Investment and consumption without commitment

By Ivar Ekeland, Traian A. Pirvu. 2007

297 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/5653992e10d9a846dcce05b7e0364dfdf70d8b9a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

1. Jiongmin Yong
2. Zhongfei Li
3. Xiang Lin
4. Yiping Qian
5. Yongwu Li
6. Tomas Björk
7. Agatha Murgoci
8. Xun Yu Zhou

In Finance

1. Jiongmin Yong
2. Xiang Lin
3. Yiping Qian
4. Zhongfei Li
5. Yongwu Li
6. Tomas Björk
7. Agatha Murgoci
8. Xun Yu Zhou

Citations in the corpus

• MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION (2014, background): “In the interesting papers [6] and [7], the authors consider optimal consumption and investment under hyperbolic discounting in deterministic and stochastic models from the above game theoretic point of view.”
• Time-Inconsistent Optimal Control Problems and the Equilibrium HJB Equation (2012, background): “ that corresponding initial pair. Some concrete examples will be presented in the next section. In recent years, time-inconsistent optimal control problems have attracted a number of researchers. See [5, 6, 2, 27, 19, 28, 7, 3, 13] and references cited therein. The purpose of this paper is to obtain”
• Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion (2013, background): “For optimal consumption and investment problems with non-exponential discounting have recently been studied from the game theoretic point of view by Ekeland and Lazrak (2006), Ekeland and Pirvu (2007) and Ekeland et al. (2012).”
• On time-inconsistent stochastic control in continuous time (2017, unknown): “– We note that the g function above appears, in a more restricted framework, already in [2, 9, 11].”
• A General Theory of Markovian Time Inconsistent Stochastic Control Problems (2010, background): “In the interesting, and mathematically very advanced, papers [3] and [4], the authors consider optimal consumption and investment under hyperbolic discounting (Problem 1 in our list above) in deterministic and stochastic models from the above game theoretic point of view.”
• Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model (2016, background): “Barro (1999), Karp (2007), Ekeland & Lazrak (2010), Ekeland & Pirvu (2008), Marin-Solano & Navas (2009, 2010), Ekeland et al. (2012) further considered time-inconsistent preferences problems in different situations.”
• A Deterministic Linear Quadratic Time-Inconsistent Optimal Control Problem (2011, background): “Recently, Ekeland– Lazrak [5] and Ekeland–Pirvu [6] continued the study of non-exponential discounting problems both for simple ODEs and SDEs.”
• Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations --- Time-Consistent Solutions (2013, background): “n the second example, the time-inconsistency is due to the appearance of the conditional expectation of the state in a nonlinear way. The non-exponential discounting situation has been discussed in 6 [7, 9, 8, 24, 25, 26, 18]. The main motivation of that is trying to catch people’s subjective prefer”