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A dynamic programming approach to a consumption/investment and retirement choice problem under borrowing constraints

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Abstract

In this paper we consider an optimization problem of an infinitely lived working agent with an option to retire who maximizes the utility from her lifetime consumption. The agent receives labor income during the period before her voluntary retirement, but suffers disutility from labor. Moreover, the agent lacks the means to borrow against her future labor income. We use the dynamic programming approach to derive the closed-form solutions and provide some numerical illustrations.

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References

  1. Bodie, Z., Detemple, J.B., Otruba, S., Walter, S.: Optimal consumption portfolio choices and retirement planning. J. Econ. Dyn. Control 28, 1115–1148 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bodie, Z., Merton, R.C., Samuelson, W.F.: Labor supply flexibility and portfolio choice in a life cycle model. J. Econ. Dyn. Control 16, 427–449 (1992)

    Article  Google Scholar 

  3. Bensoussan, A., Jang, B., Park, S.: Unemployment risks and optimal retirement in an incomplete market. Oper. Res. 64, 1015–1032 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  4. Choi, K.J., Shim, G.: Disutility, optimal retirement, and portfolio selection. Math. Financ. 16, 443–467 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Choi, K.J., Shim, G., Shin, Y.H.: Optimal portfolio, consumption-leisure and retirement choice problem with CES utility. Math. Financ. 18, 445–472 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cox, J.C., Huang, C.F.: Optimum consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49, 33–83 (1989)

    Article  MATH  Google Scholar 

  7. Duffie, D., Fleming, W., Soner, M., Zariphopoulou, T.: Hedging in incomplete markets with HARA utility. J. Econ. Dyn. Control 21, 753–782 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dybvig, P.H., Liu, H.: Lifetime consumption and investment: retirement and constrained borrowing. J. Econ. Theory 145, 885–907 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dixit, A.K., Pindyck, R.S.: Investment Under Uncertainty. Princeton University Press, New Jersey (1994)

    Google Scholar 

  10. Farhi, E., Panageas, S.: Saving and investing for early retirement: a theoretical analysis. J. Financ. Econ. 83, 87–121 (2007)

    Article  Google Scholar 

  11. He, H., Pagès, H.F.: Labor income, borrowing constraints, and equilibrium asset prices. Econ. Theory. 3, 663–696 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jang, B., Park, S., Rhee, Y.: Optimal retirement with unemployment risks. J. Bank. Financ. 37, 3585–3604 (2013)

    Article  Google Scholar 

  13. Karatzas, I., Lehoczky, J.P., Sethi, S.P., Shreve, S.E.: Explicit solution of a general consumption/investment problem. Math. Oper. Res. 11, 261–294 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25, 1557–1586 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Karatzas, I., Wang, H.: Utility maximization with discretionary stopping. SIAM J. Control Optim. 39, 306–329 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Koo, H.K.: Consumption and portfolio selection with labor income: a continuous time approach. Math. Financ. 8, 49–65 (1998)

    Article  MATH  Google Scholar 

  17. Lim, B.H., Shin, Y.H.: Optimal investment, consumption and retirement decision with disutility and borrowing constraints. Quant. Financ. 11, 1581–1592 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)

    Article  Google Scholar 

  19. Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory. 3, 373–413 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  20. Park, S., Jang, B.: Optimal retirement strategy with a negative wealth constraint. Oper. Res. Lett. 42, 208–212 (2014)

    Article  MathSciNet  Google Scholar 

  21. Shiryaev, A.N.: Optimal Stopping Rules. Springer, New York (1978)

    MATH  Google Scholar 

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Acknowledgements

We are indebted to two anonymous referees for valuable advice and useful comments, which improve our paper essentially. We also appreciate Hyeng Keun Koo and Gyoocheol Shim for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Kwangwoon University, Seoul, 01897, Republic of Korea

    Ho-Seok Lee

  2. School of Law, University of Texas at Austin, Austin, TX, 78705, USA

    Byung Lim Koo

  3. Department of Mathematics, Sookmyung Women’s University, Seoul, 04310, Republic of Korea

    Yong Hyun Shin

Authors
  1. Ho-Seok Lee
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  2. Byung Lim Koo
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  3. Yong Hyun Shin
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Corresponding author

Correspondence to Yong Hyun Shin.

Additional information

The work of the first author (H. S. Lee) was supported by the National Research Foundation of Korea Grant funded by the Korean Government (Grant No. NRF-2016R1D1A1B03933406) and by the Research Grant of Kwangwoon University in 2016. The work of the corresponding author (Y. H. Shin) was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (Grant Nos. NRF-2013R1A1A2058027, NRF-2016R1A2B4008240).

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Lee, HS., Koo, B.L. & Shin, Y.H. A dynamic programming approach to a consumption/investment and retirement choice problem under borrowing constraints. Japan J. Indust. Appl. Math. 34, 793–809 (2017). https://doi.org/10.1007/s13160-017-0264-x

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  • DOI: https://doi.org/10.1007/s13160-017-0264-x

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