance between the decision maker and the recipient. This function is likely to be influenced by culture, but research on behavioral decision making is still lacking empirical evidence. In Western societies, individuals generally perceive themselves as autonomous and independent from others, whereas the distinction between self and close others is less sharply defined by Eastern individuals where relationships and group membership are more centralized. Therefore, the social discount function should reflect this difference in the distinction of self and others by a reduced decline in generosity over close social distances. A social decision-making task was adapted to the intercultural context, and data were collected in Germany and China. For seven different social distances, we estimated how much money German and Chinese subjects were willing to forego to give a certain reward to another person. A hyperbolic model was fitted to the data. We found that other-regarding generosity declines as a function of social distance independent of cultural identity. However, German subjects showed a marked drop in generosity across close social distances, which was significantly less pronounced in Chinese participants. Copyright © 2013 John Wiley & Sons, Ltd.

Although choices of both humans and animals are more strongly influenced by immediate than delayed rewards, methodological limitations have made it difficult to estimate the precise form of temporal discounting in animals. In the present study, we sought to characterize temporal discounting in rats and to test the role of the orbitofrontal cortex (OFC) in this process. Rats were trained in a novel intertemporal choice task in which the sequence of delay durations was randomized across trials. The animals tended to choose a small immediate reward more frequently as the delay for a large reward increased, and, consistent with previous findings in other species, their choice behavior was better accounted for by hyperbolic than exponential discount functions. In addition, model comparisons showed that the animal's choice behavior was better accounted for by more complex discount functions with an additional parameter than a hyperbolic discount function. Following bilateral OFC lesions, rats extensively trained in this task showed no significant change in their intertemporal choice behavior. Our results suggest that the rodent OFC may not always play a role in temporal discounting when delays are randomized and/or after extensive training.

For a linear stochastic control system with quadratic objective functional, we introduce various generalizations of the notions of optimality on average and stochastic optimality on an infinite time interval that take into account possible degeneration of the parameter of the disturbing process with time (attenuation of the disturbances) or the presence of a discount function in the objective functional. This lets us improve upon the quality estimate for a well known optimal control in this problem from the point of view of both asymptotic behavior of the functional's expectation and its asymptotic probabilistic properties. In particular, in the considered case we have found an improvement for the well known logarithmic upper bound on the optimal control for a family of defect processes.