A consumption-investment problem modelled as a discounted Markov decision process

Hugo Cruz-Suárez; Raúl Montes-de-Oca; Gabriel Zacarías

Kybernetika (2011)

  • Volume: 47, Issue: 6, page 909-929
  • ISSN: 0023-5954

Abstract

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In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions which guarantee that a policy is optimal for the problem if and only if it satisfies the EE. The problem is exemplified in two particular cases: for a logarithmic utility and for a Cobb-Douglas utility. In both cases explicit formulas for the optimal policy and the optimal value function are supplied.

How to cite

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Cruz-Suárez, Hugo, Montes-de-Oca, Raúl, and Zacarías, Gabriel. "A consumption-investment problem modelled as a discounted Markov decision process." Kybernetika 47.6 (2011): 909-929. <http://eudml.org/doc/196815>.

@article{Cruz2011,
abstract = {In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions which guarantee that a policy is optimal for the problem if and only if it satisfies the EE. The problem is exemplified in two particular cases: for a logarithmic utility and for a Cobb-Douglas utility. In both cases explicit formulas for the optimal policy and the optimal value function are supplied.},
author = {Cruz-Suárez, Hugo, Montes-de-Oca, Raúl, Zacarías, Gabriel},
journal = {Kybernetika},
keywords = {discounted Markov decision processes; differentiable value function; differentiable optimal policy; stochastic Euler equation; consumption and investment problems; discounted Markov decision processes; differentiable value function; differentiable optimal policy; stochastic Euler equation; consumption and investment problems},
language = {eng},
number = {6},
pages = {909-929},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A consumption-investment problem modelled as a discounted Markov decision process},
url = {http://eudml.org/doc/196815},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Cruz-Suárez, Hugo
AU - Montes-de-Oca, Raúl
AU - Zacarías, Gabriel
TI - A consumption-investment problem modelled as a discounted Markov decision process
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 6
SP - 909
EP - 929
AB - In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions which guarantee that a policy is optimal for the problem if and only if it satisfies the EE. The problem is exemplified in two particular cases: for a logarithmic utility and for a Cobb-Douglas utility. In both cases explicit formulas for the optimal policy and the optimal value function are supplied.
LA - eng
KW - discounted Markov decision processes; differentiable value function; differentiable optimal policy; stochastic Euler equation; consumption and investment problems; discounted Markov decision processes; differentiable value function; differentiable optimal policy; stochastic Euler equation; consumption and investment problems
UR - http://eudml.org/doc/196815
ER -

References

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  1. Aliprantis, C. D., Burkinshaw, O., Principles of Real Analysis, Academic Press, San Diego 1998. (1998) Zbl1006.28001MR1669668
  2. Angelatos, G. M., 10.1016/j.red.2006.11.001, Rev. Econom. Dynam. 10 (2007), 1–30. (2007) DOI10.1016/j.red.2006.11.001
  3. Arrow, K. J., 10.1007/s11166-009-9065-1, J. Risk Unc. 38 (2009), 87–94. (2009) Zbl1166.91321DOI10.1007/s11166-009-9065-1
  4. Bertsekas, D. P., Dynamic Programming: Deterministic and Stochastic Models, Prentice-Hall, Belmont 1987. (1987) Zbl0649.93001MR0896902
  5. Brock, W., Mirman, L., 10.1016/0022-0531(72)90135-4, J. Econom. Theory 4 (1972), 479–513. (1972) MR0449517DOI10.1016/0022-0531(72)90135-4
  6. Cruz-Suárez, D., Montes-de-Oca, R., Salem-Silva, F., 10.1007/s001860400372, Math. Meth. Oper. Res. 60 (2004), 415–436. (2004) Zbl1104.90053MR2106092DOI10.1007/s001860400372
  7. Cruz-Suárez, H., Montes-de-Oca, R., Discounted Markov control processes induced by deterministic systems, Kybernetika 42 (2006), 647–664. (2006) Zbl1249.90312MR2296506
  8. Cruz-Suárez, H., Montes-de-Oca, R., 10.1007/s00186-007-0155-z, Math. Meth. Oper. Res. 67 (2008), 299–321. (2008) Zbl1149.90171MR2390061DOI10.1007/s00186-007-0155-z
  9. Dynkin, E. B., Yushkevich, A. A., Controlled Markov Processes, Springer-Verlag, New York 1980. (1980) MR0554083
  10. Epstein, L., Zin, S., 10.2307/1913778, Econometrica 57 (1989), 937–969. (1989) MR1006550DOI10.2307/1913778
  11. Fuente, A. De la, Mathematical Methods and Models for Economists, Cambridge University Press, Cambridge 2000. (2000) Zbl0943.91001MR1735968
  12. Gurkaynak, R. S., 10.1111/j.1467-6419.2007.00530.x, J. Econom. Surveys 22 (2008), 166–186. (2008) DOI10.1111/j.1467-6419.2007.00530.x
  13. Heer, B., Maussner, A., Dynamic General Equilibrium Modelling: Computational Method and Application, Second edition, Springer-Verlag, Berlin 2005. (2005) MR2378171
  14. Hernández-Lerma, O., Lasserre, J. B., Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York 1996. (1996) MR1363487
  15. Hernández-Lerma, O., Lasserre, J. B., 10.1006/jmaa.1993.1242, J. Math. Anal. Appl. 177 (1993), 38–55. (1993) MR1224804DOI10.1006/jmaa.1993.1242
  16. Jaskiewics, A., Nowak, A. S., 10.1016/j.jmaa.2010.08.073, J. Math. Anal. Appl. 378 (2011), 450–462. (2011) MR2773257DOI10.1016/j.jmaa.2010.08.073
  17. Korn, R., Kraft, H., 10.1137/S0363012900377791, SIAM J. Control Optim. 40 (2001), 1250–1269. (2001) Zbl1020.93029MR1882732DOI10.1137/S0363012900377791
  18. Kamihigashi, T., 10.1016/j.jmateco.2006.05.007, J. Math. Econom. 43 (2007), 477–500. (2007) MR2317118DOI10.1016/j.jmateco.2006.05.007
  19. Levhari, D., Srinivasan, T. N., 10.2307/2296834, Rev. Econom. Stud. 36 (1969), 153–163. (1969) DOI10.2307/2296834
  20. Mirman, L., Zilcha, I., 10.1016/0022-0531(75)90022-8, J. Econom. Theory 2 (1975), 329–339. (1975) Zbl0362.90024MR0414045DOI10.1016/0022-0531(75)90022-8
  21. Ramsey, F. P., A Mathematical theory of saving, Econom. J. 38 (1928), 543–559. (1928) 
  22. Stokey, N., Lucas, R., Prescott, E., Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge 1989. (1989) MR1105087

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