Do Time Preferences Matter in Intertemporal Consumption and Portfolio Decisions?

Shou Chen; Shengpeng Xiang; Hongbo He

doi:10.1515/bejte-2017-0122

Article

Do Time Preferences Matter in Intertemporal Consumption and Portfolio Decisions?

Shou Chen , Shengpeng Xiang and Hongbo He

Published/Copyright: January 26, 2019

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From the journal The B.E. Journal of Theoretical Economics Volume 19 Issue 2

Abstract

We study the intertemporal consumption and portfolio rules in the model with the general hyperbolic absolute risk aversion (HARA) utility. The equivalent approximation approach is employed to obtain the Hamilton-Jacobi-Bellman (HJB) equations, and a remarkable property is shown: portfolio rules are independent of the discount function. Moreover, both the consumption and portfolio rates are non-increasing functions of wealth. Particularly illustrative cases examined in detail are the models with the most adopted discount functions, including exponential discounting and hyperbolic discounting. Explicit solutions for intertemporal decisions are found for these special cases, revealing that individual’s time preferences affect the consumption rules only. Moreover, the time-consistent consumption rate under hyperbolic discounting is larger than its counterpart under exponential discounting.

Keywords: hyperbolic discounting; time consistency; equivalent approximation approach; HJB equation; consumption and portfolio rules

JEL Classification: C61; D91; G11

Acknowledgements

We would like to thank the editor, two anonymous referees and Ziran Zou for their helpful comments. This paper is supported by the National Natural Science Foundation of China (Grant Nos. 71790593, 71521061, 71501065 and 71573077), as well as the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No.[2015]1098).

Appendix

A Appendix

Applying Taylor’s theorem to the discount function of value function for small h0, i. e. θ(h0)=1+θ′(0)h0+o(h0), the rate of change of value function can be written as

(29)θ(h0)V(t+h0,W(t+h0))−V(t,W(t))h0=V(t+h0,W(t+h0))−V(t,x(t))h0+θ′(0)V(t+h0,W(t+h0))+o(h0)h0.

On one hand, the left-hand side of eq. (29) is

(30)maxC(s),ϕ(s)1h0E∫t+h0Tθ(h0)θ(s−t−h0)u(C(s))ds −∫tTθ(s−t)u(C(s))ds+(θ(h0)θ(T−t−h0)−θ(T−t))F(T,W(T))=maxC(s),ϕ(s)1h0E∫t+h0T[θ(h0)θ(s−t−h0)−θ(s−t)]u(C(s))ds −∫tt+h0θ(s−t)u(C(s))ds+(θ(h0)θ(T−t−h0)−θ(T−t))F(T,W(T)).

According to the mean value theorem for integrals, take the limit as h0→0, so that eq. (30) can be rewritten as

(31)limh0→0maxC(s),ϕ(s)1h0E[∫t+h0Tθ(h0)θ(s−t−h0)u(C(s))ds−∫tTθ(s−t)u(C(s))ds+(θ(h0)θ(T−t−h0)−θ(T−t))F(T,W(T))]=maxC(s),ϕ(s)−u(C(t))+K(t,W(t)),

where

K(t,W(t))=limh0→01h0E[∫tT[θ(h0)θ(s−t−h0)−θ(s−t)]u(C∗(s))ds+(θ(h0)θ(T−t−h0)−θ(T−t))⋅F(T,W(T))].

On the other hand, by Ito’s Lemma, from the right-hand side of eq. (29) as h0→0, we have

(32)limh0→0EV(t+h0,W(t+h0))−V(t,W(t))h0+θ′(0)V(t+h0,W(t+h0))=[Vt+VWf+12VWWσ2ϕ2(t)W2(t)]|W(t),C(t),ϕ(t)+θ′(0)V(t,W(t)).

Combining eqs. (31) and (32) with eq. (29), we obtain

K(t,W(t))−θ′(0)V(t,W(t))−Vt=maxC(t),ϕ(t)u(C(t))+VWf+12VWWσ2ϕ2(t)W2(t).

The HJB equations would be consistent with previous works by adjusting the discount functions in this paper.

B Appendix

Let γ→−∞ and η=1 for u(C)=1−γγ(αC1−γ+η)γ, we have u(C)=−e−αC,α>0, where α is Pratt’s measure of absolute risk-aversion. We conjecture a trial solution V(W)=−pqe−qW. Then, we have

Vt=0,VW=pe−qW,VWW=−pqe−qW.

Thus, the first-order conditions for a regular interior maximum obtained from eq. (8) are

u′(C)=VW,(μ−r)WVW=−VWWϕσ2W2,

which imply

(33)C∗(t)=1α(qW(t)+lnαp),ϕ∗(t)=−(μ−r)VWσ2WVWW=(μ−r)qσ2W(t),

and

(34)dW(t)=(μ−r)2qσ2+rW(t)−1α(qW(t)−lnpα)dt+(μ−r)qσdzt.

From eqs. (8) and (33), we have

(35)K(W)=limh0→01h0E∫0∞[θ(h0)θ(s−h0)−θ(s)]u(C∗(s))ds =limh0→01h0E∫0∞[θ(h0)θ(s−h0)−θ(s)]−pe−qW(s)αds.

Meanwhile, from eq. (34), we obtain

(36)d(e−qW(s))=−qe−qW(s)(μ−r)22qσ2+rW(s)−1α(qW(s)−lnpα)ds+(μ−r)qσdzs,

which implies

(37)E(d(e−qW(s)))=−qe−qW(s)(μ−r)22qσ2+rW(s)−1α(qW(s)−lnpα)ds,

and

(38)E(e−qW(s))=e−qW(t)exp−q∫ts(μ−r)22qσ2+rW(x)−1α(qW(x)−lnpα)dx.

Incorporating eq. (38) into eq. (35), K(W) is expressed as

(39)K(W)=−pe−qW(t)αlimh0→01h0∫0∞[θ(h0)θ(s−h0)−θ(s)] ⋅exp−q∫ts(μ−r)22qσ2+rW(x)−1α(qW(x)−lnpα)dxds.

Combining eqs. (33) and (39) with eq. (8), we find that q=αr and p must satisfy the following algebraic equation:

(40)limh0→01h0∫0∞[θ(h0)θ(s−h0)−θ(s)]exp−(μ−r)22σ2+rlnpαsds=1−(μ−r)22rσ2−lnpα+θ′(0)r.

The time-consistent decision rules for consumption and portfolio selection are

C∗(t)=rW(t)+1αlnαp,

and

ϕ∗(t)=(μ−r)αrσ2W(t).

References

Ainslie, G. 1991. “Derivation of “Rational” Economic Behavior from Hyperbolic Discount Curves.” American Economic Review 81 (2): 334–40.Search in Google Scholar

Basak, S., and G. Chabakauri. 2010. “Dynamic Mean-Variance Asset Allocation.” Review of Financial Studies 23 (8): 2970–3016.10.1093/rfs/hhq028Search in Google Scholar

Blavatskyy, P. R. 2017. “Probabilistic Intertemporal Choice.” Journal of Mathematical Economics 73: 142–48.10.1016/j.jmateco.2017.10.002Search in Google Scholar

Bond, P., and G. Sigurdsson. 2018. “Commitment Contracts.” Review of Economic Studies 85 (2): 601–46.Search in Google Scholar

Boyd III, J. H. 1990. “Symmetries, Dynamic Equilibria, and the Value Function.” In Conservation Laws and Symmetry: Applications to Economics and Finance, edited by R. Sato, and R.V. Ramachandran, 225–59. Boston: Kluwer Academic Publishers.10.1007/978-94-017-1145-6_8Search in Google Scholar

Chang, F. R. 2004. Stochastic Optimization in Continuous Time, 191–94. Cambridge, UK: Cambridge University Press.10.1017/CBO9780511616747Search in Google Scholar

Chang, H., and K. Chang. 2017. “Optimal Consumption-Investment Strategy under the Vasicek Model: HARA Utility and Legendre Transform.” Insurance: Mathematics and Economics 72: 215–27.10.1016/j.insmatheco.2016.10.014Search in Google Scholar

DellaVigna, S., and U. Malmendier. 2006. “Paying Not to Go to the Gym.” American Economic Review 96 (3): 694–719.10.1257/aer.96.3.694Search in Google Scholar

Delong, Ł., and A. Chen. 2016. “Asset Allocation, Sustainable Withdrawal, Longevity Risk and Non-Exponential Discounting.” Insurance: Mathematics and Economics 71 (1): 342–52.10.1016/j.insmatheco.2016.10.002Search in Google Scholar

Giat, Y., and A. Subramanian. 2013. “Dynamic Contracting under Imperfect Public Information and Asymmetric Beliefs.” Journal of Economic Dynamics and Control 37 (12): 2833–61.10.1016/j.jedc.2013.08.001Search in Google Scholar

Gong, L., R. Zhong, and H.-f. Zou. 2012. “On the Concavity of the Consumption Function with the Time Varying Discount Rate.” Economics Letters 117 (1): 99–101.10.1016/j.econlet.2012.04.086Search in Google Scholar

Grasselli, M. 2003. “A Stability Result for the HARA Class with Stochastic Interest Rates.” Insurance: Mathematics and Economics 33 (3): 611–27.10.1016/j.insmatheco.2003.09.003Search in Google Scholar

Grenadier, S. R., and N. Wang. 2007. “Investment under Uncertainty and Time-Inconsistent Preferences.” Journal of Financial Economics 84 (1): 2–39.10.1016/j.jfineco.2006.01.002Search in Google Scholar

Halevy, Y. 2015. “Time Consistency: Stationarity and Time Invariance.” Econometrica 83 (1): 335–52.10.3982/ECTA10872Search in Google Scholar

Harris, C., and D. Laibson. 2013. “Instantaneous Gratification.” Quarterly Journal of Economics 128 (1): 205–48.10.1093/qje/qjs051Search in Google Scholar

Karp, L. 2007. “Non-Constant Discounting in Continuous Time.” Journal of Economic Theory 132 (1): 557–68.10.1016/j.jet.2005.07.006Search in Google Scholar

Kirby, K. N., and R. J. Herrnstein. 1995. “Preference Reversals Due to Myopic Discounting of Delayed Reward.” Psychological Science 6 (2): 83–89.10.1111/j.1467-9280.1995.tb00311.xSearch in Google Scholar

Kronborg, M. T., and M. Steffensen. 2015. “Inconsistent Investment and Consumption Problems.” Applied Mathematics and Optimization 71 (3): 473–515.10.1007/s00245-014-9267-zSearch in Google Scholar

Laibson, D. 1997. “Golden Eggs and Hyperbolic Discounting.” Quarterly Journal of Economics 112 (2): 443–77.10.1162/003355397555253Search in Google Scholar

Loewenstein, G., and D. Prelec. 1992. “Anomalies in Intertemporal Choice: Evidence and an Interpretation.” Quarterly Journal of Economics 57: 573–98.10.1017/CBO9780511803475.034Search in Google Scholar

Marín-Solano, J., and J. Navas. 2009. “Non-Constant Discounting in Finite Horizon: The Free Terminal Time Case.” Journal of Economic Dynamics and Control 33 (3): 666–75.10.1016/j.jedc.2008.08.008Search in Google Scholar

Marín-Solano, J., and J. Navas. 2010. “Consumption and Portfolio Rules for Time-Inconsistent Investors.” European Journal of Operational Research 201 (3): 860–72.10.1016/j.ejor.2009.04.005Search in Google Scholar

Merton, R. C. 1969. “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case.” Review of Economics and Statistics 51 (3): 247–57.10.2307/1926560Search in Google Scholar

Merton, R. C. 1971. “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory 3 (4): 373–413.10.1016/0022-0531(71)90038-XSearch in Google Scholar

O’Donoghue, T., and M. Rabin. 1999. “Doing It Now or Later.” American Economic Review 89 (1): 103–24.10.1257/aer.89.1.103Search in Google Scholar

Palacios-Huerta, I., and A. Pérez-Kakabadse 2013. Consumption and Portfolio Rules with Stochastic Hyperbolic Discounting. Working paper, 1–32.10.2139/ssrn.1465110Search in Google Scholar

Pan, J., C. S. Webb, and H. Zank. 2015. “An Extension of Quasi-Hyperbolic Discounting to Continuous Time.” Games and Economic Behavior 89: 43–55.10.1016/j.geb.2014.11.003Search in Google Scholar

Park, H., and J. Feigenbaum. 2018. “Bounded Rationality, Lifecycle Consumption, and Social Security.” Journal of Economic Behavior and Organization 146: 65–105.10.1016/j.jebo.2017.12.005Search in Google Scholar

Pavoni, N., and H. Yazici. 2017. “Optimal Life-Cycle Capital Taxation under Self-Control Problems.” Economic Journal 127 (602): 1188–216.10.1111/ecoj.12323Search in Google Scholar

Thaler, R. H., and H. M. Shefrin. 1981. “An Economic Theory of Self-Control.” Journal of Political Economy 89 (2): 392–406.10.1086/260971Search in Google Scholar

Webb, C. S. 2016. “Continuous Quasi-Hyperbolic Discounting.” Journal of Mathematical Economics 64: 99–106.10.1016/j.jmateco.2016.04.001Search in Google Scholar

Xiang, S., S. Chen, and H. He. 2018. “An Equivalent Approximation Approach for the Hamilton-Jacobi-Bellman Equations in Intertemporal Decision Problems.” Optimal Control Applications and Methods 39 (6): 1976–88.10.1002/oca.2461Search in Google Scholar

Zhao, Q., R. Wang, and J. Wei. 2016. “Exponential Utility Maximization for an Insurer with Time-Inconsistent Preferences.” Insurance: Mathematics and Economics 70: 89–104.10.1016/j.insmatheco.2016.06.003Search in Google Scholar

Zou, Z., S. Chen, and L. Wedge. 2014. “Finite Horizon Consumption and Portfolio Decisions with Stochastic Hyperbolic Discounting.” Journal of Mathematical Economics 52 (5): 70–80.10.1016/j.jmateco.2014.03.002Search in Google Scholar

Published Online: 2019-01-26

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https://doi.org/10.1515/bejte-2017-0122

Keywords for this article

hyperbolic discounting; time consistency; equivalent approximation approach; HJB equation; consumption and portfolio rules