Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
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
Summary:
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.
References:
[1] Aliprantis, C. D., Burkinshaw, O.: Principles of Real Analysis. Academic Press, San Diego 1998. MR 1669668 | Zbl 1006.28001
[2] Angelatos, G. M.: Uninsured idiosyncratic investment risk and aggregate saving. Rev. Econom. Dynam. 10 (2007), 1–30. DOI 10.1016/j.red.2006.11.001
[3] Arrow, K. J.: A note on uncertainty and discounting in models of economic growth. J. Risk Unc. 38 (2009), 87–94. DOI 10.1007/s11166-009-9065-1 | Zbl 1166.91321
[4] Bertsekas, D. P.: Dynamic Programming: Deterministic and Stochastic Models. Prentice-Hall, Belmont 1987. MR 0896902 | Zbl 0649.93001
[5] Brock, W., Mirman, L.: Optimal economic growth and uncertainty: the discounted case. J. Econom. Theory 4 (1972), 479–513. DOI 10.1016/0022-0531(72)90135-4 | MR 0449517
[6] Cruz-Suárez, D., Montes-de-Oca, R., Salem-Silva, F.: Conditions for the uniqueness of optimal policies of discounted Markov decision processes. Math. Meth. Oper. Res. 60 (2004), 415–436. DOI 10.1007/s001860400372 | MR 2106092 | Zbl 1104.90053
[7] Cruz-Suárez, H., Montes-de-Oca, R.: Discounted Markov control processes induced by deterministic systems. Kybernetika 42 (2006), 647–664. MR 2296506 | Zbl 1249.90312
[8] Cruz-Suárez, H., Montes-de-Oca, R.: An envelope theorem and some applications to discounted Markov decision processes. Math. Meth. Oper. Res. 67 (2008), 299–321. DOI 10.1007/s00186-007-0155-z | MR 2390061 | Zbl 1149.90171
[9] Dynkin, E. B., Yushkevich, A. A.: Controlled Markov Processes. Springer-Verlag, New York 1980. MR 0554083
[10] Epstein, L., Zin, S.: Substitution, risk aversion, and the temporal behaviour of consumption and asset returns I: Theoretical framework. Econometrica 57 (1989), 937–969. DOI 10.2307/1913778 | MR 1006550
[11] Fuente, A. De la: Mathematical Methods and Models for Economists. Cambridge University Press, Cambridge 2000. MR 1735968 | Zbl 0943.91001
[12] Gurkaynak, R. S.: Econometric tests of asset price bubbles: taking stock. J. Econom. Surveys 22 (2008), 166–186. DOI 10.1111/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. MR 2378171
[14] Hernández-Lerma, O., Lasserre, J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer-Verlag, New York 1996. MR 1363487
[15] Hernández-Lerma, O., Lasserre, J. B.: Value iteration and rolling plans for Markov control processes with unbounded rewards. J. Math. Anal. Appl. 177 (1993), 38–55. DOI 10.1006/jmaa.1993.1242 | MR 1224804
[16] Jaskiewics, A., Nowak, A. S.: Discounted dynamic programming with unbounded returns: application to economic models. J. Math. Anal. Appl. 378 (2011), 450–462. DOI 10.1016/j.jmaa.2010.08.073 | MR 2773257
[17] Korn, R., Kraft, H.: A stochastic control approach to portfolio problems with stochastic interest rates. SIAM J. Control Optim. 40 (2001), 1250–1269. DOI 10.1137/S0363012900377791 | MR 1882732 | Zbl 1020.93029
[18] Kamihigashi, T.: Stochastic optimal growth with bounded or unbounded utility and bounded or unbounded shocks. J. Math. Econom. 43 (2007), 477–500. DOI 10.1016/j.jmateco.2006.05.007 | MR 2317118
[19] Levhari, D., Srinivasan, T. N.: Optimal savings under uncertainty. Rev. Econom. Stud. 36 (1969), 153–163. DOI 10.2307/2296834
[20] Mirman, L., Zilcha, I.: On optimal growth under uncertainty. J. Econom. Theory 2 (1975), 329–339. DOI 10.1016/0022-0531(75)90022-8 | MR 0414045 | Zbl 0362.90024
[21] Ramsey, F. P.: A Mathematical theory of saving. Econom. J. 38 (1928), 543–559.
[22] Stokey, N., Lucas, R., Prescott, E.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge 1989. MR 1105087
Search
Partner of
