Shin and Roh Advances in Difference Equations (2019) 2019:211 https://doi.org/10.1186/s13662-019-2144-y

R E S E A R C H Open Access

An optimal consumption and investmentproblem with stochastic hyperbolicdiscountingYong Hyun Shin1 and Kum-Hwan Roh2*

*Correspondence: khroh@hnu.kr2Department of Mathematics,Hannam University, Daejeon,Republic of KoreaFull list of author information isavailable at the end of the article

AbstractIn this paper, we analyze the optimal consumption and investment problem of anagent by incorporating the stochastic hyperbolic preferences with constant relativerisk aversion utility. Using the dynamic programming method, we deal with theoptimization problem in a continuous-time model. And we provide the closed-formsolutions of the optimization problem.

Keywords: Portfolio selection; Stochastic hyperbolic discounting; Dynamicprogramming method

1 IntroductionAfter the seminal research of Strotz [11], the exponential discount function with a constantdiscount rate has been widely used in finance/economics. He explained the phenomenonof the different preferences between current and future reward with the discount function.And much research on dynamically inconsistent preferences has been undertaken [1, 3, 4,10, 12]. Especially the hyperbolic discounting model is the most widely used in the studyof inconsistent preferences.

Harris and Laibson [4] introduced the instantaneous-gratification (IG) model of timepreferences. This model is based on a quasi-hyperbolic stochastic discount function.Palacios-Huerta and Pérez-Kakabadse [9] extended the classical problem of [7, 8] to thetime-inconsistent agent problem. They used stochastic hyperbolic preferences model (IGmodel) which is introduced in Harris and Laibson [4]. Zou et al. [13] extended the workof Palacios-Huerta and Pérez-Kakabadse [9] to an agent who has finite life time. They alsofound that stochastic hyperbolic discounting increases the optimal consumption rate.

Caillaud and Jullien [1] introduced the theoretical explanations of the model of time-inconsistent preferences. They also derived aspects of time-inconsistency from basic psy-chological phenomena. Gong et al. [2] described that the marginal effect of risk on con-sumption is always greater under hyperbolic discounting than under exponential dis-counting.

In this paper, we suggest a technical approach which is different from Palacios-Huertaand Pérez-Kakabadse [9] and Zou et al. [13] for solving an optimal consumption and in-vestment problem with a stochastic hyperbolic discounting function. A stochastic hyper-bolic discounting means that there is a jump on a subjective discounting factor. When a

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Shin and Roh Advances in Difference Equations (2019) 2019:211 Page 2 of 7

jump occurs in a subjective discounting factor, there is a change of an agent’s preference.It means that a preference of an agent is time-inconsistency. And it signifies that it is moresensitive to change the rewards that occur in the near future than those in the distantfuture. For example, the investor is willing to wait a day to receive a one more dollar ina year later. But she is not waiting to receive a one more dollar in today. We apply thedynamic programming method suggested by Karatzas et al. [6] to an optimal consump-tion and portfolio selection problem with a stochastic hyperbolic discounting function.This approach has many merits to handle the optimal consumption and portfolio selec-tion problem under infinite horizon. First, after we introduce the inverse function of theoptimal consumption X(c) = C–1(x), we can turn the nonlinear ordinary differential equa-tion (Bellman equation) into the linear ordinary differential equation (ODE). From thislinear ODE, we easily obtain the general solution and particular solution to Cauchy equa-tion. So it is possible to tackle the portfolio selection problem with the various economicconstraints. The lack of this approach is that it is not proper in a finite horizon problem.Jeon and Shin [5] suggested the Mellin transform techniques to convert the Hamilton–Jacobi–Bellman partial differential equation (PDE) into a non-homogeneous PDE. Usingthe Mellin transform method which was used in Jeon and Shin [5], it is possible to extendour result to the finite horizon.

The rest of this paper is organized as follows. In Sect. 2, we introduce the financial mar-ket setup. We use the dynamic programming approach to derive the marginal propensityto consume corresponding to a stochastic hyperbolic discounting function in Sect. 3, andSect. 4 concludes.

2 The financial market modelWe assume that there are two assets which can be invested in the financial market. Oneis a riskless asset which pays a constant interest rate r > 0 and the other is a risky asset St

which follows the geometric Brownian motion,

dSt = μSt dt + σSt dBt ,

where μ > r and σ > 0 are constants and Bt is a standard Brownian motion defined on acomplete probability space (Ω ,F ,P) and {Ft}t≥0 is the P-augmentation of the filtrationgenerated by the Brownian motion {Bt}t≥0.

Let πt be the amount invested in the risky asset at time t and ct be consumption attime t. A portfolio process {πt}t≥0 is an adapted process with respect to {Ft}t≥0, and aconsumption process {ct}t≥0 is a non-negative adapted process with respect to {Ft}t≥0.They satisfy the following mathematical conditions:

∫ t

0π2

s ds < ∞ and∫ t

0cs ds < ∞ for all t ≥ 0 a.s. (2.1)

Xt is the agent’s wealth process at time t, which evolves according to the following stochas-tic differential equation (SDE):

dXt =[rXt + πt(μ – r) – ct

]dt + σπt dBt , (2.2)

with an initial endowment X0 = x > 0.

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In this paper, we take into account the optimal consumption and portfolio selectionproblem of an agent with time-inconsistent preferences (for details, see Sect. 2.2 of Zouet al. [13]). Following Palacios-Huerta and Pérez-Kakabadse [9] and Zou et al. [13], weassume that the agent’s discount factor function D(t, s) is given by

D(t, s) =

⎧⎨⎩

e–ρ(s–t) if s ∈ [t, t + τ ),

βe–ρ(s–t) if s ∈ [t + τ ,∞),

where ρ > 0 and 0 < β ≤ 1. The discount factor function D(t, s) decays exponentially at aconstant discount rate ρ in the present interval [t, t + τ ), drops at time t + τ to a fraction β

of its level just prior to t + τ , and then decays exponentially at a rate ρ in the future interval[t + τ ,∞). The arrival of future is determined by τ which is stochastic and exponentiallydistributed with a constant intensity λ ≥ 0.

3 The optimization problem with a stochastic hyperbolic discountingNow we consider the following optimization problem. In this problem, the infinite-livedagent wants to maximize her/his expected discounted lifetime utility:

V (x) := sup(c,π )∈A(x)

E

[∫ ∞

tD(t, s)u(cs) ds

∣∣∣∣ Xt = x]

= sup(c,π )∈A(x)

E

[∫ t+τ

te–ρ(s–t)u(cs) ds + β

∫ ∞

t+τ

e–ρ(s–t)u(cs) ds∣∣∣∣ Xt = x

](3.1)

subject to the budget constraint (2.2). Here, A(x) is the class of all admissible controls(c,π ) at x, and u(·) is a constant relative risk aversion (CRRA) utility function given by

u(c) :=c1–γ

1 – γ,

where γ > 0 (γ �= 1) is the agent’s coefficient of relative risk aversion.

Definition 1 A control pair (c,π ) is said to be admissible for initial wealth Xt = x if thefollowing are satisfied:

(1) equation (2.1) is satisfied,(2) E[

∫ ∞t D(t, s)u–(cs) ds|Xt = x] < ∞, where ζ – := max(–ζ , 0).

We define Merton’s constant K such that

K := r +ρ – r

γ+

γ – 12γ 2 θ2,

where θ := (μ – r)/σ is the market price of risk. To guarantee the existence of the optimalsolution, we assume that Merton’s constant K is positive.

Remark 1 For later use, we define a quadratic equation:

g(m) :=12θ2m2 +

(ρ – r +

12θ2

)m – r = 0,

Shin and Roh Advances in Difference Equations (2019) 2019:211 Page 4 of 7

with two roots m+ > 0 and m– < –1. Also we have the following identities:

1ρ

(r –

12θ2m+

)=

m+

m+ + 1,

1ρ

(r –

12θ2m–

)=

m–

m– + 1. (3.2)

Remark 2 Note that

m– < –1γ

and γ m– + 1 < 0,

since g(–1/γ ) = –K < 0.

By the dynamic programming principle, the value function V (x) in (3.1) satisfies thefollowing Bellman equation (for details, refer to equation (20) in Sect. 4.2 of Zou et al.[13]):

ρV (x) + λ(1 – β)E[∫ ∞

0e–(ρ+λ)tu

(c∗

t)

dt]

= max(c,π )

[(rx + π (μ – r) – c

)V ′(x) +

12σ 2π2V ′′(x) + u(c)

], (3.3)

where c∗t is the optimal consumption of the optimization problem (3.1). In order to obtain

the closed-form of V (x), we conjecture that

V (x) = K–γ

Hx1–γ

1 – γ, (3.4)

where KH is constant and determined in the next theorem.

Theorem 1 Under the assumption

λ + γ K + (1 – γ )KH > 0, (3.5)

the marginal propensity to consume KH is determined implicitly by the equation

1K

+2λγ (1 – β)

(γ m+ + 1)(γ m– + 1)(λ + γ K + (1 – γ )KH )θ2 =1

KH. (3.6)

Proof Based on Karatzas et al. [6], we assume that the optimal consumption c∗ = c = C(x)is a function of wealth and X(·) = C–1(·), that is, X(c) = X(C(x)) = x. Then, from the first-order conditions (FOCs) of Bellman equation (3.3), we have

V ′(x) = c–γ , V ′′(x) = –γ c–γ –1 1X ′(c)

. (3.7)

Plugging the FOCs and equations (3.7) into equation (3.3) implies

ρV(X(c)

)+ λ(1 – β)f (c) = rc–γ X(c) +

12γ

θ2c1–γ X ′(c) +γ

1 – γc1–γ , (3.8)

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where f (c) :=∫ ∞

0 e–(ρ+λ)tE[u(c∗

t )] dt with optimal consumption c. Differentiating equation(3.8) with respect to c, we obtain the following non-homogeneous second-order ODE:

12γ

θ2c2X ′′(c) +(

r – ρ +1 – γ

2γθ2

)cX ′(c) – rγ X(c) + γ c – λ(1 – β)c1+γ f ′(c) = 0. (3.9)

Using the method of variation of parameters to equation (3.9), we can derive the solutionas follows:

X(c) =1K

c –2λ(1 – β)

(m+ – m–)θ2

{c–γ m+

∫ c

0zγ (m++1)f ′(z) dz

+ c–γ m–

∫ ∞

czγ (m–+1)f ′(z) dz

}. (3.10)

Substituting X(c) into (3.8) with using the identities in (3.2) implies

V (x) =1

1 – γ

1K

c1–γ –λ(1 – β)

ρf (c)

–2λ(1 – β)

(m+ – m–)θ2

{m+

m+ + 1c–γ (m++1)

∫ c

0zγ (m++1)f ′(z) dz

+m–

m– + 1c–γ (m–+1)

∫ ∞

czγ (m–+1)f ′(z) dz

}. (3.11)

From equation (3.4), we obtain the following relations with the optimal consumption andportfolio:

V ′(x) =1 – γ

xV (x), V ′′(x) = –

(1 – γ )γx2 V (x),

c∗ = KHX(c), π∗ =θ

σγX(c).

By using the above relations and Itô’s formula, we obtain the following equation:

dV (Xt) = V ′(Xt) dXt +12

V ′′(Xt)(dXt)2

=(

r +θ2

2γ– KH

)(1 – γ )V (Xt) dt

+θ

γ(1 – γ )V (Xt) dBt .

So we can calculate f (c) as follows:

f (c) =∫ ∞

0e–(ρ+λ)t

E[u(c∗

t)]

dt

=∫ ∞

0e–(ρ+λ)t

E[KHV (Xt)

]dt

= KHV (x)∫ ∞

0e{–(ρ+λ)+(1–γ )(r+ θ2

2γ –KH )}t dt

Shin and Roh Advances in Difference Equations (2019) 2019:211 Page 6 of 7

=c1–γ

1 – γ

∫ ∞

0e–{λ+γ K+(1–γ )KH }t dt

=1

λ + γ K + (1 – γ )KH

c1–γ

1 – γ, (3.12)

under assumption (3.5).Plugging equation (3.12) into equation (3.10) with using Remark 2 implies

X(c) =(

1K

+2λγ (1 – β)

(γ m+ + 1)(γ m– + 1)(λ + γ K + (1 – γ )KH )θ2

)c =

1KH

c. �

This result is exactly the same as the works of Palacios-Huerta and Pérez-Kakabadse[9] and Zou et al. [13]. So we omit the detailed analysis of the marginal propensity toconsumption KH .

Remark 3 If there is no hyperbolic discounting (β = 1), then the marginal propensity toconsume is equal to Merton’s constant, that is, KH = K .

Remark 4 Under assumption (3.5), equation (3.6) implies

KH > K .

This coincides with the result of Palacios-Huerta and Pérez-Kakabadse [9].

If we are able to conjecture the value function or optimal consumption policy, then wecan obtain the analytic value function as equation (3.11). It is different from the resultsof [9, 13]. It reveals that our method is useful to the portfolio selection problem with thevarious constraints.

4 Concluding remarksIn this paper we take into account a continuous-time optimal consumption and invest-ment problem with stochastic hyperbolic discounting. We suggest another mathemati-cal approach based on Karatzas et al. [6] which is different from the research works ofPalacios-Huerta and Pérez-Kakabadse [9] and Zou et al. [13]. We believe that this approachis more useful to derive the closed-form solution of the consumption and portfolio opti-mization problem with various constraints. We will consider the optimal consumptionand portfolio selection problem of an agent who receives constant labor income and willextend this problem with negative wealth constraints in the future work.

AcknowledgementsWe are indebted to an anonymous referee for valuable advices and useful comments, which have improved our paperessentially.

FundingYong Hyun Shin gratefully acknowledges the support of the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MSIP) [Grant No. NRF-2016R1A2B4008240] and Kum-Hwan Roh gratefully acknowledges thesupport of Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded bythe Korea government [Grant No. NRF-2017R1D1A3B03036548].

Competing interestsThe authors declare that they have no competing interests.

Shin and Roh Advances in Difference Equations (2019) 2019:211 Page 7 of 7

Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details1Department of Mathematics & Research Institute of Natural Sciences, Sookmyung Women’s University, Seoul, Republicof Korea. 2Department of Mathematics, Hannam University, Daejeon, Republic of Korea.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 27 November 2018 Accepted: 17 May 2019

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