A power law type of time preference on intertemporal choices

Discount utility, based on utility theory, is used to study human decision behaviors under the consideration of time preference. It assumes that by means of the axiomatic system of rationality, it is possible to quantify human beings’ utilities by some explicable models for intertemporal decision making. Recent studies have been based on two basic models: the exponential and the hyperbolic discount models. These two types of model have been proved to be either too fast for discounting or too restricted for fitting human beings’ discounting behaviors. In this study, a power law discount model is proposed. Axiomatic approach is used to ascertain the existence of the power law discount utility, and empirical investigations are implemented to verify the effectiveness of the proposed model.     

ON HYPERBOLIC TIME DISCOUNTING IN EXHAUSTIBLE RESOURCE MODELS: AN APPLICATION TO WORLD OIL RESOURCES

ABSTRACT. Recent research on discounting in long term economic models involves hyperbolic discounting, in which the marginal discount rate shrinks as time passes. To investigate hyperbolic discounting and exhaustible resource allocation, this work develops a discrete‐time world oil model and model solution procedure, then uses the model to examine the consequences of adopting conventional (constant annual) discounting when hyperbolic discounting is appropriate, of adopting one hyperbolic discount rate path when a different hyperbolic path is appropriate, and of adopting hyperbolic discounting when conventional discounting is appropriate. Five conventional and two hyperbolic discount rate paths are considered. One hyperbolic path is that used by Nordhaus and Boyer [2000]; the other is that recommended by Weitzman [2001]. The generality of the findings is also assessed.     

Asset allocation,sustainable withdrawal,longevity risk and non-exponential discounting

The present paper studies an optimal withdrawal and investment problem for a retiree who is interested in sustaining her retirement consumption above a pre-specified minimum consumption level. Apparently, the withdrawal and investment policy depends substantially on the retiree’s health condition and her time preferences (subjective discount factor). We assume that the health of the retiree can worsen or improve in an unpredictable way over her lifetime and model the retiree’s mortality intensity by a stochastic process. In order to make the decision about the consumption and investment policy more realistic, we assume that the retiree applies a non-exponential discount factor (an exponential discount factor with a small amount of hyperbolic discounting) to value her future income. In other words, we consider an optimization problem by combining four important aspects: asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting. Due to the non-exponential discount factor, we have to solve a time-inconsistent optimization problem. We derive a non-local HJB equation which characterizes the equilibrium optimal investment and consumption strategy. We establish the first-order expansions of the equilibrium value function and the equilibrium strategies by applying expansion techniques. The expansion is performed on the parameter controlling the degree of discounting in the hyperbolic discounting that is added to the exponential discount factors. The first-order equilibrium investment and consumption strategies can be calculated in a feasible way by solving PDEs.     

A consumption–investment problem with heterogeneous discounting

We analyze a stochastic continuous time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different discount rates of time preference. Within this framework we can model problems in which, when the time t$t$ approaches to the final time, the valuation of the final function increases compared with previous valuations. We study a consumption and portfolio rules problem for CRRA and CARA utility functions for time-consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of random terminal time is also discussed. Differences with both the mathematical treatment and agent’s behavior in the case of hyperbolic discounting are stressed.     

基于营运资金信息匹配平台的供应链金融动态折扣决策研究

当前逐渐受到实业界关注的一种新型供应链金融模式是基于营运资金信息匹配平台的动态折扣。本文针对两级供应链的动态折扣决策问题进行建模,揭示了动态折扣的应用对于改善供应链参与方现金流的影响机理。首先通过考虑供需双方动态折扣的独立决策,得出日折扣率的边界条件、双方效用最大时的日折扣率、买方混合还款方式下营运资金的最优准备方案和最低边界值;其次考虑供需双方动态折扣的最优决策,推导出在不同折扣率和利率关系下的最优还款策略,研究表明动态折扣可以明显提升供需双方的利润情况。     

THE INTRINSIC QUALITATIVE PROPERTIES OF THE CLASSICAL OPTIMAL STOPPING PROBLEM ARE INVARIANT TO THE FUNCTIONAL FORM OF THE DISCOUNT FUNCTION

Abstract The intrinsic qualitative properties of a generic optimal stopping model are shown to be invariant to the functional form of the discount function. If the discount function is assumed to be a member of particular infinite parametric family—a family that includes the exponential and classical hyperbolic discount functions as special cases—an additional refutable comparative statics result is produced that holds for the entire family. Consequently, if one limits econometric tests of the model to its qualitative properties, one cannot determine the form of the discount function used by the decision maker. It is also shown that the only discount function that yields a time‐consistent stopping rule is the exponential function with a constant rate of discount.     

OPTIMAL DISCOUNTING OF BENEFITS FROM CLEANUP AT WASTE SITES

Abstract This paper uses a general equilibrium optimal growth model to discuss the role of optimal discounting of future benefits from cleanup at high‐level toxic waste sites. Cleanup simultaneously generates two streams of benefits. One of these is directly from utility and the other is indirectly from the added productivity of workers. We note that the optimal discount rate is different for these two types of benefits. Along the optimal path, the former are discounted at the rate of time preference and the latter at the market rate of interest. We achieve this by identifying four components of the shadow value of the stock of toxic waste. These are the utility, productivity, cost, and abundance effects. The distinction between discount rates appears to have been overlooked in the literature but has significant implications for environmental cost‐benefit analysis due to the growing interest in applying zero time preference to environmental problems, (like waste cleanup) whose consequences extend many generations into the future. A numerical example is included to illustrate these concepts.     

基于随机双曲折现的投资消费及效用无差别定价

在随机双曲折现条件下,显式地给出了具有指数函数(CARA)效用的最优跨期消费与投资组合;在非完备市场下,显式给出了基于CARA效用的收益流的效用无差别价格.结果表明:最优投资比例以及收益流的价值不受随机双曲折现因子的影响;在低折扣阶段,本文的最优消费水平高于Merton模型下的对应值,低折扣时期越短或高低折扣值相差越大,消费差距越明显.     

Towards a constructive approach to act-conditional subjective expected utility models

In structuring a decision problem under uncertainty, the uncertain environment may be affected by the choice of an act. In decision analysis, the decision maker provides subjective probabilities and utilities through separate elicitation processes, and then both components are combined together to give an index of his preference over decision alternatives. Based upon this conceptualisation of decision analysis, a constructive approach to act-conditional subjective expected utility theory is proposed. Two utility models have been addressed: the linear utility model and the weighted utility model.     

Investment and consumption without commitment

In this paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting. This gives rise to time-inconsistency of the decision-maker. If the decision-maker at time t = 0 can commit her successors, she can choose the policy that is optimal from her point of view, and constrain the others to abide by it, although they do not see it as optimal for them. If there is no commitment mechanism, one must seek a subgame-perfect equilibrium policy between the successive decision-makers. In the line of the earlier work by Ekeland and Lazrak (Preprint, 2006) we give a precise definition of equilibrium policies in the context of the portfolio management problem, with finite horizon. We characterize them by a system of partial differential equations, and establish their existence in the case of CRRA utility. An explicit solution is provided for the case of logarithmic utility. We also investigate the infinite-horizon case and provide two different equilibrium policies for CRRA utility (in contrast with the case of exponential discounting, where there is only one optimal policy). Some of our results are proved under the assumption that the discount function h(t) is a linear combination of two exponentials, or is the product of an exponential by a linear function. I. Ekeland was supported by PIMS under NSERC grant 298427-04.