Now we consider the following optimization problem. In this problem, the infinitelived agent wants to maximize her/his expected discounted lifetime utility:
$$ \begin{aligned}[b] V(x)&:=\sup_{(c,\pi )\in \mathcal{A}(x)} \mathbb{E} \biggl[ \int _{t} ^{\infty }D(t,s)u(c_{s})\,ds\biggm X_{t}=x \biggr] \\ &=\sup_{(c,\pi )\in \mathcal{A}(x)} \mathbb{E} \biggl[ \int _{t} ^{t+\tau } e^{\rho (st)}u(c_{s})\,ds+ \beta \int _{t+\tau } ^{\infty }e ^{\rho (st)}u(c_{s})\,ds\biggm X_{t}=x \biggr] \end{aligned} $$
(3.1)
subject to the budget constraint (2.2). Here, \(\mathcal{A}(x)\) is the class of all admissible controls \((c,\pi )\) at x, and \(u(\cdot )\) is a constant relative risk aversion (CRRA) utility function given by
$$ u(c):=\frac{c^{1\gamma }}{1\gamma }, $$
where \(\gamma >0\) (\(\gamma \ne 1\)) is the agent’s coefficient of relative risk aversion.
Definition 1
A control pair \((c,\pi )\) is said to be admissible for initial wealth \(X_{t}=x\) if the following are satisfied:

(1)
equation (2.1) is satisfied,

(2)
\({\mathbb{E} [\int _{t}^{\infty }D(t,s) u^{}(c_{s})\,ds \vert X_{t}=x ]<\infty }\), where \(\zeta ^{}:=\max (\zeta ,0)\).
We define Merton’s constant K such that
$$ K:=r+\frac{\rho r}{\gamma }+\frac{\gamma 1}{2\gamma ^{2}}\theta ^{2}, $$
where \(\theta :=(\mu r)/\sigma \) is the market price of risk. To guarantee the existence of the optimal solution, we assume that Merton’s constant K is positive.
Remark 1
For later use, we define a quadratic equation:
$$ g(m):=\frac{1}{2}\theta ^{2}m^{2}+ \biggl(\rho r+ \frac{1}{2}\theta ^{2} \biggr)mr=0, $$
with two roots \(m_{+}>0\) and \(m_{}<1\). Also we have the following identities:
$$ \frac{1}{\rho } \biggl(r\frac{1}{2}\theta ^{2} m_{+} \biggr)=\frac{m _{+}}{m_{+}+1}, \qquad \frac{1}{\rho } \biggl(r \frac{1}{2}\theta ^{2} m_{} \biggr)=\frac{m _{}}{m_{} +1} . $$
(3.2)
Remark 2
Note that
$$ m_{}< \frac{1}{\gamma }\quad \text{and}\quad \gamma m_{}+1< 0, $$
since \(g(1/\gamma )=K<0\).
By the dynamic programming principle, the value function \(V(x)\) in (3.1) satisfies the following Bellman equation (for details, refer to equation (20) in Sect. 4.2 of Zou et al. [13]):
$$ \begin{aligned}[b] & \rho V(x)+\lambda (1\beta )\mathbb{E} \biggl[ \int _{0}^{\infty }e^{( \rho +\lambda )t} u\bigl(c^{*}_{t} \bigr)\,dt \biggr] \\ &\quad =\max_{(c,\pi )} \biggl[ \bigl(rx+\pi (\mu r)c \bigr)V'(x)+ \frac{1}{2}\sigma ^{2}\pi ^{2}V''(x)+u(c) \biggr], \end{aligned} $$
(3.3)
where \(c^{*}_{t}\) is the optimal consumption of the optimization problem (3.1). In order to obtain the closedform of \(V(x)\), we conjecture that
$$ V(x)=K_{H}^{\gamma }\frac{x^{1\gamma }}{1\gamma }, $$
(3.4)
where \(K_{H}\) is constant and determined in the next theorem.
Theorem 1
Under the assumption
$$ \lambda +\gamma K+(1\gamma )K_{H}>0, $$
(3.5)
the marginal propensity to consume
\(K_{H}\)
is determined implicitly by the equation
$$ \frac{1}{K}+\frac{2\lambda \gamma (1\beta )}{(\gamma m_{+}+1)(\gamma m_{}+1) (\lambda +\gamma K+(1\gamma )K_{H} )\theta ^{2}}=\frac{1}{K _{H}}. $$
(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 \(X(\cdot )=C^{1}(\cdot )\), that is, \(X(c)=X(C(x))=x\). Then, from the firstorder conditions (FOCs) of Bellman equation (3.3), we have
$$ V'(x)=c^{\gamma },\qquad V''(x)= \gamma c^{\gamma 1}\frac{1}{X'(c)}. $$
(3.7)
Plugging the FOCs and equations (3.7) into equation (3.3) implies
$$ \rho V\bigl(X(c)\bigr)+\lambda (1\beta )f(c)=rc^{\gamma }X(c)+ \frac{1}{2 \gamma }\theta ^{2} c^{1\gamma }X'(c)+ \frac{\gamma }{1\gamma }c^{1 \gamma }, $$
(3.8)
where \({f(c):=\int _{0}^{\infty }e^{(\rho +\lambda )t} \mathbb{E}[u(c^{*} _{t})]\,dt}\) with optimal consumption c. Differentiating equation (3.8) with respect to c, we obtain the following nonhomogeneous secondorder ODE:
$$ \frac{1}{2\gamma }\theta ^{2}c^{2}X''(c)+ \biggl(r\rho +\frac{1\gamma }{2\gamma }\theta ^{2} \biggr)c X'(c)r\gamma X(c)+\gamma c\lambda (1 \beta )c^{1+\gamma }f'(c)=0. $$
(3.9)
Using the method of variation of parameters to equation (3.9), we can derive the solution as follows:
$$\begin{aligned} X(c) =&\frac{1}{K}c\frac{2\lambda (1\beta )}{(m_{+}m_{})\theta ^{2}} \biggl\{ c^{\gamma m_{+}} \int _{0}^{c} z^{\gamma (m_{+}+1)}f'(z)\,dz \\ &{}+c ^{\gamma m_{}} \int ^{\infty }_{c}z^{\gamma (m_{}+1)}f'(z)\,dz \biggr\} . \end{aligned}$$
(3.10)
Substituting \(X(c)\) into (3.8) with using the identities in (3.2) implies
$$ \begin{aligned}[b] V(x) ={}&\frac{1}{1\gamma } \frac{1}{K}c^{1\gamma }\frac{\lambda (1 \beta )}{\rho }f(c) \\ &{} \frac{2\lambda (1\beta )}{(m_{+}m_{})\theta ^{2}} \biggl\{ \frac{m _{+}}{m_{+}+1}c^{\gamma (m_{+}+1)} \int _{0}^{c} z^{\gamma (m_{+}+1)}f'(z)\,dz \\ &{} +\frac{m_{}}{m_{}+1}c^{\gamma (m_{}+1)} \int ^{\infty }_{c}z ^{\gamma (m_{}+1)}f'(z)\,dz \biggr\} . \end{aligned} $$
(3.11)
From equation (3.4), we obtain the following relations with the optimal consumption and portfolio:
$$\begin{aligned} &V'(x)=\frac{1\gamma }{x}V(x),\qquad V''(x)= \frac{(1\gamma )\gamma }{x^{2}}V(x), \\ &c^{*}=K_{H} X(c),\qquad \pi ^{*}= \frac{\theta }{\sigma \gamma }X(c). \end{aligned}$$
By using the above relations and Itô’s formula, we obtain the following equation:
$$ \begin{aligned} dV(X_{t}) ={}&V'(X_{t})\,dX_{t}+ \frac{1}{2}V''(X_{t}) (dX_{t})^{2} \\ ={}& \biggl(r+\frac{\theta ^{2}}{2\gamma }K_{H} \biggr) (1\gamma )V(X _{t})\,dt\\ &{}+\frac{\theta }{\gamma }(1\gamma )V(X_{t})\,dB_{t}. \end{aligned} $$
So we can calculate \(f(c)\) as follows:
$$\begin{aligned} f(c) &= \int _{0}^{\infty }e^{(\rho +\lambda )t} \mathbb{E}\bigl[u \bigl(c^{*}_{t}\bigr)\bigr]\,dt \\ &= \int _{0}^{\infty }e^{(\rho +\lambda )t} \mathbb{E} \bigl[K_{H} V(X_{t})\bigr]\,dt \\ &=K_{H} V(x) \int _{0}^{\infty }e^{ \{(\rho +\lambda )+(1 \gamma )(r+\frac{\theta ^{2}}{2\gamma }K_{H}) \}t}\,dt \\ &=\frac{c^{1\gamma }}{1\gamma } \int _{0}^{\infty }e^{ \{\lambda + \gamma K+(1\gamma )K_{H} \}t}\,dt \\ &=\frac{1}{\lambda +\gamma K+(1\gamma )K_{H}}\frac{c^{1\gamma }}{1 \gamma }, \end{aligned}$$
(3.12)
under assumption (3.5).
Plugging equation (3.12) into equation (3.10) with using Remark 2 implies
$$ X(c)= \biggl(\frac{1}{K}+\frac{2\lambda \gamma (1\beta )}{(\gamma m _{+}+1)(\gamma m_{}+1) (\lambda +\gamma K+(1\gamma )K_{H} ) \theta ^{2}} \biggr)c=\frac{1}{K_{H}}c. $$
□
This result is exactly the same as the works of PalaciosHuerta and PérezKakabadse [9] and Zou et al. [13]. So we omit the detailed analysis of the marginal propensity to consumption \(K_{H}\).
Remark 3
If there is no hyperbolic discounting (\(\beta =1\)), then the marginal propensity to consume is equal to Merton’s constant, that is, \(K_{H}=K\).
Remark 4
Under assumption (3.5), equation (3.6) implies
This coincides with the result of PalaciosHuerta and PérezKakabadse [9].
If we are able to conjecture the value function or optimal consumption policy, then we can obtain the analytic value function as equation (3.11). It is different from the results of [9, 13]. It reveals that our method is useful to the portfolio selection problem with the various constraints.