Optimal Retirement in a General Market Environment

Abstract

We study an optimal retirement, consumption/portfolio selection problem of an economic agent in a non-Markovian environment. We show that under a suitable condition the optimal retirement decision is to retire when the individual’s wealth reaches a threshold level. We express the value and the optimal strategy by using the strong solution of the backward stochastic partial differential variational inequality (BSPDVI) associated with the dual problem. We derive properties of the value function and the optimal strategy by analyzing the strong solution and the free boundary of the BSPDVI. We also make a methodological contribution by proposing an approach to investigate properties of the strong solution and the stochastic free boundary of BSPDVI by combining a probabilistic method and the theory of backward stochastic partial differential equations (BSPDEs).

Appendices

A Few Existing Results About BSPDVI

For convenience of the reader we state the generalized Itô-Kunita-Wentzell’s formula, and results about BSPDE (refer to [8, 31]) and BSPDVI (refer to [21]) which were used in the paper.

We provide the generalized Itô–Kunita–Wentzell’s formula (refer to Theorem 3.1 in [31]) in the following⁹,

Lemma 3

Suppose that the random function $v:\varOmega \times [\,0,T\,]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ satisfies the following properties: v(x) is a continuous semimartingale and takes the following form

$$\begin{aligned} v_s(x)&=v_t(x)+\int _t^s f_u(x)\,du+\sum \limits _{i=1}^{N_2}\int _t^s w_{i,u}(x)\,dB_{i,u}\\&\text{ a.e. }\;x\in {\mathbb {R}}\; \text{ for } \text{ every }\;s\in [\,t,T\,]\; \text{ a.s. } \text{ in }\;\varOmega . \end{aligned}$$

And $v\in {\mathbb {H}}^{2,\,2}(D),\,w\in {\mathbb {H}}^{1,\,2}(D),\,f\in {\mathbb {H}}^{0,\,2}(D)$ for any compact subset D of ${\mathbb {R}}^+$. Let X be governed by (3.9), and Assumptions 2 and 3 be satisfied. Then we have

$$\begin{aligned} e^{-\beta (s-t)}v_s(X_s)&=v_t(X_t)+\int _t^s\,e^{-\beta (u-t)}\left( \,\mathcal{L} v_u +\sum \limits _{i=1}^{N_2}{{\mathcal {M}}}_i w_{i,u}+f_u\right) \,(X_u)\,du \\&\quad +\,\int _t^s\,e^{-\beta (u-t)}\Bigg [\,\sum \limits _{i=1}^{N_2}(w_{i,u}+\mathcal{M}_iv_u)\,(X_u)\,dB_{i,u}\\&\quad -\partial _xv_u(X_u)X_u\Big (\,\theta ^1_{u}\Big )^{\top }\,dW_u\,\Bigg ], \end{aligned}$$

where the differential operator $\mathcal{L}$ and ${{\mathcal {M}}}$ are defined in (3.11).

Remark 6

From the above integral representation of $e^{-\beta (\cdot -t)}v_\cdot (X_\cdot )$, we know that the stochastic process $v_\cdot (X_\cdot )$ has a continuous version.

We will now state results about a BSPDVI, which takes the following form

$$\begin{aligned} \left\{ \begin{array}{l} dv_t=-\left( \widetilde{{{\mathcal {L}}}} v_t+\sum \limits _{i=1}^{N_2}\widetilde{{{\mathcal {M}}}}_i w_{i,t}+f_t\right) \,dt +\sum \limits _{i=1}^{N_2}w_{i,t}\,dB_{i,t}\quad \text{ if }\;\;v_t>{\underline{v}}\,_t\,; \\ dv_t\le -\left( \widetilde{{{\mathcal {L}}}} v_t+\sum \limits _{i=1}^{N_2}\widetilde{{{\mathcal {M}}}}_i w_{i,t}+f_t\right) \,dt +\sum \limits _{i=1}^{N_2}w_{i,t}\,dB_{i,t}\quad \text{ if }\;\;v_t={\underline{v}}\,_t\,; \\ v_T(x)=\varphi (x)\,, \end{array} \right. \end{aligned}$$

(A.1)

where the differential operator $\widetilde{{{\mathcal {L}}}}$ and $\widetilde{{{\mathcal {M}}}}$ are defined in (4.3).

First, we state a result about the existence and uniqueness of the strong solution to the BSPDVI (see Theorem 5.4 in [21]).

Lemma 4

Let Assumptions 2 and 3 be satisfied, $\,f\in {\mathbb {H}}^{0,\,2}_\lambda ,\, \varphi \in {\mathbb {L}}^{1,\,2}_\lambda ,\,{\underline{v}}\,\in {\mathbb {H}}_\lambda ^{2,\,2}$ with some nonnegative number $\lambda$. Moreover, the lower obstacle ${\underline{v}}\,$ is a continuous semimartingale of the following form

$$\begin{aligned} d{\underline{v}}\,_t=-{\underline{g}}\,_t\,dt+\sum \limits _{i=1}^{N_2}{\underline{w}}\,_{i,t}\,dB_{i,t} \end{aligned}$$

with some $({\underline{w}}\,,\,{\underline{g}}\,)\in {\mathbb {H}}_\lambda ^{1,\,2}\times {\mathbb {H}}_\lambda ^{0,\,2}$, and ${\underline{v}}\,_T\le \varphi$. Then BSPDVI (A.1) has a unique strong solution $(v,w,k^+)\in {\mathbb {H}}_\lambda ^{2,\,2}\times {\mathbb {H}}_\lambda ^{1,\,2} \times {\mathbb {H}}_\lambda ^{0,\,2}$. Moreover, $v\in {\mathbb {S}}_\lambda ^{1,\,2}$.

Next, we provide a comparison theorem for BSPDVI (refer to Theorem 5.2 in [21]).

Lemma 5

Let the assumptions in Lemma 4 be satisfied. Let $(v_i,w^i,k_i^+)$ be the strong solution to BSPDVI (A.1) associated with $(f_i,\,\varphi _i,\,{\underline{v}}\,_i)$ for $i=1,2$. If $f_1\ge f_2,\,\varphi _1\ge \varphi _2,$ and ${\underline{v}}\,_1\ge {\underline{v}}\,_2$, then $v_1\ge v_2$ a.e. in $\varOmega \times [\,0,T\,]\times {\mathbb {R}}$.

Consider the following BSPDE,

$$\begin{aligned} \left\{ \begin{array}{l} dv_t=-\left( \widetilde{{{\mathcal {L}}}} v_t+\sum \limits _{i=1}^{N_2}\widetilde{{{\mathcal {M}}}}_i w_{i,t}+f_t\right) \,dt +\sum \limits _{i=1}^{N_2}w_{i,t}\,dB_{i,t}; \\ v_T(x)=\varphi (x)\,. \end{array} \right. \end{aligned}$$

(A.2)

In view of the results for BSPDE (refer to Lemmas 2.2 and 5.1 in [31], or Theorem 5.3 and Corollary 3.4 in [8]), we can conclude the following results for BSPDE¹⁰.

Lemma 6

Let Assumptions 2 and 3 be satisfied, $\,f\in {\mathbb {H}}^{0,\,2}_\lambda ,\, \varphi \in {\mathbb {L}}^{1,\,2}_\lambda$ with some nonnegative number $\lambda$. Then BSPDE (A.2) has a unique strong solution $(v,w)\in {\mathbb {H}}_\lambda ^{2,\,2}\times {\mathbb {H}}_\lambda ^{1,\,2}$. Moreover, $v\in {\mathbb {S}}_\lambda ^{1,\,2}$.

Lemma 7

Let the assumptions in Lemma 6 be satisfied. Assume that $(v_i,w^i)$ be the strong solution to BSPDE (A.2) associated with $(f_i,\,\varphi _i)$ for $i=1,2$. If $f_1\ge f_2$ and $\varphi _1\ge \varphi _2$, then $v_1\ge v_2$ a.e. in $\varOmega \times [\,0,T\,]\times {\mathbb {R}}$.

Finally, we give the result about the relationship between the strong solution ${\widehat{P}}$ of BSPDVI (4.4) and the value of the corresponding optimal stopping problem (refer to Theorem 5.4 in [21])¹¹.

Lemma 8

Let Assumptions 2 and 3 be satisfied. Suppose that $\widetilde{{\underline{P}}}$ and ${\widehat{P}}$ are the strong solutions to BSPDE (4.2) and BSPDVI (4.4), respectively. The terminal value and lower obstacle in BSPDVI (4.4), $\widehat{{\underline{P}}}_t({\widetilde{x}})=\widetilde{{\underline{P}}}_t({\widetilde{x}})-e^{{\widetilde{x}}}\mathcal{Y}_t$. Let $\,{\widetilde{R}}_1,{\widehat{R}}_1\in {\mathbb {H}}^{0,\,2}_\lambda ,\,{\widetilde{R}}\,_2\in {\mathbb {L}}^{1,\,2}_\lambda$ be satisfied with some nonnegative number $\lambda$. Moreover, suppose that $\widehat{{\underline{P}}}\,({\widetilde{X}}^{t,{\widetilde{x}}}),\, {\widehat{P}}\,({\widetilde{X}}^{t,{\widetilde{x}}})\in {{\mathcal {S}}}^2,\,{\widehat{R}}_{1}({\widetilde{X}}^{t,{\widetilde{x}}})\in \mathcal{L}^2$, where ${\widetilde{X}}^{t,{\widetilde{x}}}=\log X^{t,x}$¹². Then

$$\begin{aligned} \widetilde{{\underline{P}}}\,_t({\widetilde{x}})={\mathbb {E}}\left[ \, \int _t^T e^{-\beta (s-t)}{\widetilde{R}}_{1}({\widetilde{X}}_s^{t,{\widetilde{x}}})ds +e^{-\beta (T-t)}{\widetilde{R}}_2({\widetilde{X}}_T^{t,{\widetilde{x}}})\,\Big |\mathcal{F}_t\,\right] , \end{aligned}$$

and ${\widehat{P}}_t({\widetilde{x}})$ is the value of the following optimal stopping problem,

$$\begin{aligned} {\widehat{P}}_t({\widetilde{x}})=\mathop {\mathrm{ess.sup}}_{\tau \in \mathcal{U}_{t,T}}{\mathbb {E}}\left[ \, \int _t^{\tau \wedge T} e^{-\beta (s-t)}{\widehat{R}}_{1,s}({\widetilde{X}}_s^{t,{\widetilde{x}}})ds +e^{-\beta (\tau -t)}\widehat{{\underline{P}}}_{\tau \wedge T}({\widetilde{X}}_{\tau \wedge T}^{t,{\widetilde{x}}})\,\Big |\mathcal{F}_t\,\right] , \end{aligned}$$

and its optimal stopping time $\tau ^*$ can be described as

$$\begin{aligned} \tau ^*\triangleq \inf \left\{ s\in [\,t,T\,]:{\widehat{P}}_s({\widetilde{X}}^{t,{\widetilde{x}}}_s) =\underline{{\widehat{P}}}\,_s({\widetilde{X}}^{t,{\widetilde{x}}}_s)\right\} \wedge T. \end{aligned}$$

Properties of $\varDelta R$

In this section, we deduce properties about $\varDelta R$ defined in (5.3), which is important to derive the properties of the optimal retirement boundary.

Lemma 9

$\mathrm{ess.sup}\{\varDelta R_t({\widetilde{x}}):(\omega ,t)\in \varOmega \times [\,0,T\,]\}<-e^{{\widetilde{x}}}$ provided ${\widetilde{x}}$ is small enough.

Proof

We prove the conclusion by considering the property of $\varDelta R_t({\widetilde{x}})$ as ${\widetilde{x}}\rightarrow -\infty$ in six different cases.

1. Consider the case of $0<\rho <1$ and $0<\gamma \ne 1$. As ${\widetilde{x}}\rightarrow -\infty$, i.e., $x\rightarrow 0^+$, then $c^*=\mathcal{J}_{U_1}(x;l)\rightarrow +\infty$, and

$$\begin{aligned} x&=\partial _cU_1(c^*,l)\\&=\alpha (c^*)^{\rho -1}[\,\alpha (c^*)^{\rho } +(1-\alpha ) l^{\rho }\,]^{1-\gamma -\rho \over \rho } \\&= \alpha ^{1-\gamma \over \rho } (c^*)^{-\gamma }\left[ \,1 +{1-\alpha \over \alpha }{l^{\rho }\over (c^*)^\rho } \,\right] ^{1-\gamma -\rho \over \rho } \\&= \alpha ^{1-\gamma \over \rho } (c^*)^{-\gamma }\left[ \,1 +{1-\alpha \over \alpha }{1-\gamma -\rho \over \rho }\,l^{\rho }(c^*)^{-\rho }+o\left( (c^*)^{-\rho }\right) \,\right] . \end{aligned}$$

So, we deduce that as $x\rightarrow 0^+$,

$$\begin{aligned} \mathcal{J}_{U_1}(x;l)&=c^*\nonumber \\&=\alpha ^{1-\gamma \over \rho \gamma } x^{-1\over \gamma }\left[ \,1 +{1-\alpha \over \alpha }{1-\gamma -\rho \over \rho }\,l^{\rho }(c^*)^{-\rho }+o\left( (c^*)^{-\rho }\right) \,\right] ^{1\over \gamma }\nonumber \\&=\alpha ^{1-\gamma \over \rho \gamma } x^{-1\over \gamma }\left[ \,1 +{1-\alpha \over \alpha }{1-\gamma -\rho \over \rho \gamma }\,l^{\rho }(c^*)^{-\rho }+o\left( (c^*)^{-\rho }\right) \,\right] \nonumber \\&=\alpha ^{1-\gamma \over \rho \gamma } x^{-1\over \gamma }\left[ \,1 +{1-\gamma -\rho \over \rho \gamma }(1-\alpha )\,l^{\rho }\alpha ^{-1\over \gamma } x^{\rho \over \gamma }+o\left( x^{\rho \over \gamma }\right) \,\right] . \end{aligned}$$

(B.1)

And by computation we can show that as $x\rightarrow 0^+$,

$$\begin{aligned} U_1(c^*,l)-c^*x&= {\alpha ^{1-\gamma \over \rho } (c^*)^{1-\gamma }\over 1-\gamma } \left[ \,1 +{1-\alpha \over \alpha }{l^{\rho }\over (c^*)^\rho } \,\right] ^{1-\gamma \over \rho }-c^*x \\&= {\alpha ^{1-\gamma \over \rho } (c^*)^{1-\gamma }\over 1-\gamma } \left[ \,1+{1-\alpha \over \alpha }{1-\gamma \over \rho }\,l^{\rho }(c^*)^{-\rho }+o\left( (c^*)^{-\rho }\right) \,\right] \\&\quad -c^*x \\&= {\alpha ^{1-\gamma \over \rho } \over 1-\gamma } \alpha ^{(1-\gamma )^2\over \rho \gamma } x^{\gamma -1\over \gamma }\bigg [\,1 +{1-\gamma -\rho \over \rho \gamma }(1-\alpha )(1-\gamma )\,l^{\rho }\alpha ^{-1\over \gamma } x^{\rho \over \gamma }\\&\quad +o\left( x^{\rho \over \gamma }\right) \,\bigg ] \left[ \,1+{1-\alpha \over \alpha }{1-\gamma \over \rho }\,l^{\rho }\alpha ^{\gamma -1\over \gamma } x^{\rho \over \gamma }+o\left( x^{\rho \over \gamma }\right) \,\right] \\&\quad -\,\alpha ^{1-\gamma \over \rho \gamma } x^{\gamma -1\over \gamma }\left[ \,1 +{1-\gamma -\rho \over \rho \gamma }(1-\alpha )\,l^{\rho }\alpha ^{-1\over \gamma } x^{\rho \over \gamma }+o\left( x^{\rho \over \gamma }\right) \,\right] \\&= \alpha ^{1-\gamma \over \rho \gamma }x^{\gamma -1\over \gamma }\left[ \,{\gamma \over 1-\gamma } +{1-\alpha \over \rho }\,l^{\rho }\alpha ^{-1\over \gamma } x^{\rho \over \gamma }+o\left( x^{\rho \over \gamma }\right) \,\right] . \end{aligned}$$

Note that ${\overline{x}}_t$ has a positive lower bound. Hence, we have that as $x\rightarrow 0^+$,

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&= {\widehat{U}}_{1,t}(x)-{{\widetilde{U}}_{1,t}}(x)+{L_t}w_tx\\&= U_1(\mathcal{J}_{U_1}(x;{{\overline{L}}_t}),{{\overline{L}}_t})-x(\mathcal{J}_{U_1}(x;{{\overline{L}}_t})+ {{\overline{L}}_t}w_t)-U_1(\mathcal{J}_{U_1}(x;{L_t}),{L_t})\\&\quad +x\mathcal{J}_{U_1}(x;{L_t})+{L_t}w_tx \\&= {1-\alpha \over \rho }\alpha ^{1-\gamma -\rho \over \rho \gamma }x^{1-{1-\rho \over \gamma }}\left[ \, {{\overline{L}}_t}^{\rho } -{L_t}^\rho +o\left( 1\right) \,\right] +({L_t}-{{\overline{L}}_t})w_tx \\&= {1-\alpha \over \rho }\alpha ^{1-\gamma -\rho \over \rho \gamma }x^{1-{1-\rho \over \gamma }}\left[ \, {{\overline{L}}_t}^{\rho } -{L_t}^\rho +o\left( 1\right) \,\right] \\&<-x\\&=-e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough, where we have used Assumption 2.

2. In the case of $0<\rho <1$ and $\gamma =1$, (B.1) still holds. And by computation we can show that as $x\rightarrow 0^+$,

$$\begin{aligned} U_1(c^*,l)-c^*x&= {\log \alpha \over \rho } +\log c^*+{1\over \rho }\log \left[ \,1 +{1-\alpha \over \alpha }{l^{\rho }\over (c^*)^\rho } \,\right] -c^*x \\&= {\log \alpha \over \rho } -\log x -{1-\alpha \over \alpha }\,l^{\rho } x^\rho +{1-\alpha \over \rho \alpha }l^{\rho } x^{\rho } \\&\quad -\left[ \,1-{1-\alpha \over \alpha }l^{\rho }x^\rho \,\right] +o\left( x^{\rho }\right) \\&= {\log \alpha \over \rho } -\log x -1+{1-\alpha \over \rho \alpha }l^{\rho }x^{\rho }+o\left( x^{\rho }\right) , \end{aligned}$$

and

$$\begin{aligned} \varDelta R_t({\widetilde{x}}) = {1-\alpha \over \rho \alpha }x^\rho \left[ \, {{\overline{L}}_t}^{\rho } -{L_t}^\rho +o\left( 1\right) \,\right] +({L_t}-{{\overline{L}}_t})w_tx <-x=-e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough.

3. In the case of $\rho =0$ and $0<\gamma \ne 1$, the proof proceeds similarly to the above. Since

$$\begin{aligned} x=\partial _cU_1(c^*,l)=\alpha (c^*)^{\alpha (1-\gamma )-1}l^{(1-\alpha )(1-\gamma )}, \end{aligned}$$

we have that

$$\begin{aligned} \mathcal{J}_{U_1}(x;l)=c^*=\alpha ^{1\over 1-\alpha (1-\gamma )}l^{(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )} x^{1\over \alpha (1-\gamma )-1}. \end{aligned}$$

(B.2)

And by computation we can show that

$$\begin{aligned} U_1(c^*,l)-c^*x=\left[ \,{1\over 1-\gamma }\alpha ^{\alpha (1-\gamma )\over 1-\alpha (1-\gamma )} -\alpha ^{\alpha \over 1-\alpha (1-\gamma )}\,\right] \,l^{(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )} x^{\alpha (1-\gamma )\over \alpha (1-\gamma )-1}. \end{aligned}$$

Note that if $\gamma >1$, then

$$\begin{aligned} 0<{\alpha (1-\gamma )\over \alpha (1-\gamma )-1}<1,\;\; {1\over 1-\gamma }\alpha ^{\alpha (1-\gamma )\over 1-\alpha (1-\gamma )} -\alpha ^{\alpha \over 1-\alpha (1-\gamma )}<0,\;\; {(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )}<0, \end{aligned}$$

and if $0<\gamma <1$, then

$$\begin{aligned} {\alpha (1-\gamma )\over \alpha (1-\gamma )-1}<0,~ {1\over 1-\gamma }\left( \,\alpha ^{\alpha \over 1-\alpha (1-\gamma )}\,\right) ^{1-\gamma } -\alpha ^{\alpha \over 1-\alpha (1-\gamma )}>0,~ {(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )}>0. \end{aligned}$$

We can conclude that $U_1(c^*,l)-c^*x$ is strictly increasing with respect to l, and

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&= \left[ \,{1\over 1-\gamma }\alpha ^{\alpha (1-\gamma )\over 1-\alpha (1-\gamma )} -\alpha ^{\alpha \over 1-\alpha (1-\gamma )}\,\right] \,\left[ \,{{\overline{L}}_t}^{(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )} -{L_t}^{(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )} +o\left( 1\right) \,\right] \\&\quad \times x^{\alpha (1-\gamma )\over \alpha (1-\gamma )-1} \\&< -x \\&=-e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough.

4. In the case of $\rho =0$ and $\gamma =1$, (B.2) still holds and similarly to the above we have

$$\begin{aligned} U_1(c^*,l)-c^*x&= \alpha \log \alpha -\alpha \log x+(1-\alpha )\log l-\alpha , \\ \varDelta R_t({\widetilde{x}})&= (1-\alpha )(\log {{\overline{L}}_t}-\log {L_t})+({L_t}-{{\overline{L}}_t})w_tx <-x=-e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough.

5. In the case of $\rho <0$ and $0<\gamma \ne 1$, similarly to the above, as ${\widetilde{x}}\rightarrow -\infty$, i.e., $x\rightarrow 0^+$, $c^*=\mathcal{J}_{U_1}(x;l)\rightarrow +\infty$, and

$$\begin{aligned} x&= \partial _cU_1(c^*,l) \\&=\alpha (c^*)^{\rho -1}[\,\alpha (c^*)^{\rho } +(1-\alpha ) l^{\rho }\,]^{1-\gamma -\rho \over \rho } \\&= \alpha (c^*)^{\rho -1}[\,o(1) +(1-\alpha ) l^{\rho }\,]^{1-\gamma -\rho \over \rho } \\&=\alpha (1-\alpha )^{1-\gamma -\rho \over \rho }l^{1-\gamma -\rho }(c^*)^{\rho -1}[\,1+o(1)\,]. \end{aligned}$$

So we deduce that as $x\rightarrow 0^+$,

$$\begin{aligned} \mathcal{J}_{U_1}(x;l)=c^* =\alpha ^{1\over 1-\rho } (1-\alpha )^{1-\gamma -\rho \over \rho (1-\rho )}l^{1-\gamma -\rho \over 1-\rho }x^{1\over \rho -1}[\,1+o(1)\,]. \end{aligned}$$

(B.3)

And by computation we can show that as $x\rightarrow 0^+$,

$$\begin{aligned} U_1(c^*,l)-c^*x&= {(1-\alpha )^{1-\gamma \over \rho } l^{1-\gamma }\over 1-\gamma } \left[ \,1+o(1) \,\right] ^{1-\gamma \over \rho } \\&\quad -\alpha ^{1\over 1-\rho } (1-\alpha )^{1-\gamma -\rho \over \rho (1-\rho )}l^{1-\gamma -\rho \over 1-\rho }x^{\rho \over \rho -1}[\,1+o(1)\,] \\&= {(1-\alpha )^{1-\gamma \over \rho } l^{1-\gamma }\over 1-\gamma } \left[ \,1+o(1) \,\right] . \end{aligned}$$

Hence, we know that as $x\rightarrow 0^+$,

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&= {(1-\alpha )^{1-\gamma \over \rho } \left( \,{{\overline{L}}_t}^{1-\gamma }-{L_t}^{1-\gamma }\,\right) \over 1-\gamma } \left[ \,1+o(1) \,\right] +({L_t}-{{\overline{L}}_t})w_tx \\&< -x \\&=-e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough.

6. In the case of $\rho <0$ and $\gamma =1$, (B.3) still holds. Similarly to the above, as $x\rightarrow 0^+$,

$$\begin{aligned} U_1(c^*,l)-c^*x&= \left[ \,{\log (1-\alpha )\over \rho } +\log l+o(1)\,\right] \\&\quad -\left( \,{\alpha \over 1-\alpha }\,\right) ^{1\over 1-\rho } l^{\rho \over \rho -1}x^{1+{1\over \rho -1}}[\,1+o(1)\,] \\&= {\log (1-\alpha )\over \rho } +\log l+o(1), \end{aligned}$$

and

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&= \left[ \,\log {{\overline{L}}_t}-\log {L_t} +o\left( 1\right) \,\right] +({L_t}-{{\overline{L}}_t})w_tx \\&<-(1+{L_t}w_t)x \\&=-(1+{L_t}w_t)e^{{\widetilde{x}}} \end{aligned}$$

provided x is enough near zero, i.e., ${\widetilde{x}}$ is small enough. Then we have finished the proof. $\square$

Lemma 10

$\varDelta R_t({\widetilde{x}})>0$ for any ${\widetilde{x}}>\log {\overline{x}}_t$. Moreover, $\mathrm{ess.inf}\{\varDelta R_t({\widetilde{x}}):(\omega ,t)\in \varOmega \times [\,0,T\,]\}>\epsilon e^{{\widetilde{x}}}$ provided ${\widetilde{x}}$ is large enough, where $\epsilon$ is a positive constant independent of ${\widetilde{x}}$.

Proof

For any ${\widetilde{x}}>\log {\overline{x}}_t$, we have that

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&= A_t(x)-\sup \limits _{c\ge 0}\left\{ \,U_1(c,{L_t})-xc\,\right\} +{L_t}w_tx \\&= \sup \limits _{c\ge 0,l\ge 0}\left\{ \,U_1(c,l)-x(c+ w_t l)\,\right\} -\sup \limits _{c\ge 0}\left\{ \,U_1(c,{L_t})-xc\,\right\} +Lw_tx \\&> \sup \limits _{c\ge 0}\left\{ \,U_1(c,{L_t})-x(c+ {L_t}w_t)\,\right\} -\sup \limits _{c\ge 0}\left\{ \,U_1(c,{L_t})-xc\,\right\} +{L_t}w_tx \\&=0. \end{aligned}$$

Moreover, if ${\widetilde{x}}$ is large enough, we know that

$$\begin{aligned} \varDelta R_t({\widetilde{x}})&\ge U_1({c^*_t},{{\overline{L}}_t})-U_1({c^*_t},{L_t})+({L_t}-{{\overline{L}}_t})w_tx \\&\ge ({L_t}-{{\overline{L}}_t})x\left[ \,w_t-{\partial _l U_1({c^*_t},{{\overline{L}}_t})\over x}\,\right] ,\nonumber \end{aligned}$$

(B.4)

where ${c^*_t}=\mathcal{J}_{U_1}(x;{L_t})$, and we have used the fact $U_1$ is concave with respect to l. Applying the same method as in the proof of Lemma 9, we can show that as $x\rightarrow +\infty$,

$$\begin{aligned} {c^*_t} = \left\{ \begin{array}{ll} \alpha ^{1\over 1-\rho }(1-\alpha )^{1-\gamma -\rho \over \rho (1-\rho )}{L_t}^{1-\gamma -\rho \over 1-\rho }x^{1\over \rho -1}\Big [\,1+o(1)\Big ],\;\;&{} 0<\rho<1; \\ \alpha ^{1\over 1-\alpha (1-\gamma )}{L_t}^{(1-\alpha )(1-\gamma )\over 1-\alpha (1-\gamma )} x^{1\over \alpha (1-\gamma )-1},\;\;&{}\rho =0; \\ \alpha ^{1-\gamma \over \rho \gamma }x^{-1\over \gamma }\Big [\,1+o(1)\Big ],\;\;&{}\rho <0. \end{array} \right. \end{aligned}$$

So we derive that as $x\rightarrow +\infty$,

$$\begin{aligned} {\partial _l U_1({c^*_t},{{\overline{L}}_t})\over x} = \left\{ \begin{array}{ll} (1-\alpha ) {{\overline{L}}_t}^{\rho -1}[\,\alpha ({c^*_t})^{\rho } +(1-\alpha ) {{\overline{L}}_t}^{\rho }\,]^{1-\gamma -\rho \over \rho }x^{-1}&{} \\ ~=o(1),\;\;&{}1>\rho \ne 0; \\ (1-\alpha )({c^*_t})^{\alpha (1-\gamma )}l^{(1-\alpha )(1-\gamma )-1}x^{-1} =o(1),\;\;&{}\rho =0. \end{array} \right. \end{aligned}$$

Hence, by (B.4), we conclude that $\varDelta R_t({\widetilde{x}})>\epsilon e^{{\widetilde{x}}}$ provided ${\widetilde{x}}$ is large enough, where $\epsilon$ is a positive constant. $\square$

Lemma 11

If $\gamma +\rho \le 1$, then $\partial _{{\widetilde{x}}}\varDelta R\ge 0$.

Proof

From the definition of $\varDelta R$, we can show that

$$\begin{aligned} e^{-{\widetilde{x}}}\partial _{{\widetilde{x}}} \varDelta R_t({\widetilde{x}})&= \partial _x{\widehat{U}}_{1,t}(x)-{\partial _x{\widetilde{U}}_{1,t}}(x)+{L_t}w_t\\&=(-\mathcal{J}_{U_1}(x;l^*_t)-l^*_tw_t)-(-\mathcal{J}_{U_1}(x;{L_t}))+{L_t}w_t \\&= \mathcal{J}_{U_1}(x;{L_t})-\mathcal{J}_{U_1}(x;l_t^*)+({L_t}-l_t^*)w_t, \\ l^*_t&={{\overline{L}}_t}\min \left\{ 1,\left( {{\overline{x}}_t\over x}\right) ^{1\over \gamma }\right\} \in (0,{{\overline{L}}_t}\,]. \end{aligned}$$

Moreover, it is easy to check that

$$\begin{aligned} \partial _{cl} U_1(c,l)=\left\{ \begin{array}{ll} (1-\gamma -\rho )\alpha (1-\alpha ) c^{\rho -1}l^{\rho -1}&{} \\ \quad \times [\,\alpha c^{\rho } +(1-\alpha ) l^{\rho }\,]^{1-\gamma -2\rho \over \rho }, &{}1>\rho \ne 0; \\ \alpha (1-\alpha )(1-\gamma )c^{\alpha (1-\gamma )-1}l^{(1-\alpha )(1-\gamma )-1}, &{}\rho =0. \end{array} \right. \end{aligned}$$

Since $1-\gamma -\rho \ge 0$, we have that $\partial _c U_1(c,l)$ is increasing with respect to l, and $\mathcal{J}_{U_1}(x;l)$ is increasing with respect to l by $\partial _{cc} U_1(c,l)<0$. Hence, $\partial _{{\widetilde{x}}} \varDelta R\ge 0$. $\square$