Time-inconsistent consumption-investment problem for a member in a defined contribution pension plan

Qian  Zhao; Rongming  Wang; Jiaqin  Wei

doi:10.3934/jimo.2016.12.1557

Article Contents

2016, Volume 12, Issue 4: 1557-1585. Doi: 10.3934/jimo.2016.12.1557

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Time-inconsistent consumption-investment problem for a member in a defined contribution pension plan

1.
School of Business Information, Shanghai University of International Business and Economics, Shanghai 201620
2.
School of Statistics and Research Centre of International Finance and Risk Management, East China Normal University, Shanghai 200241
3.
School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai 200241

Received: September 30, 2013

Revised: September 30, 2015

Published: September 2016

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Abstract

In this paper, we investigate the consumption-investment problem for a member of the defined contribution pension plan with non-constant time preferences. The aim of the member is to maximize the discounted utility of the consumption. It leads to a time-inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, the contribution rate is assumed to be a fixed proportion of the scheme member's salary, and the pension fund can be invested in a risk-free asset, an index bond and a stock whose return follows a geometric Brownian motion. Two utility functions are considered: the power utility and the logarithmic utility. We characterize the time-consistent equilibrium consumption-investment strategies and the value function in terms of a solution of an integral equation in both situations. The existence and uniqueness of the solution is verified and the approximation of the solution is obtained. We present some numerical results of the equilibrium consumption rate and equilibrium investment policy with three types of discount functions.

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Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80.

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