Consumption and portfolio decisions with uncertain lifetimes. (English) Zbl 1437.91405
Summary: We study the consumption and portfolio decisions by incorporating mortality risk and altruistic factor in the classical model of
R. C. Merton [J. Econ. Theory 3, 373–413 (1971; Zbl 1011.91502); “Lifetime portfolio selection under uncertainty: the continuous-time case”, Rev. Econ. Stat. 51, No. 3, 247–257 (1969; doi:10.2307/1926560)] and M. E. Yaari [“Uncertain lifetime, life insurance, and the theory of the consumer”, Rev. Econ. Stud. 32, No. 2, 137–150 (1965; doi:10.2307/2296058)]. We find that besides the present-biased preference, the process of updating mortality information may be another underlying cause of dynamically time-inconsistent consumption behavior. We use the game-theoretic approach to obtain the extended Hamilton-Jacobi-Bellman equation. Furthermore, we obtain the closed-form solution for the logarithmic utility and explore comparative statics and implications for dynamic behavior. We present numerical results for the power utility that shows the sophisticated individual enjoys higher expected discounted utility than the naive. Our analytical solution enables us to generate a set of testable predictions that are consistent with existing empirical evidence. In particular, we show that for a moderate range of expected investment return, individuals will exhibit a “hump-shaped” consumption pattern, as widely documented in the empirical literature.
Keywords:
time inconsistency; equilibrium strategies; uncertain lifetime; consumption and portfolio choices; non-hyperbolic discountingCitations:
Zbl 1011.91502References:
| [1] | Angeletos, GM; Laibson, D.; Repetto, A.; Tobacman, J.; Weinberg, S., The hyperbolic consumption model: calibration, simulation, and empirical evaluation, J. Econ. Perspect., 15, 3, 47-68 (2001) |
| [2] | Akerlof, GA, Procrastination and obedience, Am. Econ. Rev., 81, 2, 1-19 (1991) |
| [3] | Azfar, O., Rationalizing hyperbolic discounting, J. Econ. Behav. Organ., 38, 2, 245-252 (1999) |
| [4] | Basak, S.; Chabakauri, G., Dynamic mean-variance asset allocation, Rev. Financ. Stud., 23, 8, 2970-3016 (2010) |
| [5] | Bommier, A., Portfolio choice under uncertain lifetime, J. Public Econ. Theory, 12, 1, 57-73 (2010) |
| [6] | Barro, RJ, Ramsey meets Laibson in the neoclassical growth model, Q. J. Econ., 114, 4, 1125-1152 (1999) · Zbl 0940.91056 |
| [7] | Bagliano, FC; Fugazza, C.; Nicodano, G., Optimal life-cycle portfolios for heterogeneous workers, Rev. Finance, 18, 6, 2283-2323 (2014) · Zbl 1417.91437 |
| [8] | Björk, T., Murgoci, A.: A general theory of Markovian time inconsistent stochastic control problems. SSRN:1694759 (2010) · Zbl 1297.49038 |
| [9] | Björk, T.; Murgoci, A., A theory of Markovian time-inconsistent stochastic control in discrete time, Finance Stoch., 18, 3, 545-592 (2014) · Zbl 1297.49038 |
| [10] | Björk, T.; Murgoci, A.; Zhou, XY, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24, 1, 1-24 (2014) · Zbl 1285.91116 |
| [11] | Björk, T.; Khapko, M.; Murgoci, A., On time-inconsistent stochastic control in continuous time, Finance Stoch., 21, 2, 331-360 (2017) · Zbl 1360.49013 |
| [12] | Beshears, J.; Choi, JJ; Laibson, D.; Madrian, BC, Does aggregated returns disclosure increase portfolio risk taking?, Rev. Financ. Stud., 30, 6, 1971-2005 (2017) |
| [13] | Chang, FR, Stochastic Optimization in Continuous Time (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1088.90041 |
| [14] | Chen, S.; Fu, R.; Wedge, L.; Zou, Z., Non-hyperbolic discounting and dynamic preference reversal, Theor. Decis., 86, 2, 283-302 (2019) · Zbl 1410.91181 |
| [15] | Chen, H.; Ju, N.; Miao, J., Dynamic asset allocation with ambiguous return predictability, Rev. Econ. Dyn., 17, 4, 799-823 (2014) |
| [16] | Chen, S.; Li, Z.; Zeng, Y., Optimal dividend strategies with time-inconsistent preferences, J. Econ. Dyn. Control, 46, 150-172 (2014) · Zbl 1402.91671 |
| [17] | Cocco, JF; Gomes, FJ; Maenhout, PJ, Consumption and portfolio choice over the life cycle, Rev. Financ. Stud., 18, 2, 491-533 (2005) |
| [18] | De Nardi, M., Wealth inequality and intergenerational links, Rev. Econ. Stud., 71, 3, 743-768 (2004) · Zbl 1103.91382 |
| [19] | DellaVigna, S.; Malmendier, U., Contract design and self control: theory and evidence, Q. J. Econ., 119, 2, 353-402 (2004) · Zbl 1090.91059 |
| [20] | Ekeland, I., Lazrak, A.: Being serious about non-commitment: subgame perfect equilibrium in continuous time. arXiv:math/0604264v1 (2006) |
| [21] | Ekeland, I.; Karp, L.; Sumaila, R., Equilibrium resource management with altruistic overlapping generations, J. Environ. Econ. Manag., 70, 1-16 (2015) |
| [22] | Epstein, LG; Zin, SE, Substitution, risk aversion and the temporal behavior of consumption and asset returns: an empirical analysis, J. Polit. Econ., 99, 2, 263-286 (1991) |
| [23] | Ekeland, I.; Pirvu, TA, Investment and consumption without commitment, Math. Financ. Econ., 2, 1, 57-86 (2008) · Zbl 1177.91123 |
| [24] | Ekeland, I.; Mbodji, O.; Pirvu, TA, Time-consistent portfolio management, SIAM J. Financ. Math., 3, 1, 1-32 (2012) · Zbl 1257.91040 |
| [25] | Ekeland, I.; Lazrak, A., The golden rule when preferences are time-inconsistent, Math. Financ. Econ., 4, 1, 29-55 (2010) · Zbl 1255.91249 |
| [26] | Frederick, S.; Loewenstein, G.; O’Donoghue, T., Time discounting and time preference: a critical review, J. Econ. Literat., 40, 2, 351-401 (2002) |
| [27] | Gollier, C., The Economics of Risk and Time (2001), Cambridge: MIT Press, Cambridge · Zbl 0991.91001 |
| [28] | Gong, LT; Smith, W.; Zou, HF, Consumption and Risk with hyperbolic discounting, Econ. Lett., 96, 2, 153-160 (2007) · Zbl 1255.91217 |
| [29] | Gul, F.; Pesendorfer, W., Temptation and self-control, Econometrica, 69, 6, 1403-1435 (2001) · Zbl 1019.91017 |
| [30] | Green, L.; Myerson, J., Exponential versus hyperbolic discounting of delayed outcomes: risk and waiting time, Am. Zool., 36, 4, 496-505 (1996) |
| [31] | Grenadier, SR; Wang, N., Investment under uncertainty and time-inconsistent preferences, J. Financ. Econ., 84, 1, 2-39 (2007) |
| [32] | Gourinchas, PO; Parker, JA, Consumption over the life cycle, Econometrica, 70, 1, 47-89 (2002) · Zbl 1137.91519 |
| [33] | Harris, C.; Laibson, D., Instantaneous gratification, Q. J. Econ., 128, 1, 205-248 (2013) · Zbl 1400.91193 |
| [34] | Halevy, Y., Time consistency: stationarity and time invariance, Econometrica, 83, 1, 335-352 (2015) · Zbl 1419.91190 |
| [35] | Halevy, Y.: Diminishing Impatience: Disentangling Time Preference from Uncertain Lifetime, unpublished working paper, Department of Economics, University of British Columbia (2005) |
| [36] | Halevy, Y., Strotz meets allais: diminishing impatience and the certainty effect, Am. Econ. Rev., 98, 3, 1145-1162 (2008) |
| [37] | He, X.D., Jiang, Z.: On the Equilibrium Stragegies for Time-Inconsistent Problems in Continuous Time. Available at SSRN: https://ssrn.com/abstract=3308274 or 10.2139/ssrn.3308274 (2019) |
| [38] | Huang, Y.J., Zhou, Z.: Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time. Available at arXiv:1809.09243v3 (2019) |
| [39] | Karp, L., Non-constant discounting in continuous time, J. Econ. Theory, 132, 1, 557-568 (2007) · Zbl 1142.91668 |
| [40] | Koopmans, TC, Stationary ordinal utility and impatience, Econometrica, 28, 2, 287-309 (1960) · Zbl 0149.38401 |
| [41] | Kotlikoff, LJ; Summers, LH, The role of intergenerational transfers in aggregate capital accumulation, J. Polit. Econ., 89, 4, 706-732 (1981) |
| [42] | Kuehlwein, M., Life-cycle and altruistic theories of saving with lifetime uncertainty, Rev. Econ. Stat., 75, 1, 38-47 (1993) |
| [43] | Kinari, Y.; Ohtake, F.; Tsutsui, Y., Time discounting: declining impatience and interval effect, J. Risk Uncertain., 39, 1, 87-112 (2009) · Zbl 1187.91169 |
| [44] | Laitner, J., Secular changes in wealth inequality and inheritance, Econ. J., 111, 474, 691-721 (2001) |
| [45] | Marín-Solano, J.; Navas, J., Non-constant discounting in finite horizon: the free terminal time case, J. Econ. Dyn. Control, 33, 3, 666-675 (2009) · Zbl 1168.49025 |
| [46] | Marín-Solano, J.; Navas, J., Consumption and portfolio rules for time-inconsistent investors, Eur. J. Oper. Res., 201, 3, 860-872 (2010) · Zbl 1180.91270 |
| [47] | McClure, SM; Laibson, D.; Loewenstein, G.; Cohen, JD, Separate neural systems value immediate and delayed monetary rewards, Science, 306, 5695, 503-507 (2004) |
| [48] | McClure, SM; Ericson, KM; Laibson, D.; Loewenstein, G.; Cohen, JD, Time discounting for primary rewards, J. Neurosci., 27, 21, 5796-5804 (2007) |
| [49] | Merton, RC, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat., 51, 247-257 (1969) |
| [50] | Merton, RC, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory, 3, 373-413 (1971) · Zbl 1011.91502 |
| [51] | Miao, J., Option exercise with temptation, Econ. Theor., 34, 3, 473-501 (2008) · Zbl 1142.91448 |
| [52] | O’Donoghue, T.; Rabin, M., Doing it now or later, Am. Econ. Rev., 89, 1, 103-124 (1999) |
| [53] | O’Donoghue, T.; Rabin, M., Present bias: lessons learned and to be learned, Am. Econ. Rev., 105, 5, 273-279 (2015) |
| [54] | Palacios-Huerta, I., Pérez-Kakabadse, A.: Consumption and portfolio rules with stochastic hyperbolic discounting, unpublished working paper, London School of Economics, London, UK (2013) |
| [55] | Phelps, ES; Pollak, RA, On second-best national saving and game-equilibrium growth, Rev. Econ. Stud., 35, 2, 185-199 (1968) |
| [56] | Read, D., Is time-discounting hyperbolic or subadditive?, J. Risk Uncertain., 23, 1, 5-32 (2001) · Zbl 0986.91013 |
| [57] | Read, D.; Roelofsma, PH, Subadditive versus hyperbolic discounting: a comparison of choice and matching, Organ. Behav. Hum. Decis. Process., 91, 2, 140-153 (2003) |
| [58] | Strotz, RH, Myopia and inconsistency in dynamic utility maximization, Rev. Econ. Stud., 23, 3, 165-180 (1955) |
| [59] | Sozou, PD, On hyperbolic discounting and uncertain hazard rates, Proc. R. Soc. Lond. (Ser. B-Biol. Sci.), 265, 1409, 2015-2020 (1998) |
| [60] | Scholten, M.; Read, D., Discounting by intervals: a generalised model of intertemporal choice, Manag. Sci., 52, 9, 1424-1436 (2006) |
| [61] | Thaler, RH; Shefrin, HM, An economic theory of self-control, J. Polit. Econ., 89, 2, 392-406 (1981) |
| [62] | Weibull, W., A statistical distribution function of wide applicability, J. Appl. Mech., 18, 3, 293-297 (1951) · Zbl 0042.37903 |
| [63] | Wang, C.; Wang, N.; Yang, J., A unified model of entrepreneurship dynamics, J. Financ. Econ., 106, 1, 1-23 (2012) |
| [64] | Wei, J.; Li, D.; Zeng, Y., Robust Optimal consumption-investment strategy with non-exponential discounting, J. Ind. Manag. Optim., 16, 1, 207-230 (2018) · Zbl 1438.90188 |
| [65] | Weil, P., Nonexpected utility in macroeconomics, Q. J. Econ., 105, 1, 29-42 (1990) |
| [66] | Yaari, ME, Uncertain lifetime, life insurance, and the theory of the consumer, Rev. Econ. Stud., 32, 2, 137-150 (1965) |
| [67] | Yong, J., Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Relat. Fields, 2, 3, 271-329 (2012) · Zbl 1251.93144 |
| [68] | Zou, Z.; Chen, S.; Wedge, L., Finite horizon consumption and portfolio decisions with stochastic hyperbolic discounting, J. Math. Econ., 52, 70-80 (2014) · Zbl 1297.91134 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
