Appendices
A-1. Proof of Lemma 1
For the retailer, for i = 1, 2, we have
$$ \mathop {Max}\limits_{{p_{i} ,v}} E[\pi_{R} |w_{i} ] = \mathop {Max}\limits_{{p_{i} ,v}} E[(p_{1} - w_{1} - c)d_{1} + (p_{2} - w_{2} - c)d_{2} |w_{i} ]. $$
(16)
From the first-order condition (FOC), we have the optimal response function for the retailer:
$$ \frac{{\partial E\left[ {\pi_{R} } \right]}}{{\partial P_{1} }} = 0,\quad \frac{{\partial E\left[ {\pi_{R} } \right]}}{{\partial P_{2} }} = 0,\quad \frac{{\partial E\left[ {\pi_{R} } \right]}}{\partial V} = 0. $$
The Hessian matrix is:
\(H = \left[ {\begin{array}{*{20}l} {\frac{{\partial_{{\pi_{R} }}^{2} }}{{\partial p_{1}^{2} }}} \hfill & {\quad \frac{{\partial_{{\pi_{R} }}^{2} }}{{\partial p_{1} p_{2} }}} \hfill \\ {\frac{{\partial_{{\pi_{R} }}^{2} }}{{\partial p_{2} p_{1} }}} \hfill & {\quad \frac{{\partial_{{\pi_{R} }}^{2} }}{{\partial p_{2}^{2} }}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - 2\beta_{1} - 2k} \hfill & {\quad 2k} \hfill \\ {2k} \hfill & {\quad - 2\beta_{1} - 2k} \hfill \\ \end{array} } \right]\),
and \(|H| = ( - 2\beta_{1} - 2k)^{2} - (2k)^{2} = 4\beta_{1}^{2} + 8\beta_{1} k > 0\). Thus, it is easily seen that the Hessian matrix H is a diagonally dominant matrix, thereby guaranteeing the joint concavity of the profit function \(\pi_{R} (p_{1} ,p_{2} )\). Consequently, we have
$$ {\text{p}}_{{1}}^{*} = \, \frac{{\beta_{1} \mu_{1} v^{2} + 2\beta_{2} v + 2\alpha_{1} + 2\beta_{1} w_{1} + 4kp_{2} + 2kw_{1} - 2kw_{2} }}{{4(\beta_{1} + k)}}, $$
(17)
$$ {\text{p}}_{2}^{*} = \frac{{\beta_{1} \mu_{1} v^{2} + 2\beta_{2} v + 2\alpha_{2} + 2\beta_{1} w_{2} + 4kp_{1} - 2kw_{1} + 2kw_{2} }}{{4(\beta_{1} + k)}}, $$
(18)
$$ v^{*} = \frac{{\beta_{2} }}{{\beta_{1} \mu_{1} }}. $$
(19)
Equation (17) is the best response function of the retailer for product A, which is affected by the retail price of product B. Then, combining (17)—(19), we can obtain that
$$ {\text{p}}_{1}^{*} = \, \frac{{3\beta_{1} \beta_{2}^{2} + 6\beta_{2}^{2} k + 2\beta_{1}^{3} \mu_{1} w_{1} + 2\alpha_{1} \beta_{1}^{2} \mu_{1} + 2\alpha_{1} \beta_{1} k\mu_{1} + 2\alpha_{2} \beta_{1} k\mu_{1} + 4\beta_{1}^{2} k\mu_{1} w_{1} }}{{4\beta_{1}^{2} \mu_{1} (\beta_{1} + 2k)}}. $$
(20)
Similarly, combining (18)—(20), we get
$$ {\text{p}}_{{2}}^{*} = \frac{{3\beta_{1} \beta_{2}^{2} + 6\beta_{2}^{2} k + 2\beta_{1}^{3} \mu_{1} w_{2} + 2\alpha_{2} \beta_{1}^{2} \mu_{1} + 2\alpha_{1} \beta_{1} k\mu_{1} + 2\alpha_{2} \beta_{1} k\mu_{1} + 4\beta_{1}^{2} k\mu_{1} w_{2} }}{{4\beta_{1}^{2} \mu_{1} (\beta_{1} + 2k)}}. $$
(21)
A-2. Proof of Lemma 2
For manufacturer 1, Eq. (9) can be simplified as
$$ \mathop {Max}\limits_{w} E\left[ {\pi_{{M_{1} }} } \right] = \alpha_{1} \varepsilon w_{1} - \varepsilon kw_{1}^{2} - \beta_{1} \varepsilon w_{1}^{2} + \varepsilon kw_{1} w_{2} - \frac{{\beta_{2}^{2} w_{1} (\ln (\overline{\mu } - \varepsilon ) - \ln (\overline{\mu } + \varepsilon ))}}{{4\beta_{1} }}. $$
(22)
For manufacturer 2, we have
$$ \mathop {Max}\limits_{w} \, E\left[ {\pi_{{M_{2} }} } \right] = \alpha_{2} \varepsilon w_{2} - \varepsilon kw_{2}^{2} - \beta_{1} \varepsilon w_{2}^{2} + \varepsilon kw_{1} w_{2} - \frac{{\beta_{2}^{2} w_{2} (\ln (\overline{\mu } + \varepsilon ) - \ln (\overline{\mu } - \varepsilon ))}}{{4\beta_{1} }}. $$
(23)
Since \(\frac{{\partial_{{\pi_{Mi} }}^{2} }}{{\partial w_{i}^{2} }} = - 2\beta_{1} \varepsilon - 2\varepsilon k < 0\), \(\pi_{{M_{i} }}\) is concave in \(w_{i}\). Thus, setting \(\frac{{\partial E[\pi_{Mi} ]}}{{\partial W_{i} }} = 0\), we obtain the following optimal wholesale prices under no information sharing scenario:
$$ w_{1}^{N*} = \frac{{2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} k\varepsilon + 4\alpha_{2} \beta_{1} k\varepsilon }}{{4\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}, $$
(24)
$$ w_{2}^{N*} = \frac{{2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{2} \beta_{1}^{2} \varepsilon + 8\alpha_{2} \beta_{1} k\varepsilon + 4\alpha_{1} \beta_{1} k\varepsilon }}{{4\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}. $$
(25)
It is easily seen that \(\frac{{\partial w_{i}^{N*} }}{\partial \varepsilon } = \frac{{\beta_{2}^{2} (2\beta_{1} + 3k)\left[ {\varepsilon^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) - \overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 2\varepsilon \overline{\mu }} \right]}}{{4\beta_{1} \varepsilon^{2} (\overline{\mu }^{2} - \varepsilon^{2} )(4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}\). Since \(\left( {\overline{\mu } - \varepsilon } \right) > 0\), we have that \(\left( {\overline{\mu }^{{2}} - \varepsilon^{{2}} } \right) > 0\). Next, let
$$ \Upsilon (\varepsilon ) = \varepsilon^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) - \overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 2\varepsilon \overline{\mu }. $$
Then,\(\frac{\partial \Upsilon (\varepsilon )}{{\partial \varepsilon }} = 2\varepsilon \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} > 0\), and \(\Upsilon (0) = {0}\). Hence \(\Upsilon (\varepsilon ) > {0}\) for all \(\varepsilon > 0\). Thus, \(\frac{{\partial w_{i}^{N*} }}{\partial \varepsilon } > 0\).
A-3. Proof of Lemma 3
For manufacturer 1, Eq. (10) can be simplified as
$$ \mathop {Max}\limits_{w} E\left[ {\pi_{{M_{1} }} } \right] = \, \frac{{\alpha_{2} \varepsilon w_{1} }}{2} + \frac{{\varepsilon kw_{1} w_{2} }}{2} - \frac{{\beta_{1} \varepsilon w_{1}^{2} }}{2} - \frac{{\varepsilon kw_{1}^{2} }}{2} + \frac{{\beta_{2}^{2} w_{1} (\ln (\overline{\mu }) - \ln (\overline{\mu } - \varepsilon ))}}{{4\beta_{1} }} $$
(26)
Moreover, for manufacturer 2, we have
$$ \mathop {Max}\limits_{w} E\left[ {\pi_{{M_{2} }} } \right] = \frac{{\alpha_{2} \varepsilon w_{2} }}{2} + \frac{{\varepsilon kw_{1} w_{2} }}{2} - \frac{{\beta_{1} \varepsilon w_{2}^{2} }}{2} - \frac{{\varepsilon kw_{2}^{2} }}{2} + \frac{{\beta_{2}^{2} w_{2} (\ln (\overline{\mu }) - \ln (\overline{\mu } - \varepsilon ))}}{{4\beta_{1} }} $$
(27)
Since \(\frac{{\partial_{{\pi_{Mi} }}^{2} }}{{\partial w_{i}^{2} }} = - 2\beta_{1} \varepsilon - 2\varepsilon k < 0\), \(\mathop \pi \nolimits_{{M_{i} }}\) is concave in \(w_{i}\), using \(\frac{{\partial E[\pi_{Mi} ]}}{{\partial W_{i} }} = 0\), we obtain the optimal wholesale prices under full information sharing scenario as:
$$ w_{1}^{Sh*} = \, \frac{{2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} k\varepsilon + 2\alpha_{2} \beta_{1} k\varepsilon }}{{2\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}, $$
(28)
$$ w_{2}^{Sh*} = \, \frac{{2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} k\varepsilon + 4\alpha_{2} \beta_{1} k\varepsilon }}{{2\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}. $$
(29)
Moreover, \(\frac{{\partial W_{i}^{N*} }}{\partial \varepsilon } = \frac{{ - \beta_{2}^{2} \left[ {\varepsilon - \overline{\mu }\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \varepsilon \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right)} \right]}}{{2\beta_{1} \varepsilon^{2} (\overline{\mu } - \varepsilon )(2\beta_{1} + k)}}\). Since \(\left( {\overline{\mu } - \varepsilon } \right) > 0\), in order to see \(\frac{{\partial w_{i}^{N*} }}{\partial \varepsilon } < 0\), we only need to show that
$$ \Theta (\varepsilon ): = \varepsilon - \overline{\mu }\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \varepsilon \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) > 0. $$
For this, note that \(\frac{\partial \Theta (\varepsilon )}{{\partial \varepsilon }} = \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} > 0\). Then, since \(\Theta (0) = 0\), we have that \(\Theta (\varepsilon ) > 0\).
A-4. Proof of Lemma 4
This proof is similar to that of Lemma 3.
A-5. Proof of Lemma 5
For manufacturer 1, Eq. (12) can be simplified as
$$ \mathop {Max}\limits_{w} E\left[ {\pi_{{M_{1} }} } \right] = \, \frac{{\alpha_{1} \varepsilon w_{1} }}{2} + \frac{{\varepsilon kw_{1} w_{2} }}{2} - \frac{{\beta_{1} \varepsilon w_{1}^{2} }}{2} - \frac{{\varepsilon kw_{1}^{2} }}{2} + \frac{{\beta_{2}^{2} w_{1} (\ln (\overline{\mu }) - \ln (\overline{\mu } - \varepsilon ))}}{{4\beta_{1} }}. $$
(30)
Also, for manufacturer 2, Eq. (13) can be simplified as
$$ \mathop {Max}\limits_{w} E\left[ {\pi_{{M_{2} }} } \right] = \alpha_{2} \varepsilon w_{2} - \varepsilon kw_{2}^{2} - \beta_{1} \varepsilon w_{2}^{2} + \varepsilon kw_{1} w_{2} + \frac{{\beta_{2}^{2} w_{2} (\ln (\overline{\mu }) - \ln (\overline{\mu } - \varepsilon ))}}{{4\beta_{1} }}. $$
(31)
Note that \(\frac{{\partial_{{\pi_{{M_{i} }} }}^{2} }}{{\partial w_{i}^{2} }} = - 2\beta_{1} \varepsilon - 2\varepsilon k < 0\); thus, \(\pi_{{M_{i} }}\) is concave in wi. Therefore, \(\frac{{\partial E[\pi_{Mi} ]}}{{\partial w_{i} }} = 0\) gives the following optimal wholesale prices under the full information sharing scenario.
$$ w_{1}^{Ph*} = \frac{{4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} k\varepsilon + 2\alpha_{2} \beta_{1} k\varepsilon }}{{4\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}, $$
(32)
$$ w_{2}^{Ph*} = \frac{{\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} k\varepsilon + 4\alpha_{2} \beta_{1} k\varepsilon }}{{2\beta_{1} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )}}. $$
(33)
Moreover, we can get that
$$ \frac{{\partial W_{i}^{Ph*} }}{\partial \varepsilon } = \frac{{\beta_{2}^{2} \left[ {\varepsilon^{2} k + \beta_{1} \varepsilon^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \varepsilon^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) - \beta_{1} \overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 2\beta_{1} \varepsilon \overline{\mu } - k\overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3k\varepsilon \overline{\mu } + \varepsilon^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) - k\overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)} \right]}}{{2\beta_{1} \varepsilon^{2} (\overline{\mu }^{2} - \varepsilon^{2} )(4\beta_{2}^{2} + 3k^{2} + 8\beta_{1} k)}} $$
Let \(\Psi (\varepsilon ): = \varepsilon^{2} k + \beta_{1} \varepsilon^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \varepsilon^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) - \beta_{1} \overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 2\beta_{1} \varepsilon \overline{\mu } - k\overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3k\varepsilon \overline{\mu } + \varepsilon^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) - k\overline{\mu }^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)\), Then, we have \(\frac{\partial \Psi (\varepsilon )}{{\partial \varepsilon }} = \varepsilon \left( {k + 2k\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 2\beta_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} + 2k\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) > 0\). Since \(\Psi (0) = 0\), we can deduce that \(\Psi (\varepsilon ) > 0\) for all \(\varepsilon { > 0}\). Finally, since \(\left( {\overline{\mu }^{2} - \varepsilon^{2} } \right) > 0\), we obtain that \(\frac{{\partial W_{i}^{Ph*} }}{\partial \varepsilon } > 0\).
A-6. Proof of Lemma 6
This proof is similar to that of Lemma 5.
A-7. Proof of Proposition 1
Substituting the optimal prices in profit expressions, we can derive that
$$ \pi_{{M{1}}}^{N} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{16\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(34)
$$ \pi_{{M{1}}}^{Sh} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 2\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{8\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(35)
$$ \pi_{{M{1}}}^{Sl} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 2\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{8\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(36)
$$ \pi_{{M{1}}}^{Ph} = \, \frac{{(\beta_{1} + k)\left[ {4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{32\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(37)
$$ \pi_{{M{1}}}^{Pl} = \frac{{(\beta_{1} + k)\left[ {4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{32\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}. $$
(38)
First, some basic algebraic comparison of the profit expressions yields \(\pi_{M1}^{Sh} \ge \pi_{M1}^{Sl}\).
Now let \(\pi_{M1}^{N} : = \frac{{(\beta_{1} + k)A_{1}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\) and \(\pi_{M1}^{Sh} : = \frac{{(\beta_{1} + k)B_{1}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\),
where
$$ \begin{aligned} A_{1} & = 2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k,\quad {\text{and}} \\ B_{1} & = \sqrt 2 \left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1}^{2} \varepsilon k + 2\alpha_{2} \beta_{1} \varepsilon k} \right]. \\ \end{aligned} $$
It is easy to see that for sufficiently large \(\alpha_{i}\) (i = 1, 2), \({A}_{1}\ge {B}_{1}\). Thus,\(\pi_{M1}^{N} \ge \pi_{M1}^{Sh}\)\(.\) So, we obtain the desired result, i.e.,\(\pi_{M1}^{N} \ge \pi_{M1}^{S}\).
Again, for the second part, it is an easy exercise to see that \(\pi_{M1}^{Ph} \ge \pi_{M1}^{Pl}\).
Let \(\pi_{{M_{1} }}^{N} : = \frac{{(\beta_{1} + k)C_{1}^{2} }}{{32\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\) and \(\pi_{{M_{1} }}^{Ph} : = \frac{{(\beta_{1} + k)D_{1}^{2} }}{{32\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\),
where
$$ \begin{aligned} C_{1} & = \sqrt 2 \left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right],\quad {\text{and}} \\ D_{1} & = 4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k. \\ \end{aligned} $$
Assuming that \(\alpha_{i}\) is large enough, it is easy to conclude that \(C_{1} > D_{1}\), implying that \(\pi_{M1}^{N} \ge \pi_{M1}^{Ph}\). Thus, we have that \(\pi_{M1}^{N} \ge \pi_{M1}^{P}\).
A-8. Proof of Proposition 2
For the first part, let \(\pi_{M1}^{Sh} : = \frac{{(\beta_{1} + k)A_{2}^{2} }}{{32\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\) and \(\pi_{M1}^{Ph} : = \frac{{(\beta_{1} + k)B_{2}^{2} }}{{32\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\),
where
$$ \begin{aligned} A_{2} & : = 2\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 2\alpha_{2} \beta_{1} \varepsilon k} \right],\quad {\text{and}} \\ B_{2} & : = 4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k. \\ \end{aligned} $$
Since \(\left( {A_{2} - B_{2} } \right) = \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }^{2} }}{{\overline{\mu }^{2} - \varepsilon^{2} }}} \right) \ge 0\), \({A}_{2}\) is larger than\({B}_{2}\). Thus,\(\pi_{M1}^{Sh} \ge \pi_{M1}^{Ph}\).
For the second part, let \(\pi_{M1}^{Sl} : = \frac{{(\beta_{1} + k)C_{2}^{2} }}{{32\varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\) and \(\pi_{M1}^{Pl} : = \frac{{(\beta_{1} + k)D_{2}^{2} }}{{32\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\),
where
$$ \begin{aligned} C_{2} & : = 2\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 4\alpha_{1} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 2\alpha_{2} \beta_{1} \varepsilon k} \right],\quad {\text{and}} \\ D_{2} & : = 4\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 4\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{1} \beta_{1}^{2} \varepsilon + 8\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k. \\ \end{aligned} $$
Here \(\left( {D_{2} - C_{2} } \right) = \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }^{2} }}{{\overline{\mu }^{2} - \varepsilon^{2} }}} \right) \ge 0\) implying that \({D}_{2}\) is larger than\({C}_{2}\). Therefore, \(\pi_{M1}^{Pl} \ge \pi_{M1}^{Sl}\).
A-9. Proof of Proposition 3
Here also, substituting the optimal prices in profit expressions, we can derive that
$$ \pi_{M2}^{N} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{2} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 8\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{16\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(39)
$$ \pi_{M2}^{Sh} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{8\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(40)
$$ \pi_{M2}^{Sl} = \, \frac{{(\beta_{1} + k)\left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{8\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(41)
$$ \pi_{M2}^{Ph} = \, \frac{{(\beta_{1} + k)\left[ {\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{4\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}, $$
(42)
$$ \pi_{M2}^{Ph} = \, \frac{{(\beta_{1} + k)\left[ {\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]^{2} }}{{4\beta_{{1}}^{{2}} \varepsilon \left( {{4}\beta_{{1}}^{{2}} { + 8}\beta_{{1}} k + 3k^{2} } \right)^{2} }}. $$
(43)
First, let \(\pi_{M2}^{N} : = \frac{{(\beta_{1} + k)A_{3}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\) and \(\pi_{M2}^{Sh} : = \frac{{(\beta_{1} + k)B_{3}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}\),
where
$$ \begin{aligned} A_{3} & : = \, 2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{2} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 8\alpha_{2} \beta_{1} \varepsilon k,{\text{ and}} \\ B_{3} & : = \sqrt 2 \left[ {2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]. \\ \end{aligned} $$
Here also it is easy to see that for sufficiently large \(\alpha_{i}\) (i = 1, 2), \({A}_{3}\ge {B}_{3}\), and therefore, \(\pi_{M2}^{N} \ge \pi_{M2}^{Sh}\). A similar argument yields \(\pi_{M2}^{N} \ge \pi_{M2}^{Sl}\). Thus, we have \(\pi_{M2}^{N} \ge \pi_{M2}^{S}\).
The proof \(\pi_{M2}^{P} \ge \pi_{M2}^{S}\) is similar and hence, we have omitted it.
A-10. Proof of Proposition 4
It can be shown that
$$ \pi_{M2}^{N} = \frac{{(\beta_{1} + k)A_{4}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }};\quad \pi_{M2}^{Ph} = \frac{{(\beta_{1} + k)C_{4}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }};\quad \pi_{M2}^{Pl} = \frac{{(\beta_{1} + k)D_{4}^{2} }}{{16\beta_{1}^{2} \varepsilon (4\beta_{1}^{2} + 8\beta_{1} k + 3k^{2} )^{2} }}, $$
where
$$ \begin{aligned} A_{4} & : = 2\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 3\beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 8\alpha_{2} \beta_{1}^{2} \varepsilon + 4\alpha_{1} \beta_{1} \varepsilon k + 8\alpha_{2} \beta_{1} \varepsilon k, \\ C_{4} & : = 2\left[ {\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right],\quad {\text{and}} \\ D_{4} & : = 2\left[ {\beta_{1} \beta_{2}^{2} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + \beta_{2}^{2} k\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha_{2} \beta_{1}^{2} \varepsilon + 2\alpha_{1} \beta_{1} \varepsilon k + 4\alpha_{2} \beta_{1} \varepsilon k} \right]. \\ \end{aligned} $$
As in the proofs of the previous propositions, it is easy to see that \({C}_{4}\ge {A}_{4}\) and \({A}_{4}\)≥ \({D}_{4}.\) Thus, we have the desired result, i.e., \(\pi_{M2}^{Ph} \ge \pi_{M2}^{N}\) and \(\pi_{M2}^{N} \ge \pi_{M2}^{Pl}\).
A-11. Proof of Proposition 5
It can be shown that
$$ \pi_{R}^{N} = \frac{{\left( {4\beta^{2} \varepsilon + 2\beta \varepsilon k - \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha \varepsilon k\mu_{1} + 4\alpha \beta \varepsilon \mu_{1} } \right)^{2} }}{{{32}\beta \varepsilon^{{2}} \mu_{{1}}^{{2}} ({2}\beta + {\text{k)}}^{2} }}, $$
(44)
$$ \pi_{R}^{{S{\text{h}}}} = \frac{{\left( {2\beta^{2} \varepsilon + \beta \varepsilon k - \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) + 2\alpha \varepsilon k\mu_{1} + 2\alpha \beta \varepsilon \mu_{1} } \right)^{2} }}{{{8}\beta \varepsilon^{{2}} \mu_{{1}}^{{2}} ({2}\beta + {\text{k)}}^{2} }}, $$
(45)
$$ \pi_{R}^{{S{\text{l}}}} = \frac{{\left( {2\beta^{2} \varepsilon + \beta \varepsilon k - \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 2\alpha \varepsilon k\mu_{1} + 2\alpha \beta \varepsilon \mu_{1} } \right)^{2} }}{{{8}\beta \varepsilon^{{2}} \mu_{{1}}^{{2}} ({2}\beta + {\text{k)}}^{2} }}, $$
(46)
$$ \pi_{R}^{Ph} = \frac{{(\beta + 2k)^{2} A}}{{64\beta \varepsilon^{2} \mu_{1}^{2} (4\beta^{3} + 16\beta^{2} k + 19\beta k^{2} + 6k^{3} )^{2} }}, $$
(47)
$$ \pi_{R}^{Pl} = \frac{{(\beta + 2k)^{2} B}}{{64\beta \varepsilon^{2} \mu_{1}^{2} (4\beta^{3} + 16\beta^{2} k + 19\beta k^{2} + 6k^{3} )^{2} }}, $$
(48)
where
$$ \begin{aligned} {\text{A}} & = {\text{128}}\beta ^{6} \varepsilon ^{2} + 512\beta ^{5} \varepsilon ^{2} {\text{k}} + 4\beta ^{6} \varepsilon ^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + 16\beta ^{6} \varepsilon ^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + {\text{72}}\beta ^{2} \varepsilon ^{2} {\text{k}}^{4} + 384\beta ^{{\text{3}}} \varepsilon ^{2} {\text{k}}^{3} + 704\beta ^{{\text{4}}} \varepsilon ^{2} {\text{k}}^{2} \\ & \quad + \beta ^{{\text{3}}} k^{3} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + 9\beta ^{4} k^{2} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} - 64\beta ^{6} \varepsilon \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 4\beta ^{3} k^{3} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + 36\beta ^{{\text{4}}} k^{2} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} \\ & \quad + {\text{128}}\alpha ^{2} \beta ^{{\text{4}}} \varepsilon ^{2} \mu _{{\text{1}}}^{2} + {\text{288}}\alpha ^{2} \varepsilon ^{2} k^{4} \mu _{{\text{1}}}^{2} + 256\alpha \beta ^{5} \varepsilon ^{2} \mu _{1} + 12\beta ^{5} \alpha \mu _{1} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} - 32\beta ^{6} \varepsilon \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 48\beta ^{5} k\mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} \\ & \quad + 288\alpha \beta \varepsilon ^{2} k^{4} \mu _{1} + 1152\alpha \beta ^{4} \varepsilon ^{2} k\mu _{1} + 1184\alpha ^{2} \beta ^{{\text{4}}} \varepsilon ^{2} k^{2} \mu _{{\text{1}}}^{2} - 112\beta ^{5} \varepsilon k\mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 1248\alpha \beta ^{2} \varepsilon ^{2} k^{3} \mu _{1} + 1856\alpha \beta ^{{\text{3}}} \varepsilon ^{2} k^{2} \mu _{1} \\ & \quad - 224\beta ^{5} \varepsilon k\mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 32\alpha \beta ^{5} \varepsilon \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 36\beta ^{{\text{3}}} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 120\beta ^{{\text{4}}} \varepsilon k^{2} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 960\alpha ^{2} \beta \varepsilon ^{2} k^{3} \mu _{{\text{1}}}^{2} \\ & \quad + 640\alpha ^{2} \beta ^{{\text{3}}} \varepsilon ^{2} k\mu _{{\text{1}}}^{2} - 64\alpha \beta ^{5} \varepsilon \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 4\beta ^{{\text{3}}} k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 72\beta ^{{\text{3}}} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 240\beta ^{{\text{4}}} \varepsilon k^{2} \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 128\alpha \beta ^{{\text{4}}} \varepsilon k\mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 256\alpha \beta ^{{\text{4}}} \varepsilon k\mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 168\alpha \beta ^{{\text{3}}} \varepsilon k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 144\alpha \beta ^{2} \varepsilon k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ \quad - 336\alpha \beta ^{{\text{3}}} \varepsilon k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}, \\ {\text{B}} & = {\text{128}}\beta ^{6} \varepsilon ^{2} + 512\beta ^{5} \varepsilon ^{2} {\text{k}} + 4\beta ^{6} \mu _{{\text{1}}}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + 16\beta ^{6} \mu _{{\text{1}}}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + {\text{72}}\beta ^{2} \varepsilon ^{2} {\text{k}}^{4} + 384\beta ^{{\text{3}}} \varepsilon ^{2} {\text{k}}^{3} + 704\beta ^{{\text{4}}} \varepsilon ^{2} {\text{k}}^{2} \\ & \quad + \beta ^{{\text{3}}} k^{3} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + 9\beta ^{4} k^{2} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} - 64\beta ^{6} \varepsilon \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + {\text{16}}\beta ^{3} k^{3} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} + {\text{48}}\beta ^{{\text{4}}} k^{2} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} \\ & \quad + {\text{128}}\alpha ^{2} \beta ^{{\text{4}}} \varepsilon ^{2} \mu _{{\text{1}}}^{2} + {\text{288}}\alpha ^{2} \varepsilon ^{2} k^{4} \mu _{{\text{1}}}^{2} + 256\alpha \beta ^{5} \varepsilon ^{2} \mu _{1} + 12\beta ^{5} {\text{k}}\mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} - 32\beta ^{6} \varepsilon \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 48\beta ^{5} k\mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} \\ & \quad + 4\beta ^{3} k^{3} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}} \right)} \right)^{2} + 4\beta ^{4} k^{2} \mu _{1}^{2} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}} \right)} \right)^{2} + 288\alpha \beta \varepsilon ^{2} k^{4} \mu _{1} + 1152\alpha \beta ^{4} \varepsilon ^{2} k\mu _{1} + 8\beta ^{5} k\mu _{1}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} \\ & \quad - 32\beta ^{5} \varepsilon k\mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} + 1184\alpha ^{2} \beta ^{2} \varepsilon ^{2} k^{2} \mu _{{\text{1}}}^{2} - 112\beta ^{5} \varepsilon k\mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 1248\alpha \beta ^{2} \varepsilon ^{2} k^{3} \mu _{1} + 1856\alpha \beta ^{{\text{3}}} \varepsilon ^{2} k^{2} \mu _{1} \\ & \quad - 8\beta ^{5} k\mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } + \varepsilon }} + 4\beta ^{3} k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } + \varepsilon }} + 16\beta ^{4} k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } + \varepsilon }} - 192\beta ^{5} \varepsilon k\mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 24\beta ^{3} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} - 64\beta ^{4} \varepsilon k^{2} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} - 32\alpha \beta ^{5} \varepsilon \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 36\beta ^{{\text{3}}} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 120\beta ^{{\text{4}}} \varepsilon k^{2} \mu _{1} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} \\ & \quad - 16\beta ^{{\text{3}}} k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 16\beta ^{4} k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 960\alpha ^{2} \beta \varepsilon ^{2} k^{3} \mu _{{\text{1}}}^{2} + 640\alpha ^{2} \beta ^{{\text{3}}} \varepsilon ^{2} k\mu _{{\text{1}}}^{2} - 64\alpha \beta ^{5} \varepsilon \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 8\beta ^{{\text{3}}} k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}\ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 16\beta ^{{\text{4}}} k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}}\ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 48\beta ^{{\text{3}}} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 176\beta ^{{\text{4}}} \varepsilon k^{3} \mu _{1} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 32\alpha \beta ^{4} \varepsilon k\mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} - 128\alpha \beta ^{{\text{4}}} \varepsilon k\mu _{1}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 224\alpha \beta ^{4} \varepsilon k\mu _{1}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 48\alpha \beta ^{2} \varepsilon k^{3} \mu _{1}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} \\ & \quad - 80\alpha \beta ^{3} \varepsilon k^{2} \mu _{1}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu }}} - 72\alpha \beta ^{2} \varepsilon k^{3} \mu _{1}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 168\alpha \beta ^{{\text{3}}} \varepsilon k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 96\alpha \beta ^{2} \varepsilon k^{3} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 256\alpha \beta ^{{\text{3}}} \varepsilon k^{2} \mu _{{\text{1}}}^{2} \ln \frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}. \\ \end{aligned}
$$
Let \(\pi_{R}^{N} = \frac{{A_{5}^{2} }}{{32\beta \varepsilon^{{2}} \mu_{{1}}^{{2}} ({2}\beta + k)^{2} }}\) and \(\pi_{R}^{Sl} = \frac{{B_{5}^{2} }}{{8\beta \varepsilon^{2} \mu_{1}^{2} (2\beta + k)^{2} }}\),
where
$$ \begin{aligned} A_{{5}} & = {4}\beta^{{2}} \varepsilon + {2}\beta \varepsilon {\text{k}} - \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right) + 4\alpha \varepsilon k\mu_{1} + 4\alpha \beta \varepsilon \mu_{1} , \\ B_{{5}} & = {2}\left[ {{2}\beta^{2} \varepsilon + \beta \varepsilon k - \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right) + 2\alpha \varepsilon k\mu_{1} + 2\alpha \beta \varepsilon \mu_{1} } \right]. \\ \end{aligned} $$
Since \(\left( {B_{5} - A_{5} } \right) = \beta^{2} \mu_{1} \ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right) \ge 0\),\({B}_{5}\ge {A}_{5}\). Thus,\(\pi_{R}^{Sl} \ge \pi_{R}^{N}\).
Next, note that \(\pi_{R}^{N} - \pi_{R}^{Pl} = \, \frac{{\beta (\beta + 2k)^{2} C_{5} }}{{64\varepsilon^{2} \mu_{1} (4\beta^{3} + 16\beta^{2} + 19\beta^{2} k + 6k^{3} )^{2} }}\),
where
$$ \begin{aligned} C_{5} & = 64\beta^{4} \varepsilon \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} - 32\beta^{4} \varepsilon \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} + 16\beta^{4} \mu_{1} \left( {\ln \left( {\frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right)} \right)^{2} + 4\beta^{4} \mu_{1} \left( {\ln \left( {\frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)} \right)^{2} + 48\beta \varepsilon k^{3} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} \\ & \quad + 192\beta^{3} \varepsilon k\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 64\beta^{2} \varepsilon k^{2} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} - 4\beta k^{3} \mu_{1} \left( {\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right)^{2} - 120\beta^{2} \varepsilon k^{2} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} - \beta k^{3} \mu_{1} \left( {\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)^{2} \\ & \quad + 12\beta^{3} k\mu_{1} \left( {\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)^{2} + 176\beta^{2} \varepsilon k^{2} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} - 16\beta k^{3} \mu_{1} \left( {\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right)^{2} - 48\beta^{3} k\mu_{1} \left( {\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right)^{2} - 4\beta^{2} k^{2} \mu_{1} \left( {\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}} \right)^{2} \\ & \quad + 9\beta^{2} k^{2} \mu_{1} \left( {\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}} \right)^{2} + 24\beta \varepsilon k^{3} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} + 32\beta^{3} \varepsilon k\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} - 36\beta \varepsilon k^{3} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} - 112\beta^{3} \varepsilon k\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} \\ & \quad - 48\beta^{2} k^{2} \mu_{1} \left( {\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}} \right)^{2} + 16\beta k^{3} \mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} + 8\beta k^{3} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} + 8\beta^{3} k\mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} \\ & \quad - 16\beta^{2} k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} + 48\alpha \varepsilon k^{3} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} - 32\alpha \beta^{3} \varepsilon \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} - 72\alpha \varepsilon k^{3} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} \\ & \quad + 16\beta^{2} k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 64\alpha \beta^{3} \varepsilon \mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 16\beta^{2} k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 64\alpha \beta^{3} \varepsilon \mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} \\ & \quad + 16\beta^{2} k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}\ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 96\alpha k^{3} \mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} - 4\beta k^{3} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} - 8\beta^{3} k\mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}}\ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} \\ & \quad + 256\alpha \beta \varepsilon k^{2} \mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 224\alpha \beta^{2} \varepsilon k\mu_{1} \ln \frac{{\overline{\mu }}}{{\overline{\mu } - \varepsilon }} + 80\alpha \beta \varepsilon k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} + 32\alpha \beta^{2} \varepsilon k\mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu }}} \\ & \quad - 168\alpha \beta \varepsilon k^{2} \mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }} - 128\alpha \beta^{2} \varepsilon k\mu_{1} \ln \frac{{\overline{\mu } + \varepsilon }}{{\overline{\mu } - \varepsilon }}. \\ \end{aligned} $$
Again, for sufficiently large values of \(\alpha\), it can be shown that \({C}_{5}\)≥0, and thus, we have \(\pi_{R}^{N} \ge \pi_{R}^{Pl}\).
A-12. Proof of Proposition 6
This proof is similar to that of Proposition 5.
A-13. Proof of \((\pi_{M1}^{N} - \pi_{M1}^{Ph} ) + (\pi_{R}^{N} - \pi_{R}^{Ph} ) > (\pi_{M2}^{Ph} - \pi_{M2}^{N} )\).
To prove this result, we show that \((\pi_{M2}^{Ph} - \pi_{M2}^{N} ) - (\pi_{M1}^{N} - \pi_{M1}^{Ph} ) - (\pi_{R}^{N} - \pi_{R}^{Ph} ) < 0\). Using some basic algebra, we obtain
\((\pi_{M2}^{Ph} - \pi_{M2}^{N} ) - (\pi_{M1}^{N} - \pi_{M1}^{Ph} ) - (\pi_{R}^{N} - \pi_{R}^{Ph} ) = \, \frac{{(\beta + 2k)^{2} A_{6} }}{{64\varepsilon^{2} \mu_{1} (4\beta^{3} + 16\beta^{2} k + 19\beta k^{2} + 6k^{3} )^{2} }}\),
where
$$ \begin{aligned} A_{6} & = 16\beta ^{5} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)^{2} + 32\beta ^{5} \varepsilon \ln \frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 64\beta ^{5} \varepsilon {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 4\beta ^{5} \mu _{1} \left( {\ln \left( {\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)} \right)^{2} - 256\alpha ^{2} \beta ^{3} \varepsilon ^{3} \mu _{1} - 576\alpha ^{2} \varepsilon ^{3} k^{3} \mu _{1} \\ & \quad - 224\beta ^{4} \varepsilon k{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 256\beta ^{4} \varepsilon \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 36\beta ^{2} \varepsilon k^{3} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 120\beta ^{3} \varepsilon k^{2} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 16\beta ^{5} \varepsilon \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} - 12\beta ^{4} k\mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} \\ & \quad - 72\beta ^{2} \varepsilon k^{3} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 240\beta ^{3} \varepsilon k^{2} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 48\beta ^{4} k\mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)^{2} + 512\alpha \beta ^{3} \varepsilon ^{3} \mu _{1} + \beta ^{2} k^{3} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} - 9\beta ^{3} k^{2} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} \\ & \quad + 576\alpha \varepsilon ^{2} k^{3} \mu _{1} + 112\beta ^{4} \varepsilon k{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 4\beta ^{2} k^{3} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)^{2} + 36\beta ^{3} k^{2} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)^{2} - 56\beta ^{2} k^{3} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} \\ & \quad + 1920\alpha \beta \varepsilon ^{2} k^{2} \mu _{1} + 1792\alpha \beta ^{2} \varepsilon ^{2} k\mu _{1} + 32\alpha \beta ^{4} \varepsilon \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 16\beta ^{2} \varepsilon k^{3} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right)^{2} + 16\beta ^{3} \varepsilon k^{2} \mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}} \right) \\ & \quad - 1344\alpha ^{2} \beta \varepsilon ^{2} k^{2} \mu _{1} - 1024\alpha ^{2} \beta ^{2} \varepsilon ^{3} k\mu _{1} - 64\alpha \beta ^{4} \varepsilon \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 4\beta ^{2} k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 288\beta \varepsilon k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad + 896\beta ^{3} \varepsilon k\mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 128\alpha \beta ^{4} \varepsilon ^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 80\beta ^{4} \varepsilon k\mu _{1} \left( {{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }}} \right)^{2} + 960\beta ^{2} \varepsilon k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 144\alpha \beta k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad - 256\alpha \beta ^{3} \varepsilon k\mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 168\alpha \beta ^{2} \varepsilon k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 384\alpha \beta \varepsilon ^{2} k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 576\alpha \beta ^{3} \varepsilon ^{2} k\mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} - 336\alpha \beta ^{2} \varepsilon k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} \\ & \quad + 192\alpha \beta \varepsilon ^{2} k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 128\alpha \beta ^{3} \varepsilon ^{2} k\mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} - 832\alpha \beta ^{2} \varepsilon ^{2} k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 32\beta ^{4} \varepsilon k\mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} \\ & \quad + 320\alpha \beta ^{2} \varepsilon ^{2} k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 72\alpha \beta \varepsilon k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 128\alpha \beta ^{3} \varepsilon k\mu _{1} {\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} + 32\beta ^{2} \varepsilon k^{3} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }} + 64\beta ^{3} \varepsilon k^{2} \mu _{1} {\text{ln}}\frac{{\bar{\mu }}}{{\bar{\mu } - \varepsilon }}{\text{ln}}\frac{{\bar{\mu } + \varepsilon }}{{\bar{\mu } - \varepsilon }} \\ \end{aligned} $$
For sufficiently large \(\alpha\), it can be shown that \(A_{6} < 0\). Hence, we have the desired result.
A-14. Proof of Proposition 9
For given w1 and w2, we know that \(\pi_{{M_{1} }}^{RSC} = {\text{w}}_{1} d_{1}\), \(\pi_{{M_{{2}} }}^{RSC} = {\text{w}}_{{2}} d_{{2}}\),
\(\pi_{R}^{RSC} = ({\text{p}}_{1} - w_{1} - c_{i(v)} )d_{1} + ({\text{p}}_{2} - w_{2} - c_{i(v)} )d_{2}\) and \(\pi_{SC}^{RSC} = \pi_{R}^{RSC} + \pi_{{M_{1} }}^{RSC} + \pi_{{M_{2} }}^{RSC}\).
Thus, we have the following results:
\(\frac{{\partial \pi_{SC}^{RSC} }}{{\partial w_{1} }} \le 0\), we can get that \(w_{1} \ge a + bw_{2}\), that is \(\pi_{SC}^{RSC}\) decreases in w1.
\(\frac{{\partial \pi_{SC}^{RSC} }}{{\partial w_{2} }} \le 0\), we can get that \(w_{2} \ge a + bw_{1}\), that is \(\pi_{SC}^{RSC}\) decreases in \(w_{2}\).
The Hessian matrix is:
$$ H = \left[ {\begin{array}{*{20}l} {\frac{{\partial_{{\pi_{SC}^{RSC} }}^{2} }}{{\partial_{{{\text{w}}_{1}^{2} }} }}} \hfill & {\quad \frac{{\partial_{{\pi_{SC}^{RSC} }}^{2} }}{{\partial_{{w_{1} w_{2} }} }}} \hfill \\ {\frac{{\partial_{{\pi_{SC}^{RSC} }}^{2} }}{{\partial_{{w_{2} w_{1} }} }}} \hfill & {\quad \frac{{\partial_{{\pi_{SC}^{RSC} }}^{2} }}{{\partial_{{{\text{w}}_{2}^{2} }} }}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - \frac{{\beta_{1} }}{2} - \frac{k}{2}} \hfill & {\quad \frac{k}{2}} \hfill \\ \frac{k}{2} \hfill & {\quad - \frac{{\beta_{1} }}{2} - \frac{k}{2}} \hfill \\ \end{array} } \right] $$
and
$$ |H| = \left( { - \frac{{\beta_{1} }}{2} - \frac{{\text{k}}}{2}} \right)^{2} - \left( \frac{k}{2} \right)^{2} = \frac{{\beta_{1}^{2} }}{4} + \frac{{\beta_{1} k}}{2} > o $$
Therefore, we can show that the Hessian matrix H is a diagonally dominant matrix, which guarantees joint concavity of the function \(\pi_{SC}^{RSC} (w_{1} ,w_{2} )\).
We can get that
$$ \left\{ {\begin{array}{*{20}l} {\frac{a}{1 - b} \le w_{1} \le w_{1}^{N*} } \hfill \\ {a + bw_{1} \le w_{2} \le \frac{{w_{1} - a}}{b}} \hfill \\ \end{array} } \right. $$
where \(a = \frac{{\beta_{1}^{2} \mu_{1}^{2} v^{2} + 2\beta_{1} \beta_{2} \mu_{1} v - 3\beta_{2}^{2} }}{{2\beta_{1} \mu_{1} (k + \beta_{1} )}}\) and \(b = \frac{k}{{k + \beta_{1} }}\).
In that case, the profit of the supply chain decreases in \(w_{1}\) and \(w_{2}\). And we prove that \(\pi_{SC}^{RSC} \ge \pi_{SC}^{N}\), that is the revenue sharing contract realizes supply chain coordination under complete efficient cost sharing scenario. Then we distribute the profits according to a certain proportion, purposing to achieve the optimal profit for each participant.
A-15. Proof of condition 2.
This proof is similar to that of Proposition 9
A-16. The retailer’s profit as ‘k’ increases
See Fig. 7.