Closed-loop supply chain coordination through incentives with asymmetric information

Abstract

A closed-loop supply chain seeks to enhance the consumers’ environmental consciousness to increase both the profits and the return of past-sold products. Even though, firms have misaligned interests for closing the loop: while all firms exploit consumers environmental consciousness to increase sales, only manufacturers use it for appropriating of returns’ residual value. Starting from a benchmark (no-incentive) scenario where a manufacturer (M) is the leader and a retailer (R) is the follower, we develop two incentive games through a profit-sharing contract to align firms’ motivations for closing the loop. In both incentive games, the incentive takes the form of a share of profits that M transfers to R. Our question is how the sharing fraction should be determined to make both players economically better-off. The first incentive game assumes that R has no-information on the sharing parameter, which is determined by M after R sets her strategies; thus the incentive has an endogenous nature. In the second incentive game the sharing parameter is common knowledge and both players know its values before the game starts, thus the incentive has an exogenous nature. We find that an endogenous incentive is never more economically and environmentally convenient than a no-incentive game. In contrast, an exogenous incentive can make both players economically better-off inside specific sharing parameter ranges. Nevertheless, when other forces (e.g., competition or legislation) impose the adoption of a profit-sharing contract, M should supply an endogenous incentive when the exogenous share is either too high or too low.

Appendix

Proof of Proposition 1

We need to establish the existence of bounded and continuously differentiable value functions $V_M \left( G \right) ,V_R \left( G \right)$ such that there exists a unique solution G(t) to Eq. (1) and the HJB equations. The players’ HJBs in the benchmark scenario are:

$$\begin{aligned} \rho V_M= & {} \left( {\theta \sqrt{G}-\beta p} \right) \left( {\omega +K\sqrt{G}} \right) -\frac{\mu _M A_M^2}{2}+V_M^{\prime } \left[ {aA_M +bA_R -\delta G} \right] \end{aligned}$$

(25)

$$\begin{aligned} \rho V_R= & {} \left( {\theta \sqrt{G}-\beta p} \right) \left( {p-\omega } \right) -\frac{\mu _R A_R^2}{2}+V_R^{\prime } \left[ {aA_M +bA_R -\delta G} \right] \end{aligned}$$

(26)

Since the game is played à la Stackelberg and M is the leader, we first determine the necessary conditions of R as:

$$\begin{aligned} p=\frac{\theta \sqrt{G}+\beta \omega }{2\beta }\quad \hbox {and}\quad A_R =\frac{bV_R^{\prime }}{\mu _R} \end{aligned}$$

(27)

Substituting (27) into M’s HBJ equation, Eq. (25) becomes:

$$\begin{aligned} \rho V_M =\left( {\frac{\theta \sqrt{G}-\beta \omega }{2}} \right) \left( {\omega +K\sqrt{G}} \right) -\frac{\mu _M A_M^2 }{2}+V_M^{\prime } \left[ {aA_M +\frac{b^{2}V_R^{\prime }}{\mu _R }-\delta G} \right] \end{aligned}$$

(28)

and therefore M’s necessary conditions are:

$$\begin{aligned} \omega =\frac{\left( {\theta -\beta K} \right) \sqrt{G}}{2\beta }\quad \hbox {and}\quad A_M =\frac{aV_M^{\prime }}{\mu _M} \end{aligned}$$

(29)

Using Eqs. (29), the pricing strategy is given by:

$$\begin{aligned} p=\frac{\left( {3\theta -\beta K} \right) \sqrt{G}}{4\beta } \end{aligned}$$

(30)

Replacing the players’ strategies inside Eqs. (26) and (28), we obtain:

$$\begin{aligned} 8\beta \mu _M \mu _R V_M= & {} \mu _M \mu _R \left( {\theta +\beta K} \right) ^{2}G+4\beta V_M^{\prime } \left[ \left( \mu _R a^{2}V_M^{\prime } \right. \right. \nonumber \\&\left. \left. +\,2\mu _M b^{2}V_R^{\prime } \right) -2\mu _M \mu _R \delta G \right] \end{aligned}$$

(31)

$$\begin{aligned} 16\beta \mu _M \mu _R V_R= & {} \mu _M \mu _R \left( {\theta +\beta K} \right) ^{2}G+8V_R^{\prime } \left[ \left( 2\mu _R a^{2}V_M^{\prime }\right. \right. \nonumber \\&\left. \left. +\,\mu _M b^{2}V_R^{\prime } \right) -2\mu _M \mu _R \delta G \right] \end{aligned}$$

(32)

We conjecture linear value functions, $V_M =l_1 G+l_2$ and $V_R =l_3 G+l_4$ where $l_1, l_2, l_3$, and $l_4$ are the constant parameters to be identified. Substituting $V_M$ and $V_R$ and their derivatives inside Eqs. (31) and (32) it gives:

$$\begin{aligned} 8\beta \mu _M \mu _R \left( {l_1 G+l_2} \right)= & {} \mu _M \mu _R \left( {\theta +\beta K} \right) ^{2}G+4\beta l_1 \left[ \left( {\mu _R a^{2}l_1 +2\mu _M b^{2}l_3} \right) \right. \nonumber \\&\left. -\,2\mu _M \mu _R \delta G \right] \end{aligned}$$

(33)

$$\begin{aligned} 16\beta \mu _M \mu _R \left( {l_3 G+l_4} \right)= & {} \mu _M \mu _R \left( {\theta +\beta K} \right) ^{2}G+8\beta l_3 \left[ \left( {2\mu _R a^{2}l_1 +\mu _M b^{2}l_3} \right) \right. \nonumber \\&\left. -\,2\mu _M \mu _R \delta G \right] \end{aligned}$$

(34)

while we can identify the parameter values such as:

$$\begin{aligned} l_1= & {} \frac{\left( {\theta +\beta K} \right) ^{2}}{8\beta \left( {\rho +\delta } \right) }, \quad l_2 =\frac{l_1 \left( {\mu _R a^{2}l_1 +2\mu _M b^{2}l_3} \right) }{2\rho \mu _M \mu _R}, \nonumber \\ l_3= & {} \frac{\left( {\theta +\beta K} \right) ^{2}}{16\beta \left( {\rho +\delta } \right) },\quad \hbox { and}\quad l_4 =\frac{l_3 \left( {2\mu _R a^{2}l_1 +\mu _M b^{2}l_3} \right) }{2\rho \mu _M \mu _R} \end{aligned}$$

(35)

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. $\square$

Proof of Proposition 2

We need to establish the existence of bounded and continuously differentiable value functions $V_M^I \left( {G^{I}} \right) ,V_R^I \left( {G^{I}} \right)$ such that there exists a unique solution $G^{I}(t)$ to Eq. (1) and the HJB equations. The players’ HJBs in Scenario I are:

$$\begin{aligned} \rho V_M^I= & {} \left( {\theta \sqrt{G^{I}}-\beta p^{I}} \right) \left( {\omega +K^{I}\sqrt{G^{I}}} \right) \left( {1-\phi } \right) -\frac{\mu _M A_M^{I^{2}}}{2}+V_M^{{\prime }} \left[ {aA_M^I +bA_R^I -\delta G^{I}} \right] \nonumber \\ \end{aligned}$$

(36)

$$\begin{aligned} \rho V_R^I= & {} \left( {\theta \sqrt{G^{I}}-\beta p^{I}} \right) \left( p^{I}-\omega +\phi \left( {\omega +K\sqrt{G^{I}}} \right) \right. \nonumber \\&\left. -\,sc_L \sqrt{G^{I}} \right) -\frac{\mu _R A_R^{I^{2}} }{2}+V_R^{{\prime }} \left[ {aA_M^I +bA_R^I -\delta G^{I}} \right] \end{aligned}$$

(37)

As in the benchmark scenario, the game is played à la Stackelberg and M is the leader. First we substitute the wholesale price strategy of the benchmark scenario inside Eqs. (36) and (37), thus we determine the necessary conditions of R as:

$$\begin{aligned} p^{I}=\frac{\left[ {\left( {3-\phi } \right) \theta -\beta K\left( {1+\phi } \right) +2\beta sc_L} \right] \sqrt{G^{I}}}{4\beta }\quad \hbox { and}\quad A_R^I =\frac{bV_R^{{\prime }}}{\mu _R} \end{aligned}$$

(38)

Substituting (38) into M’s HBJ equation, Eq. (36) becomes:

$$\begin{aligned} V_M^I= & {} \left( {\frac{\left[ {\left( {\theta +\beta K} \right) \left( {1+\phi } \right) -2\beta sc_L} \right] }{4}} \right) \left( {\frac{\left( {\theta +\beta K} \right) }{2\beta }} \right) \left( {1-\phi } \right) G^{I}\nonumber \\&-\,\frac{\mu _M A_M^{I^{2}} }{2}+V_M^{{\prime }} \left[ {aA_M^I +\frac{b^{2}V_R^{{\prime }I} }{\mu _R}-\delta G^{I}} \right] \end{aligned}$$

(39)

and therefore M’s necessary conditions are:

$$\begin{aligned} \phi =\frac{\beta sc_L}{\theta +\beta K}\quad \hbox { and}\quad A_M^I =\frac{aV_M^{{\prime }}}{\mu _M} \end{aligned}$$

(40)

Then, the pricing strategy becomes:

$$\begin{aligned} p^{I}=\frac{3\theta -\beta K+\beta sc_L}{4\beta }\sqrt{G^{I}} \end{aligned}$$

(41)

Replacing Eqs. (40) and (41) inside Eq. (37) and Eq. (39), we obtain:

$$\begin{aligned} 8\beta \rho \mu _M \mu _R V_M^I= & {} \mu _M \mu _R \left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}G^{I}\nonumber \\&+\,4V_M^{{\prime }} \left[ {\mu _R a^{2}V_M^{{\prime }} +2\mu _M b^{2}V_R^{{\prime }} -2\mu _M \mu _R \delta G^{I}} \right] \end{aligned}$$

(42)

$$\begin{aligned} 16\beta \rho \mu _M \mu _R V_R^I= & {} \mu _M \mu _R \left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}G^{I}\nonumber \\&+\,8\beta V_R^{{\prime }} \left[ {2\mu _R a^{2}V_M^{{\prime }} +\mu _M b^{2}V_R^{{\prime }} -2\mu _M \mu _R \delta G^{I}} \right] \end{aligned}$$

(43)

We conjecture linear value functions, $V_M^I =n_1 G^{I}+n_2$ and $V_R^I =n_3 G^{I}+n_4$ where $n_1, n_2, n_3$, and $n_4$ are the constant parameters to be identified. Substituting $V_M^I$ and $V_R^I$ and their derivatives inside Eq. (42) and Eq. (43) it gives:

$$\begin{aligned} 8\beta \rho \mu _M \mu _R \left( {n_1 G^{I}+n_2} \right)= & {} \mu _M \mu _R \left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}G^{I}\nonumber \\&+\,4n_1 \left[ {\mu _R a^{2}n_1 +2\mu _M b^{2}n_3 -2\mu _M \mu _R \delta G^{I}} \right] \end{aligned}$$

(44)

$$\begin{aligned} 16\beta \rho \mu _M \mu _R \left( {n_3 G^{I}+n_4} \right)= & {} \mu _M \mu _R \left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}G^{I}\nonumber \\&+\,8\beta n_3 \left[ {2\mu _R a^{2}n_1 +\mu _M b^{2}n_3 -2\mu _M \mu _R \delta G^{I}} \right] \end{aligned}$$

(45)

while we can identify the parameter values such as:

$$\begin{aligned} n_1= & {} \frac{\left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}}{8\beta \left( {\rho +\delta } \right) }, \quad n_2 =\frac{n_1 \left[ {\mu _R a^{2}n_1 +2\mu _M b^{2}n_3} \right] }{2\rho \mu _M \mu _R}, \nonumber \\ n_3= & {} \frac{\left[ {\theta +\beta \left( {K-sc_L} \right) } \right] ^{2}}{16\beta \left( {\rho +\delta } \right) },\quad \hbox { and}\quad n_4 =\frac{n_3 \left[ {2\mu _R a^{2}n_1 +\mu _M b^{2}n_3} \right] }{2\rho \mu _M \mu _R}. \end{aligned}$$

(46)

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. $\square$

Proof of Proposition 3

Hereby we follow the same procedure of the previous proof with the difference that the sharing parameter is exogenous. We need to establish the existence of bounded and continuously differentiable value functions $V_M^{II} \left( {G^{II}} \right) ,V_R^{II} \left( {G^{II}} \right)$ such that there exists a unique solution $G^{II}(t)$ to Eq. (1) and the HJB equations. The players’ HJBs in Scenario II are:

$$\begin{aligned} \rho V_M^{II}= & {} \left( {\theta \sqrt{G^{II}}-\beta p^{II}} \right) \left( {\omega +K\sqrt{G^{II}}} \right) \left( {1-\phi ^{II}} \right) \nonumber \\&-\,\frac{\mu _M A_M^{II^{2}}}{2}+V_M^{II{\prime }} \left[ {aA_M^{II} +bA_R^{II} -\delta G^{II}} \right] \end{aligned}$$

(47)

$$\begin{aligned} \rho V_R^{II}= & {} \left( {\theta \sqrt{G^{II}}-\beta p^{II}} \right) \left( {p^{II}-\omega +\phi ^{II}\left( {\omega +K\sqrt{G^{II}}} \right) -sc_L \sqrt{G^{II}}} \right) \nonumber \\&-\,\frac{\mu _R A_R^{II^{2}}}{2}+V_R^{II{\prime }} \left[ {aA_M^{II} +bA_R^{II} -\delta G^{II}} \right] \end{aligned}$$

(48)

As in the benchmark scenario, the game is played à la Stackelberg and M is the leader. First the wholesale price in the benchmark scenario inside Eqs. (47) and (48), thus we determine the necessary conditions of R as:

$$\begin{aligned} p^{II}=\frac{\left[ {\left( {3-\phi ^{II}} \right) \theta -\beta K\left( {1+\phi ^{II}} \right) +2\beta sc_L} \right] \sqrt{G^{II}}}{4\beta }\quad \hbox { and}\quad A_R^{II} =\frac{bV_R^{II{\prime }}}{\mu _R} \end{aligned}$$

(49)

Substituting (49) into M’s HBJ equations Eq. (48) becomes:

$$\begin{aligned} V_M^{II}= & {} \left( {\frac{\left[ {\left( {\theta +\beta K} \right) \left( {1+\phi } \right) -2\beta sc_L} \right] }{4}} \right) \left( {\frac{\left( {\theta +\beta K} \right) }{2\beta }} \right) \left( {1-\phi } \right) G^{II}\nonumber \\&-\,\frac{\mu _M A_M^{II^{2}} }{2}+V_M^{II{\prime }} \left[ {aA_M^{II} +\frac{b^{2}V_R^{{\prime }II} }{\mu _R}-\delta G^{II}} \right] \end{aligned}$$

(50)

and therefore M’s necessary condition is:

$$\begin{aligned} A_M^{II} =\frac{aV_M^{II{\prime }}}{\mu _M} \end{aligned}$$

(51)

Replacing the players’ strategies inside Eqs. (48) and (50), we obtain:

$$\begin{aligned} 8\beta \mu _M \mu _R \rho V_M^{II}= & {} \mu _M \mu _R \left[ {\left( {\theta +\beta K} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] \left( {\theta +\beta K} \right) \left( {1-\phi ^{II}} \right) G^{II} \nonumber \\&+\,4\beta V_M^{II{\prime }} \left[ {\mu _R a^{2}V_M^{II{\prime }} +2\mu _M b^{2}V_R^{II{\prime }} -2\delta \mu _M \mu _R G^{II}} \right] \end{aligned}$$

(52)

$$\begin{aligned} 16\beta \mu _M \mu _R \rho V_R^{II}= & {} \left[ {\left( {\theta +\beta K} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] ^{2}G^{II}\nonumber \\&+\,8\beta V_R^{II{\prime }} \left[ {2\mu _R a^{2}V_M^{II{\prime }} +\mu _M b^{2}V_R^{II{\prime }} -2\mu _M \mu _R \delta G^{II}} \right] \end{aligned}$$

(53)

We conjecture linear value functions, $V_M^{II} =m_1 G^{II}+m_2$ and $V_R^{II} =m_3 G^{II}+m_4$where $m_1, m_2, m_3$, and $m_4$ are the constant parameters to be identified. Substituting $V_M^{II}$ and $V_R^{II}$ and their derivatives inside Eqs. (52) and (53), it gives:

$$\begin{aligned} 8\beta \mu _M \mu _R \rho \left( {m_1 G^{II}+m_2} \right)= & {} \mu _M \mu _R \left[ {\left( {\theta +\beta K} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] \left( {\theta +\beta K} \right) \left( {1-\phi ^{II}} \right) G^{II} \nonumber \\&+\,4\beta m_1 \left[ {\mu _R a^{2}m_1 +2\mu _M b^{2}m_3 -2\delta \mu _M \mu _R G^{II}} \right] \end{aligned}$$

(54)

$$\begin{aligned} 16\beta \mu _M \mu _R \rho \left( {m_3 G^{II}+m_4} \right)= & {} \left[ {\left( {\theta +\beta K} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] ^{2}G^{II}\nonumber \\&+\,8\beta m_3 \left[ {2\mu _R a^{2}m_1 +\mu _M b^{2}m_3 -2\mu _M \mu _R \delta G^{II}} \right] \end{aligned}$$

(55)

while we can identify the parameter values such as:

$$\begin{aligned}&\displaystyle m_1 =\frac{\left[ {\left( {\theta +\beta K^{C}} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] \left( {\theta +\beta K^{C}} \right) \left( {1-\phi ^{II}} \right) }{8\beta \left( {\rho +\delta } \right) }, \nonumber \\&\displaystyle m_2 =\frac{m_1 \left[ {\mu _R a^{2}m_1 +2\mu _M b^{2}m_3} \right] }{2\mu _M \mu _R \rho }, \nonumber \\&\displaystyle m_3 =\frac{\left[ {\left( {\theta +\beta K^{C}} \right) \left( {1+\phi ^{II}} \right) -2\beta sc_L} \right] ^{2}}{16\beta \left( {\rho +\delta } \right) },\quad \hbox { and}\quad m_4 =\frac{m_3 \left[ {2\mu _R a^{2}m_1 +\mu _M b^{2}m_3} \right] }{2\mu _M \mu _R \rho }.\nonumber \\ \end{aligned}$$

(56)

which are all strictly positive; thus, the result shows concave profit functions with respect to the decision variables and an unique equilibrium that maximizes players’ objective functions. $\square$

Proof of Proposition 9

Compute the difference $V_M -V_M^I$ to show that $V_M -V_M^I =\frac{c_L s\left[ {\left( {b^{2}\mu _M +a^{2}\mu _R} \right) \left( {\delta +\rho } \right) +a^{2}\mu _R \rho } \right] \Omega }{128\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}$ with

$$\begin{aligned} \Omega= & {} \left\{ \begin{array}{l} 4\left( {s\beta \Delta +\theta } \right) ^{3}+6c_R^2 h^{2}\beta ^{2}\left( {2\left( {s\beta \Delta +\theta } \right) -3c_L s\beta } \right) +c_L^2 s^{2}\beta ^{2}\left( {s\beta \Delta +\theta } \right) \left[ {28c_L s\beta -18\left( {s\beta \Delta +\theta } \right) } \right] \\ -\beta ^{3}\left( {4c_R^3 h^{3}+15c_L ^{3}s^{3}} \right) +4c_L h\beta \left( {9c_L s\beta \left( {s\beta \Delta +\theta } \right) -3\left( {s\beta \Delta +\theta } \right) ^{2}-7c_L^2 s^{2}\beta ^{2}} \right) \\ \end{array} \right\} >0. \end{aligned}$$

To proof that $V_M -V_M^I >0$, we solve the third degree polynomial $\Omega$ with respect to $\theta +\beta s\Delta$. Among the three solutions, only one is feasible, that is $\theta +\beta s\Delta =\frac{\beta \left( {3c_L s+2c_R h} \right) }{2}$. Therefore, the result $V_M -V_M^I >0$ always holds because it meets the assumption $\psi _1 \ge \beta c_L s$; further, computing the difference $V_M -V_M^{II}$, we can demonstrate that:

• (a) when $\phi ^{II}=0,\quad V_M -V_M^{II} =\frac{\psi _1 \left[ {\mu _M b^{2}\left( {\rho +\delta } \right) \left( {\psi _1^3 -\psi _2^3} \right) +\mu _R a^{2}\left( {2\rho +\delta } \right) \psi _1 \left( {\psi _1^2 +\psi _2^2} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$;

• (b) when $\phi ^{II}=1,\quad V_M -V_M^{II} =\frac{\psi _1^4 \left[ {\left( {\rho +\delta } \right) \left( {\mu _M b^{2}+\mu _R a^{2}} \right) +\rho \mu _R a^{2}} \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$;

• (c) when

$$\begin{aligned}&V_M -V_M^{II} =\\&\frac{\psi _1 \left\{ {\left( {V_M -V_M^{II}} \right) _{\left| {\phi ^{II}=1} \right. } -\left( {1-\phi ^{II}} \right) \left( {\psi _2 +\phi ^{II}\psi _1} \right) ^{2}\left[ {\left( {2\rho +\delta } \right) a^{2}\mu _R \psi _1 \left( {1-\phi ^{II}} \right) +\left( {\delta +\rho } \right) b^{2}\mu _M \left( {\psi _2 +\psi _1 \phi ^{II}} \right) } \right] } \right\} }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}\le 0 \end{aligned}$$

with $\psi _2 =\psi _1 -2c_L s\beta \ge 0$, where define the interval inside which $V_M -V_M^{II} \le 0$ holds.

Finally, compute the difference $V_M^I -V_M^{II}$, we demonstrate that:

• (a) when $\phi ^{II}=0,V_M^I -V_M^{II} =\frac{\left( {V_M^I -V_M^{II}} \right) _{\left| {\phi ^{II}=1} \right. } -\psi _1 \psi _2^2 \left[ {a^{2}\rho \mu _R \psi _1 +\left( {\rho +\delta } \right) \left( {a^{2}\mu _R \psi _1 +b^{2}\mu _M \psi _2} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$;

• (b) when $\phi ^{II}=1,V_M^I -V_M^{II} =\frac{\left( {\psi _1 -c_L s\beta } \right) ^{4}\left[ {\left( {\rho +\delta } \right) \left( {\mu _M b^{2}+\mu _R a^{2}} \right) +\rho \mu _R a^{2}} \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$;

• (c) when ,

  $$\begin{aligned}&V_M^I -V_M^{II} \\&=\frac{\left( {V_M^I -V_M^{II}} \right) _{\left| {\phi ^{II}=1} \right. } -\psi _1 \left( {1-\phi ^{II}} \right) \left( {\psi _2 +\phi ^{II}\psi _1} \right) ^{2}\left[ {\left( {2\rho +\delta } \right) a^{2}\mu _R \psi _1 \left( {1-\phi ^{II}} \right) +\left( {\delta +\rho } \right) b^{2}\mu _M \left( {\psi _2 +\psi _1 \phi ^{II}} \right) } \right] }{128\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}\le 0, \end{aligned}$$

  where define the interval inside which $V_M^I -V_M^{II} \le 0$ holds. $\square$

Proof of Proposition 10

Compute the difference $V_R -V_R^I$ to show that $V_R -V_R^I =\frac{c_L s\left[ {\left( {b^{2}\mu _M +4a^{2}\mu _R} \right) \left( {\delta +\rho } \right) +b^{2}\mu _M \rho } \right] \Omega }{512\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$ because $\Omega >0$. The sign of the difference $V_R -V_R^{II}$ depends on $\phi ^{II}$. Fixing $\psi _3 =4a^{2}\mu _R \left( {\delta +\rho } \right) +b^{2}\mu _M \left( {\delta +2\rho } \right) >0$, the following cases can be displayed:

• (1) when $\phi ^{II}=0, \quad V_R -V_R^{II} =\frac{\psi _3 \psi _1^4 -\psi _2^3 \left[ {4a^{2}\mu _R \psi _1 \left( {\rho +\delta } \right) +b^{2}\mu _M \psi _2 \left( {\delta +2\rho } \right) } \right] }{512\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0$, as $\psi _1 >\psi _2$;

• (2) when $\phi ^{II}=1, V_R -V_R^{II} =\frac{\psi _3 \psi _1^4 -\left( {\psi _2 +\psi _1} \right) ^{4}b^{2}\mu _M \left( {\delta +2\rho } \right) }{512\beta \delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}<0$. Assume that $\delta =\rho =\mu _M =\mu _R =a=b=1$; substituting for $\psi _2 =\psi _1 -2c_L s\beta$, the numerator becomes $11\psi _1^4 -48\left( {\psi _1 -c_L s\beta } \right) ^{4}\le 0$. Solving this polynomial with respect to $\psi _1$ gives four solutions: the first two are not feasible, while the second two are feasible and negative. Therefore, $V_R -V_R^{II} <0$ always holds.

Finally, cases (1) and (2) allow one to verify that $V_R -V_R^{II} \left\{ {\begin{array}{ll} \ge 0&{}\quad \phi ^{II}\in \left[ {0,\bar{{\phi }}^{II}} \right] \\ <0&{}\quad \phi ^{II}\in \left( {\bar{{\phi }}^{II},1} \right] \\ \end{array}} \right.$.

The sign of the difference $V_R^I -V_R^{II}$ also depends on $\phi ^{II}$ and it results:

$$\begin{aligned} V_R^I -V_R^{II} =\frac{\psi _3 \left( {\psi _1-c_L s\beta } \right) ^{4}-\left( {\psi _2 +\phi ^{II}\psi _1} \right) ^{3}\left[ {4a^{2}\mu _R \psi _1 \left( {1-\phi ^{II}} \right) \left( {\rho +\delta } \right) +b^{2}\mu _M \left( {\psi _2 +\psi _1 \phi ^{II}} \right) \left( {\delta +2\rho } \right) } \right] }{516\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}} \end{aligned}$$

According to $\phi ^{II}$, three cases may be displayed:

• (1) when $\phi ^{II}=0, V_R^I -V_R^{II} =\frac{\psi _3 \left( {\psi _1 -c_L s\beta } \right) ^{4}-\psi _2^3 \left[ {4a^{2}\mu _R \psi _1 \left( {\rho +\delta } \right) +b^{2}\mu _M \psi _2 \left( {\delta +2\rho } \right) } \right] }{516\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}>0\, \psi _1 -c_L s\beta >\psi _2$;

• (2) when $\phi ^{II}=1, V_R^I -V_R^{II} =\frac{\psi _3 \left( {\psi _1 -c_L s\beta } \right) ^{4}-\left( {\psi _2 +\psi _1 } \right) ^{4}b^{2}\mu _M \left( {\delta +2\rho } \right) }{516\beta ^{2}\delta \rho \mu _M \mu _R \left( {\delta +\rho } \right) ^{2}}<0$. Solving the numerator with respect to $\psi _1$ gives four solutions: two are not feasible, two are feasible and negative, $V_R^I -V_R^{II} <0$ always holds;

• (3) Cases (1) and (2) allow one to verify that $V_R^I -V_R^{II} \left\{ {\begin{array}{ll} \ge 0&{}\quad \phi ^{II}\in \left[ {0,\bar{{\bar{{\phi }}}}^{II}} \right] \\ <0&{}\quad \phi ^{II}\in \left( {\bar{{\bar{{\phi }}}}^{II},1} \right] \\ \end{array}} \right.$. $\square$