An anticipative hyperbolic discount utility on intertemporal decision making

Discount utility is facilitated to scrutinize human decision behaviors for different points in time under the consideration of time preference. Recent studies are extended from two basic models, which are the exponential and the hyperbolic discount models, and the more popular hyperbolic discount model is well recognized for resolving the effects of preference reversal and is less steeply discounting than the exponential discount model. However, to enhance the capability of explaining intertemporal decision making behavior, an anticipative hyperbolic discount utility model was proposed which revised the conventional hyperbolic discount utility by introducing the anticipative parameters under the consideration of anticipation of future gains or losses. An empirical investigation was employed to validate the effectiveness of the proposed anticipative hyperbolic discounting utility model, which was able to empirically go beyond the traditional hyperbolic discounting utility model. Also, the proposed anticipative hyperbolic model is capable of dealing with the anomalies of preference reverse and framing effects.     

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 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.     

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.     

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。     

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.     

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.     

Time consistent pension funding in a defined benefit pension plan with non-constant discounting

We consider the time consistent management of a defined benefit stochastic pension plan where the participants have different rates of time preference and the fund manager collects this heterogeneity when discounting the future. The main objective is to select the amortization rate and the investment strategy minimizing both the contribution rate risk and the solvency risk. The problem is formulated as a stochastic control problem with non-constant rate of discount and is solved analytically by means of the dynamic programming approach and the technical interest rate is selected in order to keep stable the fund evolution within prescribed targets. A numerical illustration shows a comparative of the stability of the fund assets and the rate of contribution for a convex combination of exponential functions as discount function and for the constant discount case.     

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.     

Harvesting of a phytoplankton–zooplankton model

Considering that some phytoplankton and zooplankton are harvested for food, a phytoplankton–zooplankton model with harvesting is proposed and investigated. First, stability conditions of equilibria and existence conditions of a Hopf-bifurcation are established. Our results indicate that over exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population which is in line with reality. Furthermore, the existence of bionomic equilibria and the optimal harvesting policy are discussed. The present value of revenues is maximized by using Pontryagin’s maximum principle subject to the state equations and the control constraints. We discussed the case of optimal equilibrium solution. It is found that the shadow prices remain constant over time in optimal equilibrium when they satisfy the transversality condition. It is established that the zero discounting leads to the maximization of economic revenue and that an infinite discount rate leads to complete dissipation of economic rent. Finally, some numerical simulations are given to illustrate our results.     

Impatience and dynamic optimal behavior: A bifurcation analysis of the Robinson-Solow-Srinivasan model

This paper provides a study of optimal economic growth under discounting in discrete time in a two-sector version of the Robinson-Solow-Srinivasan model. It examines how dynamic optimal behavior changes in response to increasing impatience. The optimal policy function is a pan map for high discount factors and a check map for low discount factors. It is shown that the transformation from the pan map to the check map, for the intermediate range of discount factors, can be quite intricate. This is demonstrated by proving the existence of two bifurcation values of the discount factor in the borderline case of the model, which establishes the possibility of reswitching of optimal actions as the discount factor varies.     

A Differential Games Solution to a Logarithmic Advertising Model

A logarithmic excess-advertising model of a duopoly is presented, and Nash optimal open-loop advertising strategies are determined. It turns out that if the two firms use different discount rates, then the optimal strategies will be exponentially decreasing. However, in this case the state equation has no nice solution and must be solved by numerical methods. When both firms use the same discount rate, then the state equation has a simple solution. This solution is also valid for the case where no discounting is performed. Furthermore, when no discounting is performed, the optimal strategies will be simple time-linear decreasing strategies. Finally, it is studied how the optimal strategies and trajectories depend on the parameters of the model.     

How to correctly calculate discounted healthcare costs and benefits

A feature of a healthcare policy (such as screening) with interventions at specific ages is that when it is introduced, part of the population is too old to participate in the full programme. This fact changes the formulae to be used for cost and benefit discounting in a non-intuitive way. General formulae are derived for the expected discounted costs and benefits of such health promotion policies, for a stationary population. Correct ways to calculate discounted costs and benefits via simulation are also described. The formulae have some surprising properties, for example the relative cost of two health policies does not depend on the discounting rate. They are also relevant to the ongoing debate over the correct discounting rate for benefits. It is shown that when health benefits follow quickly on treatments, varying the discounting rate for health benefits is merely equivalent to rescaling the cash value of a benefit. It is only when benefit follows long after treatment that the problem of choosing an appropriate discount rate for benefits cannot be simplified.     

Inventory model for deteriorating items with freshness and price dependent demand: Optimal discounting and ordering policies

A major part of retail industry deals with items whose freshness declines with time, resulting in lower demand at the same price. The item may later begin to deteriorate, when it is customary to offer discount in order to boost sales. A discounting policy may bring many benefits for the retailer, if correctly chosen. Motivated by this we have developed and analyzed an inventory model when demand for a deteriorating item depends initially only upon its selling price and later also on the freshness condition. We consider general demand function and general deterioration distribution for an inventory model with lost sales shortage. It is shown that net profit is a concave function of the period with positive inventory and conditionally concave function of discount. Important managerial insights obtained from sensitivity analysis suggest some policies counter to those commonly practiced by the retailers while others are in concurrence with the strategies in vogue.     

Unbounded growth in the Neoclassical growth model with non-constant discounting

For a Neoclassical growth model, the literature highlights that exponential discounting is observationally equivalent to quasi-hyperbolic discounting, if the instantaneous discount rate decreases asymptotically towards a positive value. Conversely, in this paper a zero long-run value allows a solution without stagnation. We prove that a less than exponential but unbounded growth can be attained, even without technological progress. The growth rate of consumption decreases asymptotically towards zero, although so slowly that consumption grows unboundedly. The asymptotic convergence towards a non-hyperbolic steady-state which saving rate matches the intertemporal elasticity of substitution and the speed of convergence towards this equilibrium are analyzed.     

Extreme events,discounting and stochastic optimization

The paper analyzes the implications of extreme events on the proper choice of discounting. Any discounting with constant or declining rates can be linked to random “stopping time” events, which define the internal discount-related horizons of evaluations. Conversely, any stopping time induces a discounting, in particular, with the standard discount rates. The expected duration of the stopping time horizon for discount rates obtained from capital markets does not exceed a few decades and, as such, these rates may significantly underestimate the net benefits of long-term decisions. The alternative undiscounted stopping time criterion allows to induce social discounting focusing on arrival times of potential extreme events rather then horizons of market interests. Induced discount rates are conditional on the degree of social commitment to mitigate risk. In general, extreme events affect these rates, which alter the optimal mitigation efforts that, in turn, change events. The use of undiscounted stopping time criteria requires stochastic optimisation methods.     

考虑污染控制、技术进步和人力资本积累下的经济可持续增长

构建一个带有环境污染、污染控制、内生人力资本积累和内生技术进步的动态经济增长模型,并运用最优控制方法探讨经济最优增长路径.研究表明:人力资本积累效率、消费跨期替代弹性和时间贴现率决定着经济最优增长路径.消费跨期替代弹性、时间贴现率、人力资本积累效率、物质产品和技术研发部门的投入产出弹性、污染排放的产出和控制弹性会对稳态中的人均经济增长率、人均污染排放增长率产生影响.     

WHY INCREASED UNCERTAINTY MAY LEAD TO MORE RISKY BEHAVIOR

The purpose of this article is to investigate circumstances under which it may be optimal to deliberately harvest a fish stock to extinction applying a stochastic surplus growth model. It is known from the literature that deliberate extinction may result when there is critical depensation or when the discount rate is high compared to the intrinsic growth rate. Here it is shown that deliberate extinction may also be optimal when the degree of stochasticitry is high even with zero discounting. A high degree of stochasticity may have the same effect as critical depensation even though it is not present in the biological model. In other words, high uncertainty, instead of leading to more conservative harvesting as is usually expected, in this model result in more aggressive harvesting and more risky behavior. The main message is therefore always to try to keep the stock well above any critical limit.     

OPTIMAL DISCOUNTING IN CONTROL PROBLEMS THAT SPAN MULTIPLE GENERATIONS

ABSTRACT. The principal contribution of this paper is the linking together of separate control problems across multiple generations using the bequest motive, intergenerational altruism, rational expectations, and solution boundary conditions. We demonstrate that discounting at the market rate of interest is an endogenous characteristic of a general equilibrium, optimal control problem that spans multiple generations. Within the confines of our model, we prove that it is optimal to discount at the market rate of interest the social benefits to distant generations from immediate clean up at toxic waste sites if the current generation that bears the cleanup cost is perfectly altruistic towards future generations. Also, we show that this result holds for alternative assumptions regarding pure time preference. Moreover, the result holds regardless of whether selfish interim generations attempt to undo the provisions made for distant generations. In our distortion‐free deterministic model, the evidence for intergenerational discounting at the market rate of interest is compelling.     

The golden rule when preferences are time inconsistent

We investigate the classical Ramsey problem of economic growth when the planner uses non-constant discounting. It is well-known that this leads to time inconsistency, so that optimal strategies are no longer implementable. We then define equilibrium strategies to be such that unilateral deviations occurring during a small time interval are penalized. Non-equilibrium strategies are not implementable, so only equilibrium strategies should be considered by a rational planner. We show that there exists such strategies which are (a) smooth, and (b) lead to stationary growth, as in the classical Ramsey model. Finally, we prove an existence and multiplicity result: for logarithmic utility and quasi-exponential discount, there is an interval I such that, for every k in I, there is an equilibrium strategy converging to k. We conclude by giving an example where the planner is led to non-constant discount rates by considerations of intergenerational equity.