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October  2016, 12(4): 1557-1585. doi: 10.3934/jimo.2016.12.1557

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, China

2. 

School of Statistics and Research Centre of International Finance and Risk Management, East China Normal University, Shanghai 200241, China

3. 

School of Statistics, Faculty of Economics and Management, East China Normal University, Shanghai 200241, China

Received  October 2013 Revised  October 2015 Published  January 2016

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.
Citation: Qian Zhao, Rongming Wang, Jiaqin Wei. Time-inconsistent consumption-investment problem for a member in a defined contribution pension plan. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1557-1585. doi: 10.3934/jimo.2016.12.1557
References:
[1]

G. Ainslie, Picoeconomics, Cambridge University Press, Cambridge, UK, 1992.

[2]

R. Barro, Ramsey meets Laibson in the neoclassical growth model, Quarterly Journal of Economics, 114 (1999), 1125-1152. doi: 10.1162/003355399556232.

[3]

T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, 2010,, Working Paper, (). 

[4]

T. Björk, A. Murgoci and X. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[5]

Z. Bodiei, A. Marcus and R. Merton, Defined benefit versus defined contribution pension plans: What are the real trade-offs?, in Pensions in the US Economy, University of Chicago Press, 1988, 139-162.

[6]

A. Cairns, D. Blake and K. Dowd, Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, 30 (2006), 843-877. doi: 10.1016/j.jedc.2005.03.009.

[7]

I. Ekeland and A. Lazrak, Being serious about non-commitment: Subgame perfect equilibrium in continuous time, 2006,, Preprint. University of British Columbia., (). 

[8]

I. Ekeland, O. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32. doi: 10.1137/100810034.

[9]

I. Ekeland and T. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

[10]

P. Emms, Lifetime investment and consumption using a defined-contribution pension scheme, Journal of Economic Dynamics and Control, 36 (2012), 1303-1321. doi: 10.1016/j.jedc.2012.01.012.

[11]

S. Goldman, Consistent plans}, Review of Financial Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[12]

S. Haberman and E. Vigna, Optimal investment strategies and risk measures in defined contribution pension schemes, Insurance: Mathematics and Economics, 31 (2002), 35-69. doi: 10.1016/S0167-6687(02)00128-2.

[13]

L. He and Z. Liang, Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance: Mathematics and Economics, 52 (2013), 404-410. doi: 10.1016/j.insmatheco.2013.02.005.

[14]

D. Laibson, Golden eggs and hyperbolic discounting, Quarterly Journal of Economics, 112 (1997), 443-478. doi: 10.1162/003355397555253.

[15]

D. Laibson, Life-cycle consumption and hyperbolic discount functions, European Economic Review, 42 (1998), 861-871. doi: 10.1016/S0014-2921(97)00132-3.

[16]

D. Laibson, A. Repetto and J. Tobacman, Self-control and saving for retirement, Brookings Papers on Economic Activity, 1998 (1998), 91-196. doi: 10.2307/2534671.

[17]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, Quarterly Journal of Economics, 107 (1992), 573-597. doi: 10.2307/2118482.

[18]

J. Marín-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872. doi: 10.1016/j.ejor.2009.04.005.

[19]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[20]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[21]

B. Peleg and M. Yaari, On the existence of a consistent course of action when tastes are changing, Review of Financial Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[22]

E. Phelps and R. Pollak, On second-best national saving and game-equilibrium growth, Review of Economic Studies, 35 (1968), 185-199.

[23]

R. Pollak, Consistent planning, Review of Financial Studies, 35 (1968), 201-208. doi: 10.2307/2296548.

[24]

T. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473. doi: 10.1007/s10479-012-1211-5.

[25]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Review of Economic Studies, 23 (1955), 165-180. doi: 10.2307/2295722.

[26]

R. Thaler, Some empirical evidence on dynamic inconsistency, Economics Letters, 8 (1981), 201-207. doi: 10.1016/0165-1765(81)90067-7.

[27]

A. Zhang, R. Korn and C. Ewald, Optimal management and inflation protection for defined contribution pension plans, Blätter der DGVFM, 28 (2007), 239-258. doi: 10.1007/s11857-007-0019-x.

[28]

Q. Zhao, Y. Shen and J. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835. doi: 10.1016/j.ejor.2014.04.034.

show all references

References:
[1]

G. Ainslie, Picoeconomics, Cambridge University Press, Cambridge, UK, 1992.

[2]

R. Barro, Ramsey meets Laibson in the neoclassical growth model, Quarterly Journal of Economics, 114 (1999), 1125-1152. doi: 10.1162/003355399556232.

[3]

T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, 2010,, Working Paper, (). 

[4]

T. Björk, A. Murgoci and X. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[5]

Z. Bodiei, A. Marcus and R. Merton, Defined benefit versus defined contribution pension plans: What are the real trade-offs?, in Pensions in the US Economy, University of Chicago Press, 1988, 139-162.

[6]

A. Cairns, D. Blake and K. Dowd, Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, 30 (2006), 843-877. doi: 10.1016/j.jedc.2005.03.009.

[7]

I. Ekeland and A. Lazrak, Being serious about non-commitment: Subgame perfect equilibrium in continuous time, 2006,, Preprint. University of British Columbia., (). 

[8]

I. Ekeland, O. Mbodji and T. Pirvu, Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32. doi: 10.1137/100810034.

[9]

I. Ekeland and T. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86. doi: 10.1007/s11579-008-0014-6.

[10]

P. Emms, Lifetime investment and consumption using a defined-contribution pension scheme, Journal of Economic Dynamics and Control, 36 (2012), 1303-1321. doi: 10.1016/j.jedc.2012.01.012.

[11]

S. Goldman, Consistent plans}, Review of Financial Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[12]

S. Haberman and E. Vigna, Optimal investment strategies and risk measures in defined contribution pension schemes, Insurance: Mathematics and Economics, 31 (2002), 35-69. doi: 10.1016/S0167-6687(02)00128-2.

[13]

L. He and Z. Liang, Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance: Mathematics and Economics, 52 (2013), 404-410. doi: 10.1016/j.insmatheco.2013.02.005.

[14]

D. Laibson, Golden eggs and hyperbolic discounting, Quarterly Journal of Economics, 112 (1997), 443-478. doi: 10.1162/003355397555253.

[15]

D. Laibson, Life-cycle consumption and hyperbolic discount functions, European Economic Review, 42 (1998), 861-871. doi: 10.1016/S0014-2921(97)00132-3.

[16]

D. Laibson, A. Repetto and J. Tobacman, Self-control and saving for retirement, Brookings Papers on Economic Activity, 1998 (1998), 91-196. doi: 10.2307/2534671.

[17]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, Quarterly Journal of Economics, 107 (1992), 573-597. doi: 10.2307/2118482.

[18]

J. Marín-Solano and J. Navas, Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872. doi: 10.1016/j.ejor.2009.04.005.

[19]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[20]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[21]

B. Peleg and M. Yaari, On the existence of a consistent course of action when tastes are changing, Review of Financial Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[22]

E. Phelps and R. Pollak, On second-best national saving and game-equilibrium growth, Review of Economic Studies, 35 (1968), 185-199.

[23]

R. Pollak, Consistent planning, Review of Financial Studies, 35 (1968), 201-208. doi: 10.2307/2296548.

[24]

T. Siu, A BSDE approach to risk-based asset allocation of pension funds with regime switching, Annals of Operations Research, 201 (2012), 449-473. doi: 10.1007/s10479-012-1211-5.

[25]

R. Strotz, Myopia and inconsistency in dynamic utility maximization, Review of Economic Studies, 23 (1955), 165-180. doi: 10.2307/2295722.

[26]

R. Thaler, Some empirical evidence on dynamic inconsistency, Economics Letters, 8 (1981), 201-207. doi: 10.1016/0165-1765(81)90067-7.

[27]

A. Zhang, R. Korn and C. Ewald, Optimal management and inflation protection for defined contribution pension plans, Blätter der DGVFM, 28 (2007), 239-258. doi: 10.1007/s11857-007-0019-x.

[28]

Q. Zhao, Y. Shen and J. Wei, Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835. doi: 10.1016/j.ejor.2014.04.034.

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