Access pricing in network industries with mixed oligopoly

Abstract

We characterize optimal regulated access prices in mixed oligopoly network industries where a private, profit-maximizing firm competes against a public enterprise after purchasing an essential input (e.g., network access). Optimal access prices often are lower for the private firm than for the public enterprise, and can be particularly low for a relatively efficient private supplier. In contrast to a private, profit-maximizing input supplier, the regulator reduces the access price charged to a private supplier as it becomes more efficient. The optimal access price for a private firm is the same whether it competes against another private firm or a public enterprise.

Appendix

Proof of Lemma 1

⁴⁷

We first derive the demand functions for the products of the downstream suppliers. If downstream supplier $i\in \{1,2\}$ seeks to maximize its profit plus the fraction $\alpha _{i}\in \left[ \,0,1\,\right]$ of consumer surplus, the supplier’s problem, [Pi], is:

$$\begin{aligned} \underset{p_{i}}{Maximize} \left[ \,p_{i}-w_{i}-c_{i}\,\right] X_{i}({\mathbf {p}})+\alpha _{i}\,[\,U(X_{i}( {\mathbf {p}}),X_{j}({\mathbf {p}}))-\mathop {\sum }\limits _{i\,=\,1}^{2}p_{i}\,X_{i}( {\mathbf {p}})\,]\,\text {.} \end{aligned}$$

The necessary conditions for a solution to [Pi] are, for $\,i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&X_{i}({\mathbf {p}})+\left[ \,p_{i}-w_{i}-c_{i}\,\right] \frac{ \partial X_{i}({\mathbf {p}})}{\partial p_{i}} \nonumber \\&\qquad +\,\,\alpha _{i}\left[ \,\frac{\partial U(X_{i}({\mathbf {p}} ),X_{j}({\mathbf {p}}))}{\partial p_{i}}-X_{i}({\mathbf {p}})-p_{i}\,\frac{X_{i}( \mathbf {p)}}{\partial p_{i}}-p_{j}\,\frac{X_{j}(\mathbf {p)}}{\partial p_{i}} \,\right] \,\,=0 \nonumber \\&\quad \Leftrightarrow \,\left[ \,1-\alpha _{i}\,\right] X_{i}({\mathbf {p}})+\left[ \,p_{i}-w_{i}-c_{i}\,\right] \frac{\partial X_{i}( {\mathbf {p}})}{\partial p_{i}}=0\, . \end{aligned}$$

(9)

The equivalence in (9) holds because consumer utility maximization ensures $\,\frac{\partial U(\cdot )}{\partial X_{i}({\mathbf {p}})} =p_{i}$ for $i\in \{1,2\}$. This equality and (1) imply that for $\,i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&u_{i}-\frac{1}{1+b}\left[ \,x_{i}+b\,x_{j}\,\right] \,\,=p_{i}\Rightarrow \left[ \,1+b\,\right] \left[ \,u_{i}-p_{i}\,\right] \,\,=x_{i}+b\,x_{j} \nonumber \\&\quad \Rightarrow x_{i}=\left[ \,1+b\, \right] \left[ \,u_{i}-p_{i}\,\right] -b\left[ \,\left( \,1+b\,\right) \left( \,u_{j}-p_{j}\,\right) -b\,x_{i}\,\right] \nonumber \\&\quad \Rightarrow x_{i}\left[ \,1-b^{2}\,\right] \,\,=\,\,\left[ \,1+b\,\right] \left[ \,u_{i}-p_{i}\,\right] -b\left[ \,1+b\, \right] \left[ \,u_{j}-p_{j}\,\right] \nonumber \\&\quad \Rightarrow X_{i}({\mathbf {p}})= \frac{u_{i}-b\,u_{j}}{1-b}-\left[ \,\frac{1}{1-b}\,\right] p_{i}+\left[ \, \frac{b}{1-b}\,\right] p_{j}{. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(10)

We now specify equilibrium prices as a function of the established retail prices. (9) and (10) imply:

$$\begin{aligned}&\left[ \,1-\alpha _{i}\,\right] \left[ \,\frac{u_{i}-b\,u_{j}}{1-b}-\frac{ p_{i}}{1-b}+\frac{b\,p_{j}}{1-b}\,\right] +\,\left[ \,p_{i}-w_{i}-c_{i}\, \right] \left[ \,-\,\frac{1}{1-b}\,\right] \,=0 \nonumber \\&\quad \Rightarrow p_{i}=\frac{w_{i}+c_{i} }{2-\alpha _{i}}+\left[ \,\frac{1-\alpha _{i}}{2-\alpha _{i}}\,\right] \left[ \,u_{i}-b\,u_{j}\,\right] +\frac{b\left[ \,1-\alpha _{i}\,\right] }{2-\alpha _{i}}\,p_{j}{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \nonumber \\&\quad \Rightarrow p_{i}({\mathbf {w}})= \frac{1}{D\,}\,\{\,\left[ \,2-\alpha _{j}\,\right] \left[ \,w_{i}+c_{i}\, \right] +b\left[ \,1-\alpha _{i}\,\right] \left[ \,w_{j}+c_{j}\,\right] \nonumber \\&\qquad +\,\left[ \,1-\alpha _{i}\,\right] \left[ \,2-\alpha _{j}\,\right] \left[ \,u_{i}-b\,u_{j}\,\right] +b\left[ \,1-\alpha _{i}\,\right] \left[ \,1-\alpha _{j}\,\right] \left[ \,u_{j}-b\,u_{i}\,\right] \,\} \end{aligned}$$

(11)

where $\,\mathbf {w\,\equiv }\,(w_{1},w_{2})$ and:

$$\begin{aligned} D\equiv \,\,\left[ \,2-\alpha _{i}\,\right] \left[ \,2-\alpha _{j}\, \right] -b^{2}\left[ \,1-\alpha _{i}\,\right] \left[ \,1-\alpha _{j}\,\right] \,\,>0\,. \end{aligned}$$

(12)

Equations (11) and (12) imply that when $\,\alpha _{i}\,=\,\alpha _{j}\,=0$:

$$\begin{aligned} p_{i}^{0*}=w_{i}+c_{i}+\frac{1}{4-b^{2}}\left[ \,\left( 2-b^{2}\right) \left( \,u_{i}-w_{i}-c_{i}\,\right) -b\left( \,u_{j}-w_{j}-c_{j}\,\right) \,\right] \text {.} \end{aligned}$$

(13)

(11) and (12) imply that when $\,\alpha _{1}\,=\,1\,$ and $\,\alpha _{2}\,=\,0$:

$$\begin{aligned} p_{1}^{*}=\frac{1}{2\,}\,\{\,2\left[ \,w_{1}+c_{1}\, \right] \,\}=w_{1}+c_{1}\,\text {; and } \end{aligned}$$

(14)

$$\begin{aligned} p_{2}^{*}=\frac{1}{2}\left[ \,u_{2}+w_{2}+c_{2}-b\left( \,u_{1}-w_{1}-c_{1}\,\right) \right] {. \ } \ \ \ \ \ \end{aligned}$$

(15)

We now proceed to characterize optimal access prices. (10) and ( 11) imply:

$$\begin{aligned} X_{i}({\mathbf {w}})=\frac{u_{i}-b\,u_{j}}{1-b}-\left[ \,\frac{1 }{1-b}\,\right] p_{i}({\mathbf {w}})+\left[ \,\frac{b}{1-b}\,\right] p_{j}( {\mathbf {w}})\,\text {.} \end{aligned}$$

(16)

It is readily verified that industry profit, given access prices ${\mathbf {w}}$ is:

$$\begin{aligned} \Pi ({\mathbf {w}})=\,\,\left[ \,p_{1}({\mathbf {w}})-c_{u}-c_{1}\,\right] X_{1}({\mathbf {w}})+\left[ \,p_{2}({\mathbf {w}})-c_{u}-c_{2}\,\right] X_{2}( {\mathbf {w}})-F\,\text {.} \end{aligned}$$

(17)

Recall $\lambda$ is the Lagrange multiplier associated with (4) and $v=\gamma +\lambda$. (3), (4), and (17 ) imply that the necessary conditions for a solution to [RP] include, for $i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&\frac{\partial U(X_{i}({\mathbf {w}}),X_{j}({\mathbf {w}}))}{\partial X_{i}( {\mathbf {w}})}\,\frac{\partial X_{i}({\mathbf {w}})}{\partial w_{i} }+\frac{\partial U(X_{i}({\mathbf {w}}),X_{j}({\mathbf {w}}))}{\partial X_{j}( {\mathbf {w}})}\,\frac{\partial X_{j}({\mathbf {w}})}{\partial w_{i} }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad -\,\left[ \,p_{i}({\mathbf {w}})\,\frac{\partial X_{i}({\mathbf {w}} )}{\partial w_{i}}+\frac{\partial \,p_{i}({\mathbf {w}})}{\partial w_{i}} \,X_{i}({\mathbf {w}})+p_{j}({\mathbf {w}})\,\frac{\partial X_{j}({\mathbf {w}})}{ \partial w_{i}}+\frac{\partial p_{j}({\mathbf {w}})}{\partial w_{i}}\,X_{j}( {\mathbf {w}})\,\right] \ \ \ \ \ \ \ \nonumber \\&\quad +\,\,v\left[ \,\left( \,p_{i}({\mathbf {w}})-c_{u}-c_{i}\,\right) \,\frac{ \partial X_{i}({\mathbf {w}})}{\partial w_{i}}+\frac{\partial \,p_{i}({\mathbf {w}} )}{\partial w_{i}}\,X_{i}({\mathbf {w}})\right. \nonumber \\&\quad \left. +\,\left( \,p_{j}({\mathbf {w}})-c_{u}-c_{j}\,\right) \,\frac{\partial X_{j}({\mathbf {w}})}{ \partial w_{i}}+\frac{\partial p_{j}({\mathbf {w}})}{\partial w_{i}}\,X_{j}( {\mathbf {w}})\,\right] \nonumber \\&=\ v\left[ \,\left( \,p_{i}({\mathbf {w}})-c_{i}-c_{u}\,\right) \,\frac{\partial X_{i}({\mathbf {w}})}{\partial w_{i}}+\left( \,p_{j}({\mathbf {w}} )-c_{j}-c_{u}\,\right) \,\frac{\partial X_{j}({\mathbf {w}})}{\partial w_{i}}\, \right] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad +\,\left[ \,v-1\,\right] \left[ \,\frac{\partial \,p_{i}( {\mathbf {w}})}{\partial w_{i}}\,X_{i}({\mathbf {w}})+\frac{ \partial p_{j}({\mathbf {w}})}{\partial w_{i}}\,X_{j}({\mathbf {w}} )\,\right] \,\,=0{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(18)

where, from (11) and (16):

$$\begin{aligned}&\frac{\partial p_{i}({\mathbf {w}})}{\partial w_{i}}\ =\ \frac{2-\alpha _{j}}{ D\,}\,\text {; }\frac{\partial p_{j}({\mathbf {w}})}{\partial w_{i}} =\frac{b\left[ \,1-\alpha _{j}\,\right] }{D}\,\text {;}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\frac{\partial X_{i}({\mathbf {w}})}{\partial w_{i}}\ =\frac{b^{2}\left[ \,1-\alpha _{j}\,\right] -\left[ \,2-\alpha _{j}\,\right] }{D\left[ \,1-b\, \right] }\,\text {;}\ \ \text {and }\frac{\partial X_{j}({\mathbf {w}})}{ \partial w_{i}}=\frac{b}{D\left[ \,1-b\,\right] \,}\,\text {.} \end{aligned}$$

(19)

Equations (10) and (19) imply that (18) can be written as:

$$\begin{aligned}&0=\frac{v}{D\left[ \,1-b\,\right] }\,\{\,\left[ \,p_{i}( {\mathbf {w}})-c_{i}-c_{u}\,\right] \left[ \,b^{2}\left( \,1-\alpha _{j}\,\right) \,-\,\left( \,2-\alpha _{j}\,\right) \,\right] +\,b\left[ \,p_{j}({\mathbf {w}})-c_{j}-c_{u}\,\right] \,\} \nonumber \\&\quad +\,\,\frac{\,v-1}{D}\left\{ \,\frac{\left[ \,2-\alpha _{j}\,\right] }{1-b} \left[ \,u_{i}-b\,u_{j}-p_{i}({\mathbf {w}})+b\,p_{j}({\mathbf {w}})\,\right] \right. \, \nonumber \\&\quad \left. +\,\,\frac{b\left[ \,1-\alpha _{j}\,\right] }{1-b} \left[ \,u_{j}-b\,u_{i}-p_{j}({\mathbf {w}})+b\,p_{i}({\mathbf {w}})\,\right] \,\right\} .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(20)

Tedious calculations reveal that (11) and (20) imply the optimal prices are:

$$\begin{aligned} w_{i}= & {} \frac{T_{i}}{b^{2}\,\gamma _{1}\,\gamma _{3}-\gamma _{2}\,\gamma _{4}}\,\text { and }w_{j}=\frac{T_{j} }{b^{2}\,\gamma _{1}\,\gamma _{3}-\gamma _{2}\,\gamma _{4}}\text {, where} \end{aligned}$$

(21)

$$\begin{aligned} \gamma _{1}= & {} \left[ \,2\,v-1\,\right] \left\{ \,2-\alpha _{i}+\left[ \,b^{2}\left( \,1-\alpha _{j}\,\right) -\left( \,2-\alpha _{j}\,\right) \,\right] \left[ \,1-\alpha _{i}\,\right] \,\right\} , \nonumber \\ \gamma _{2}= & {} \left[ \,2\,v-1\,\right] \left\{ \,b^{2}\left[ \,1-\alpha _{j}\,\right] \left[ \,3-\alpha _{j}\,\right] - \left[ \,2-\alpha _{j}\,\right] ^{2}\,\right\} , \,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\ \gamma _{3}= & {} \left[ \,2\,v-1\,\right] \left\{ \,2-\alpha _{j}+\left[ \,b^{2}\left( \,1-\alpha _{i}\,\right) -\left( \,2-\alpha _{i}\,\right) \,\right] \left[ \,1-\alpha _{j}\,\right] \,\right\} , \nonumber \\ \gamma _{4}= & {} \left[ \,2\,v-1\,\right] \left\{ \,b^{2}\left[ \,1-\alpha _{i}\,\right] \left[ \,3-\alpha _{i}\,\right] - \left[ \,2-\alpha _{i}\,\right] ^{2}\,\right\} , \nonumber \\ T_{i}= & {} \left[ \,\gamma _{1}\,\theta _{i}-\gamma _{4}\,\right] v\,b\,D\,c_{j}+\left[ \,\gamma _{4}\,\gamma _{2}-\gamma _{4}\,v\,D\,\theta _{j}\,-\,\gamma _{1}\,b^{2}\,\gamma _{3}+\gamma _{1}\,b^{2}\,v\,D\,\right] c_{i}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&-\,\,\left[ \,\gamma _{4}\left( \,\theta _{j}+b\,\right) -\gamma _{1}\,b\left( \,\theta _{i}+b\,\right) \,\right] v\,D\,c_{u}\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&+ \left\{ \,\left[ \,\gamma _{2}\,\gamma _{4}-b^{2}\,\gamma _{1}\,\gamma _{3}\,\right] \left[ \,1-\alpha _{i}\,\right] +\left[ \,\gamma _{4}\left( \,2-\alpha _{j}\,\right) -b^{2}\,\gamma _{1}\left( \,1-\alpha _{i}\,\right) \,\right] \left[ \,v-1\,\right] D\, \right\} \, \left[ \,u_{i}-b\,u_{j}\,\right] \nonumber \\&+\left[ \,\gamma _{4}\left( \,1-\alpha _{j}\,\right) -\gamma _{1}\left( \,2-\alpha _{i}\,\right) \,\right] \left[ \,v-1\,\right] D\,b\left[ \,u_{j}-b\,u_{i}\,\right] ,\ \ \text {and}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\ T_{j}= & {} \left[ \,\gamma _{3}\,\theta _{j}-\gamma _{2}\,\right] v\,b\,D\,c_{i}+\left[ \,\gamma _{4}\,\gamma _{2}-\gamma _{2}\,v\,D\,\theta _{i}\,-\,\gamma _{1}\,b^{2}\,\gamma _{3}+\gamma _{3}\,b^{2}\,v\,D\,\right] c_{j}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&-\,\,\left[ \,\gamma _{2}\left( \,\theta _{i}+b\,\right) -\gamma _{3}\,b\left( \,\theta _{j}+b\,\right) \,\right] v\,D\,c_{u}\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&+ \left\{ \,\left[ \,\gamma _{2}\,\gamma _{4}-b^{2}\,\gamma _{1}\,\gamma _{3}\,\right] \left[ \,1-\alpha _{j}\,\right] +\left[ \,\gamma _{2}\left( \,2-\alpha _{i}\,\right) -b^{2}\,\gamma _{3}\left( \,1-\alpha _{j}\,\right) \,\right] \left[ \,v-1\,\right] D\, \right\} \, \left[ \,u_{j}-b\,u_{i}\,\right] \nonumber \\&+\left[ \,v-1\,\right] D\,b\left[ \,u_{i}-b\,u_{j}\, \right] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\text {where }\theta _{i}=b^{2}\left[ \,1-\alpha _{i}\, \right] -\left[ \,2-\alpha _{i}\,\right] \text { and }\theta _{j} =b^{2}\left[ \,1-\alpha _{j}\,\right] -\left[ \,2-\alpha _{j}\,\right] . \end{aligned}$$

(22)

When firm 1 is a public enterprise (so $\alpha _{1}\,=\,1\,$ and $\,\alpha _{2}\,=\,0$):

$$\begin{aligned}&\gamma _{1}=\gamma _{3}=\ 2\,v-1\,\text {;}\ \ \ \gamma _{2}=\,\,\left[ \,2\,v-1\,\right] \left[ \,3\,b^{2}-4\,\right] \text {;}\ \ \ \gamma _{4}=\,\,-\,\left[ \,2\,v-1\,\right] \text {;} \nonumber \\&\theta _{i}=\,\,-\,1\,\text {;}\ \ \theta _{j} =b^{2}-2\,\text {;}\ \ \text {and }\ D=2\, \text {.}\, \end{aligned}$$

(23)

Tedious calculations using (21), (22 ), and (23) reveal that when $v=v^{*}\ne \frac{1}{2 }$:

$$\begin{aligned} w_{1}^{*}=c_{u}+\frac{v^{*}-1}{2\,v^{*}-1} \,\Delta _{1}\text { and }w_{2}^{*}=c_{u}-\frac{ \Delta _{2}-b\,v^{*}\,\Delta _{1}}{2\,v^{*}-1\,}\,\text {.} \end{aligned}$$

(24)

When firm 1 is a private firm (so $\alpha _{1}\,=\,\alpha _{2}\,=\,0$):

$$\begin{aligned} \gamma _{1}=\gamma _{3}=\,\,\left[ \,2\,v-1\,\right] b^{2}\ \text {;}\ \ \ \gamma _{2}=\gamma _{4}=\,\,\left[ \,2\,v-1\,\right] \left[ \,3\,b^{2}-4\,\right] \text {;} \nonumber \\ \theta _{i}=\,\theta _{j}=\,b^{2}-2\,\text {;} \ \ \text { and }\ D=4-b^{2}\,{. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(25)

Tedious calculations using (21), (22 ), and (25) reveal that when $v=v^{0*}\ne \frac{1}{ 2}$:

$$\begin{aligned} w_{1}^{0*}=c_{u}-\frac{\Delta _{1}-b\,v^{0*}\,\Delta _{2}}{2\,v^{0*}-1\,}\text { and }w_{2}^{0*}=c_{u}- \frac{\Delta _{2}-b\,v^{0*}\,\Delta _{1}}{2\,v^{0*}-1\,}\,\text {.} \end{aligned}$$

(26)

Equations (13) and (26) imply that for $i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned} p_{i}^{0*}= & {} w_{i}^{0*}+c_{i}+\frac{1}{ 4-b^{2}}\left[ \,\left( 2-b^{2}\right) \left( \,u_{i}-w_{i}^{0*}-c_{i}\,\right) -b\left( \,u_{j}-w_{j}^{0*}-c_{j}\,\right) \,\right] \nonumber \\= & {} c_{u}+c_{i}+\frac{\left[ \,v^{0*}-1\,\right] \Delta _{i}}{2\,v^{0*}-1}\,{. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(27)

Equations (14) and (24) imply:

$$\begin{aligned} p_{1}^{*}=w_{1}^{*}+c_{1}= c_{u}+c_{1}+\frac{\left[ \,v^{*}-1\,\right] \Delta _{1}}{ 2\,v^{*}-1}\,\text {.} \end{aligned}$$

(28)

Equations (15) and (24) imply:

$$\begin{aligned} p_{2}^{*}=\frac{1}{2}\left[ \,u_{2}+w_{2}^{*}+c_{2}-b\left( \,u_{1}-w_{1}^{*}-c_{1}\,\right) \,\right] \,\,= c_{u}+c_{2}+\frac{\left[ \,v^{*}-1\,\right] \Delta _{2}}{2\,v^{*}-1} \,\text {.} \end{aligned}$$

(29)

Equations (16) and (27) imply that for $i,j\in \{1,2\}$ ($i\ne j$):

$$\begin{aligned} X_{i}^{0*}= & {} \frac{1}{1-b}\left[ \,u_{i}-b\,u_{j}-p_{i}^{0 *}+b\,p_{j}^{0*}\,\right] \,{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \nonumber \\= & {} \frac{1}{1-b}\left[ \,u_{i}-b\,u_{j}- \left( c_{u}+c_{i}+\frac{\left[ \,v^{0*}-1\,\right] \Delta _{i}}{ 2\,v^{0*}-1}\right) +b\left( c_{u}+c_{j}+\frac{\left[ \,v^{0*}-1\, \right] \Delta _{j}}{2\,v^{0*}-1}\right) \,\right] \nonumber \\= & {} \frac{1}{1-b}\left[ \,\Delta _{i}-\frac{\left[ \,v^{0*}-1\,\right] \Delta _{i}}{2\,v^{0*}-1}-b\,\Delta _{j}+b\, \frac{\left[ \,v^{0*}-1\,\right] \Delta _{j}}{2\,v^{0*}-1}\,\right] { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \nonumber \\= & {} \frac{1}{\left[ \,1-b\,\right] \left[ \,2\,v^{0*}-1\,\right] }\left[ \,v^{0*}\,\Delta _{i}-b\,v^{0*}\,\Delta _{j}\, \right] \,\,=\frac{v^{0*}\left[ \,\Delta _{i}-b\,\Delta _{j}\, \right] }{\left[ \,1-b\,\right] \left[ \,2\,v^{0*}-1\,\right] }\,{. \ \ \ \ \ \ \ \ \ \ \ } \end{aligned}$$

(30)

Similarly, (16), (28), and (29) imply that for $i,j\in \{1,2\}$ ($i\ne j$):

$$\begin{aligned} X_{i}^{*}=\frac{1}{1-b}\left[ \,u_{i}-b\,u_{j}-p_{i}^{*}+b\,p_{j}^{*}\,\right] = \frac{v^{*}\left[ \,\Delta _{i}-b\,\Delta _{j}\,\right] }{\left[ \,1-b\, \right] \left[ \,2\,v^{*}-1\,\right] }\, . \end{aligned}$$

(31)

Equations (17), (28), (29), and (31) imply that when firm 1 is a public enterprise, industry profit is:

$$\begin{aligned}&\Pi ({\mathbf {w}})= \,\,\left[ \,p_{1}^{*}-c_{u}-c_{1}\,\right] X_{1}^{*}+\left[ \,p_{2}^{*}-c_{u}-c_{2}\,\right] X_{2}^{*}-F \nonumber \\&\quad =\frac{\left[ \,v^{*}-1\,\right] \Delta _{1}}{ 2\,v^{*}-1}\,\frac{v^{*}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] }{\left[ \,1-b\,\right] \left[ \,2\,v^{*}-1\,\right] }+\frac{\left[ \,v^{*}-1\,\right] \Delta _{2}}{2\,v^{*}-1}\,\frac{v^{*}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] }{\left[ \,1-b\,\right] \left[ \,2\,v^{*}-1\,\right] }-F\, \nonumber \\&\quad =\ g(v^{*})\,H-F \end{aligned}$$

(32)

$$\begin{aligned}&\text {where }g(v)\ \equiv \ \frac{v\left[ \,v-1\,\right] }{\left[ \,1-b\, \right] \left[ \,2\,v-1\,\right] ^{2}} > reqless 0\ \ \Leftrightarrow \ \ v > reqless 1\, \end{aligned}$$

(33)

$$\begin{aligned}&\text {and }H\ \equiv \ \Delta _{1}\left[ \,\Delta _{1}\,-b\,\Delta _{2}\, \right] +\Delta _{2}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \,\,=\,\, \left[ \,\Delta _{1}\,\right] ^{2}-2\,b\,\Delta _{1}\,\Delta _{2}+\left[ \,\Delta _{2}\,\right] ^{2}\,>0\,\text {.} \qquad \end{aligned}$$

(34)

Corresponding calculations using (17), (27), and (30) imply that when firm 1 is a private firm, industry profit is $g(v^{0*})\,H-F$.

Case (i). $\,\gamma \,\in \,[\,0,1)$.

Suppose $\lambda =0$ at the solution to [RP]. Then (4) and ( 33) imply that for $v\in \{v^{*},v^{0*}\}$:

$$\begin{aligned} v=\gamma \,\in \,[\,0,1)\ \ \Rightarrow \ \ g(v)<0\ \ \Rightarrow \ \ \Pi ({\mathbf {w}}) <0\text {.} \end{aligned}$$

(35)

Equation (35) implies (4) is violated. Therefore, by contradiction, $\lambda >0$, and so $\Pi ({\mathbf {w}})=0$ by complementary slackness. Consequently, (32) implies:

$$\begin{aligned} g(v)=\frac{F}{H}\,\text {.} \end{aligned}$$

(36)

Case (ii). $\,\gamma \,=\,1$.

If $\,\Pi ({\mathbf {w}})>0$, then (4) implies that $g(v)>\frac{F }{H}\ge 0$ for $v\in \{v^{*},v^{0*}\}$. Therefore, $v>1$ from (33). $\lambda >0$ because $\gamma =1$ and $v=\gamma +\lambda >1$. Therefore, $\Pi ({\mathbf {w}})=0$ by complementary slackness. This contradiction ensures $\Pi ({\mathbf {w}})=0$, so (36) holds.

Equation (36) implies that if $F=0$, then $g(v)=\frac{F}{H}\,=0$, so $v=1$, from (33). (36) also implies that if $F>0$, then $g(v)=\frac{F}{H}\,>0$, so $v>1$, from (33). $\square$

Proof of Corollary 1

Equations (6) and ( 7) imply $\,p_{1}^{*}\,=\,w_{1}^{*}+c_{1}\,$ and:

$$\begin{aligned}&p_{2}^{\bullet *}>w_{2}^{\bullet *}+c_{2}\Leftrightarrow c_{u}+c_{2}+\frac{\left[ \,v^{\bullet *}-1\,\right] \Delta _{2}}{2\,v^{\bullet *}-1}> c_{u}-\frac{\Delta _{2}-b\,v^{\bullet *}\,\Delta _{1}}{ 2\,v^{\bullet *}-1\,}+c_{2} \nonumber \\&\quad \Leftrightarrow \frac{\Delta _{2}-b\,v^{\bullet *}\,\Delta _{1}}{2\,v^{\bullet *}-1\, }+\frac{\left[ \,v^{\bullet *}-1\,\right] \Delta _{2}}{2\,v^{\bullet *}-1}>0\Leftrightarrow \Delta _{2} >b\,\Delta _{1}\,\text {.}\, \end{aligned}$$

(37)

The last inequality here reflects Assumption 1. $\square$

Proof of Corollary 2

$v^{*}=v^{0*}=1$ when $F=0$, from Lemma 1. Therefore, the Corollary follows immediately from (6), (7), and Assumption 1. $\square$

Proof of Lemma 2

Equation (33) implies that for $v\in \{v^{*},v^{0*}\}$:

$$\begin{aligned} \frac{\partial g(v)}{\partial v}= & {} \frac{\left[ \,2\,v-1\, \right] ^{2}\left[ \,2\,v-1\,\right] -4\left[ \,v^{2}-v\,\right] \left[ \,2\,v-1\,\right] }{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{4}} =\frac{\left[ \,2\,v-1\,\right] ^{2}-4\left[ \,v^{2}-v\, \right] }{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}} \nonumber \\= & {} \frac{4\,v^{2}-4\,v+1-4\,v^{2}+4\,v}{\left[ \,1-b\, \right] \left[ \,2\,v-1\,\right] ^{3}}=\frac{1}{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}}>0\,\text {; and } \end{aligned}$$

(38)

$$\begin{aligned} \frac{\partial g(v)}{\partial b}= & {} \frac{v\left[ \,v-1\,\right] }{\left[ \,1-b\,\right] ^{2}\left[ \,2\,v-1\,\right] ^{2}}>0 \,\text {for }\,v\,>\,1\, . \end{aligned}$$

(39)

Equation (34) implies that for $\,i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&\frac{\partial H}{\partial b}=\,\,-\,2\,\Delta _{i}\,\Delta _{j} <0\,\text {; }\frac{\partial H}{\partial F}= 0\, ; \end{aligned}$$

(40)

$$\begin{aligned}&\frac{\partial H}{\partial u_{i}}=2\left[ \Delta _{i}-b\,\Delta _{j}\,\right] \,\,>0\, ;\frac{ \partial H}{\partial c_{i}}=\,\,-\,\frac{\partial H}{\partial u_{i}}<0\,\text {; and } \nonumber \\&\frac{\partial H}{\partial c_{u}}=\,\,\Delta _{i} \left[ \,-\,1+b\,\right] -\left[ \,\Delta _{i}-b\,\Delta _{j}\,\right] +\Delta _{j}\left[ \,-\,1+b\,\right] -\left[ \,\Delta _{j}-b\,\Delta _{i}\, \right] \nonumber \\&\qquad \,\,\,=\,\,-\,2\left[ \,1-b\,\right] \left[ \,\Delta _{i}+\Delta _{j}\,\right] \,\,<0\, . \end{aligned}$$

(41)

Equations (36)–(40) imply:

$$\begin{aligned} \frac{\partial g(\cdot )}{\partial v}\,\,dv= \frac{1}{H}\,\,dF\Rightarrow \frac{dv}{dF}= \frac{1}{H\,\frac{\partial g(\cdot )}{\partial v}}> 0\,\text {. } \end{aligned}$$

(42)

Equations (36)–(40) also imply:

$$\begin{aligned}&\frac{\partial g(\cdot )}{\partial v}\,\,dv+\left[ \,\frac{\partial g(\cdot ) }{\partial b}+\frac{F}{H^{2}}\,\,\frac{\partial H}{\partial b}\,\right] db =0\Rightarrow \frac{dv}{db} =\,\,-\,\frac{\frac{\partial g(\cdot )}{\partial b}+\frac{F}{H^{2}}\,\,\frac{ \partial H}{\partial b}}{\frac{\partial g(\cdot )}{\partial v}} \nonumber \\&\quad \Rightarrow \frac{dv}{db}=\,\,-\,\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}\left\{ \,\frac{v\left[ \,v-1\, \right] }{\left[ \,1-b\,\right] ^{2}\left[ \,2\,v-1\,\right] ^{2}}-2\,\, \frac{F}{H^{2}}\,\Delta _{1}\,\Delta _{2}\,\right\} \nonumber \\&\quad =\,\,-\,\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}\left\{ \,\frac{v\left[ \,v-1\,\right] }{\left[ \,1-b\,\right] ^{2}\left[ \,2\,v-1\,\right] ^{2}}-\frac{2}{H}\,\,g_{1}(v)\,\Delta _{1}\,\Delta _{2}\,\right\} \nonumber \\&\quad =\,\,-\,\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}\left\{ \,\frac{v\left[ \,v-1\,\right] }{\left[ \,1-b\, \right] ^{2}\left[ \,2\,v-1\,\right] ^{2}}-\frac{2}{H}\,\,\frac{v\left[ \,v-1\,\right] \Delta _{1}\,\Delta _{2}}{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{2}}\,\right\} \nonumber \\&\quad =\,\,-\,v\left[ \,v-1\,\right] \left[ \,2\,v-1\,\right] \left[ \,\frac{1}{ 1-b\,}-\frac{2}{H}\,\Delta _{1}\,\Delta _{2}\,\right] \text {.}\, \end{aligned}$$

(43)

Equation (34) implies:

$$\begin{aligned}&\frac{1}{1-b\,}-\frac{2}{H}\,\Delta _{1}\,\Delta _{2}=\frac{ H-2\left[ \,1-b\,\right] \Delta _{1}\,\Delta _{2}}{H\left[ \,1-b\,\right] } \nonumber \\&\quad =\frac{1}{H\left[ \,1-b\,\right] }\,\{\,\Delta _{1} \left[ \,\Delta _{1}\,-b\,\Delta _{2}\,\right] +\Delta _{2}\left[ \,\Delta _{2}\,-b\,\Delta _{1}\,\right] -\,\,2\left[ \,1-b\,\right] \Delta _{1}\,\Delta _{2}\,\} \nonumber \\&\quad =\frac{1}{H\left[ \,1-b\,\right] }\,\{\,\left[ \,\Delta _{1}\,\right] ^{2}+\left[ \,\Delta _{2}\,\right] ^{2}-2\,\Delta _{1}\,\Delta _{2}\,\}=\frac{\left[ \,\Delta _{1}-\Delta _{2}\,\right] ^{2}}{H\left[ \,1-b\,\right] }\,\text {. } \end{aligned}$$

(44)

Equations (43) and (44) imply:

$$\begin{aligned}&\frac{dv}{db}=\,\,-\,\frac{v\left[ \,v-1\,\right] \left[ \,2\,v-1\, \right] \left[ \,\Delta _{1}-\Delta _{2}\,\right] ^{2}}{H\left[ \,1-b\, \right] }\left\{ \begin{array}{c} =0\text { if }\,v\,=\,1\,\text { or }\,\Delta _{1}\,=\,\Delta _{2} \\ <0\text { if }\,v\,>\,1\,\text { and }\,\Delta _{1}\,\ne \,\Delta _{2}\,\text {.} \end{array} \right. \end{aligned}$$

(45)

Equations (36) and (38) imply that for $x\in \{\,u_{i},\,u_{j},\,c_{i},\,c_{j},c_{u}\,\}$:

$$\begin{aligned} \frac{\partial g(\cdot )}{\partial v}\,\,dv+\frac{F}{H^{2}}\,\,\frac{ \partial H}{\partial x}\,\,dx=0\Rightarrow \frac{dv}{dx}=\,\,-\,\frac{\frac{F}{H^{2}}\,\,\frac{\partial H}{ \partial x}}{\frac{\partial g(\cdot )}{\partial v}}=\frac{F}{ H^{2}\,\frac{\partial g(\cdot )}{\partial v}}\,\left[ \,-\,\frac{\partial H}{ \partial x}\,\right] \text {.} \qquad \end{aligned}$$

(46)

Equations (33), (36), and (38) imply:

$$\begin{aligned}&\frac{F}{H^{2}\,\frac{\partial g(\cdot )}{\partial v}}=\frac{ F}{H}\,\,\frac{1}{H\,\frac{\partial g(\cdot )}{\partial v}}= g(v)\,\,\frac{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}}{H} \nonumber \\&\quad =\frac{v\left[ \,v-1\,\right] }{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{2}}\,\,\frac{\left[ \,1-b\,\right] \left[ \,2\,v-1\,\right] ^{3}}{H}=\frac{1}{H}\,v\left[ \,v-1\,\right] \left[ \,2\,v-1\,\right] \text {.} \qquad \end{aligned}$$

(47)

Equations (46) and (47) imply that for $x\in \{\,u_{i},\,u_{j},\,c_{i},\,c_{j},c_{u}\,\}$:

$$\begin{aligned} \frac{dv}{dx}=\frac{1}{H}\,v\left[ \,v-1\,\right] \left[ \,2\,v-1\,\right] \left[ \,-\,\frac{\partial H}{\partial x}\,\right] \,\, \overset{s}{=}\,\,-\,\frac{\partial H}{\partial x}\text { if }v >1\,\text {.} \end{aligned}$$

(48)

The observations in the Lemma follow directly from (40) to (42), (45), and (48). $\square$

Proof of Corollary 3

Lemma 1 implies that $v>1$ when $F>0$, for $v\in \{v^{*},v^{0*}\}$. Therefore, (7) implies $p_{1}^{\bullet *}>c_{u}+c_{1}$ and $p_{2}^{\bullet *}> c_{u}+c_{2}$.

Equation (6) implies that for $\,i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned} \frac{\partial w_{i}^{0*}}{\partial v^{0*}}=b\, \Delta _{j}\,\frac{2\,v^{0*}-1-2\,v^{0*}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}+\frac{2\,\Delta _{i}\,}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}=\frac{2\,\Delta _{i}\,-b\,\Delta _{j}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}>0\,\text {.} \end{aligned}$$

(49)

The inequality in (49) reflects Assumption 1. (42) and (49) imply:

$$\begin{aligned} \frac{dw_{i}^{0*}}{dF}=\frac{\partial w_{i}^{0*}}{ \partial F}+\frac{\partial w_{i}^{0*}}{\partial v^{0*}}\,\,\frac{ \partial v^{0*}}{\partial F}=\frac{\partial w_{i}^{0*}}{\partial v^{0*}}\,\,\frac{\partial v^{0*}}{\partial F}> 0\,\text {.} \end{aligned}$$

(50)

Equations (6), (41), (48), and (49) imply:

$$\begin{aligned}&\frac{dw_{i}^{0*}}{dc_{u}}=\frac{\partial w_{i}^{0*} }{\partial c_{u}}+\frac{\partial w_{i}^{0*}}{\partial v^{0*}}\,\, \frac{\partial v^{0*}}{\partial c_{u}} \nonumber \\&\quad =1+\frac{1-b\,v^{0*}}{2\,v^{0*}-1}+\,\frac{ 2\,\Delta _{i}\,-b\,\Delta _{j}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}\,\, \frac{2}{H}\,v^{0*}\left[ \,v^{0*}-1\,\right] \left[ \,2\,v^{0*}-1\,\right] \left[ \,1-b\,\right] \left[ \,\Delta _{i}\,+\,\Delta _{j}\, \right] \nonumber \\&\quad =\frac{v^{0*}\left[ \,2-b\,\right] }{2\,v^{0*}-1}+\frac{2\,\Delta _{i}\,-b\,\Delta _{j}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}\,\,\frac{2}{H}\,\,v^{0*}\left[ \,v^{0*}-1\,\right] \left[ \,2\,v^{0*}-1\,\right] \left[ \,1-b\,\right] \left[ \,\Delta _{i}\,+\,\Delta _{j}\,\right] \,\,>0\,\text {.}\nonumber \\ \end{aligned}$$

(51)

Equations (8) and (42) imply:

$$\begin{aligned} \frac{dX_{i}^{0*}}{dF}=\frac{\partial X_{i}^{0*}}{\partial v^{0*}}\,\,\frac{\partial v^{0*}}{\partial F }\overset{s}{=}\frac{2\,v^{0*}-1-2\,v^{0*}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}=\,\,-\,\frac{1}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}<0\,\text {.} \end{aligned}$$

Equations (8) and (48) imply:

$$\begin{aligned}&\frac{dX_{i}^{0*}}{dc_{u}}=\frac{\partial X_{i}^{0*} }{\partial c_{u}}+\frac{\partial X_{i}^{0*}}{\partial v^{0*}}\,\, \frac{\partial v^{0*}}{\partial c_{u}}=\,\,-\,\frac{v^{0*}}{ 2\,v^{0*}-1\,}+\frac{\partial X_{i}^{0*}}{\partial v^{0*}}\,\, \frac{\partial v^{0*}}{\partial c_{u}}<\frac{\partial X_{i}^{0*}}{\partial v^{0*}}\,\,\frac{\partial v^{0*}}{\partial c_{u}} \nonumber \\&\overset{s}{=}\frac{2\,v^{0*}-1-2\,v^{0*}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}\,\left[ \,-\, \frac{\partial H}{\partial c_{u}}\,\right] \,\,\overset{s}{=}\,\,-\,\frac{1}{ \left[ \,2\,v^{0*}-1\,\right] ^{2}}<0\,\text {.} \end{aligned}$$

(52)

The last $\,\overset{s}{=}\,$ in (52) reflects (41).

Equations (7) and (42) imply:

$$\begin{aligned} \frac{dp_{i}^{0*}}{dF}=\frac{\partial p_{i}^{0*}}{ \partial v^{0*}}\,\,\frac{\partial v^{0*}}{\partial F} \overset{s}{=}\frac{\partial p_{i}^{0*}}{\partial v^{0*}} \overset{s}{=}\frac{2\,v^{0*}-1-2\left[ \,v^{0*}-1\, \right] }{\left[ \,2\,v^{0*}-1\,\right] ^{2}}=\frac{1}{ \left[ \,2\,v^{0*}-1\,\right] ^{2}}>0\,\text {.} \end{aligned}$$

(53)

Equations (7), (41), (48), and (53) imply:

$$\begin{aligned}&\frac{dp_{i}^{0*}}{dc_{u}}=1-\frac{v^{0*}-1}{ 2\,v^{0*}-1}+\frac{\partial p_{i}^{0*}}{\partial v^{0*}}\,\, \frac{\partial v^{0*}}{\partial c_{u}}=\frac{v^{0*}}{ 2\,v^{0*}-1}+\frac{\partial p_{i}^{0*}}{\partial v^{0*}}\,\, \frac{\partial v^{0*}}{\partial c_{u}} \\&\quad>\frac{\partial p_{i}^{0*}}{\partial v^{0*}} \,\,\frac{\partial v^{0*}}{\partial c_{u}}\overset{s}{=} \frac{\partial p_{i}^{0*}}{\partial v^{0*}}\,\left[ \,-\,\frac{ \partial H}{\partial c_{u}}\,\right] \,\,\overset{s}{=}\frac{ \partial p_{i}^{0*}}{\partial v^{0*}}>0\, . \end{aligned}$$

The calculations for the public enterprise are analogous, and so are omitted. $\square$

Proof of Proposition 1

Equation (6) implies:

$$\begin{aligned}&w_{2}^{*}<w_{1}^{*} \Leftrightarrow c_{u}-\frac{\Delta _{2}-b\,v^{*}\,\Delta _{1}}{ 2\,v^{*}-1\,}<\,c_{u}+\frac{\left[ \,v^{*}-1\,\right] \Delta _{1}}{2\,v^{*}-1} \nonumber \\&\quad \Leftrightarrow \Delta _{2}>\,\,\left[ \,1-v^{*}\left( 1-b\,\right) \,\right] \Delta _{1}\,\text { . } \end{aligned}$$

(54)

The last inequality in (54) holds if $\,1-v^{*}\left[ \,1\,-\,b\,\right] \,\le \,0$. The inequality also holds if $\,1-v^{*} \left[ \,1\,-\,b\,\right] \,>\,0\,$ and $\,\Delta _{1}\,<\,\frac{\Delta _{2} }{1\,-\,v^{*}\left[ \,1\,-\,b\,\right] }\,$.

Equation (6) also implies:

$$\begin{aligned} w_{1}^{0*}\lesseqgtr w_{2}^{0*} \Leftrightarrow \Delta _{2}-b\,v^{0*}\,\Delta _{1} \lesseqgtr \Delta _{1}-b\,v^{0*}\,\Delta _{2} \\ \Leftrightarrow \,\left[ \,1+b\,v^{0*}\,\right] \Delta _{2}\lesseqgtr \,\,\left[ \,1+b\,v^{0*}\,\right] \Delta _{1}\Leftrightarrow \Delta _{1} > reqless \Delta _{2}\,\text {. } \end{aligned}$$

$\square$

Proof of Proposition 2

$\Pi ({\mathbf {w}} )=0$, from Lemma 1. Therefore, (17) implies $g(\cdot )=\frac{F}{H}$. (33) implies $\frac{\partial g(\cdot )}{ \partial \Delta _{i}}=0$ for $i\in \{1,2\}$. Therefore, (34), (38), and (47) imply that for $\,i,j\in \{1,2\}$ ($j\ne i$ ):

$$\begin{aligned}&\frac{dv^{\bullet *}}{d\Delta _{i}}=\,\,-\,\frac{ \frac{F}{H^{2}}\,\frac{\partial H}{\partial \Delta _{i}}}{\frac{\partial g(\cdot )}{\partial v}}=\,\,-\,\frac{1}{H}\,v^{\bullet *}\left[ \,v^{\bullet *}-1\,\right] \left[ \,2\,v^{\bullet *}-1\,\right] \, \frac{\partial H}{\partial \Delta _{i}} \nonumber \\&\qquad \,\,\,\,\,=\,\,-\,\frac{2}{H}\,v^{\bullet *}\left[ \,v^{\bullet *}-1\,\right] \left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,\Delta _{i}-b\,\Delta _{j}\,\right] \,\,<0\,\text {.} \end{aligned}$$

(55)

Equation (6) implies:

$$\begin{aligned} \frac{dw_{2}^{\bullet *}}{d\Delta _{2}}=\frac{\partial w_{2}^{\bullet *}}{\partial \Delta _{2}}+\frac{\partial w_{2}^{\bullet *}}{\partial v^{\bullet *}}\,\,\frac{\partial v^{\bullet *}}{ \partial \Delta _{2}}=\,\,-\,\frac{1}{2\,v^{\bullet *}-1}+\frac{ \partial w_{2}^{\bullet *}}{\partial v^{\bullet *}}\,\,\frac{ \partial v^{\bullet *}}{\partial \Delta _{2}}<0\, . \end{aligned}$$

(56)

The inequality in (56) holds because $\frac{\partial v^{\bullet *}}{\partial \Delta _{2}}<0$ from (55) and $\frac{\partial w_{2}^{\bullet *}}{\partial v^{\bullet *}}>0$ from ( 6) and (49).

Equation (6) implies:

$$\begin{aligned} \frac{dw_{1}^{*}}{d\Delta _{2}}=\frac{ \partial w_{1}^{*}}{\partial \Delta _{2}}+\frac{\partial w_{1}^{*}}{ \partial v^{*}}\,\,\frac{\partial v^{*}}{\partial \Delta _{2}}=\frac{\partial w_{1}^{*}}{\partial v^{*}}\,\,\frac{ \partial v^{*}}{\partial \Delta _{2}}<0\, . \end{aligned}$$

(57)

The inequality in (57) holds because: (i) $\frac{\partial v }{\partial \Delta _{2}}<0$ from (55); and (ii) from (6):

$$\begin{aligned} \frac{\partial w_{1}^{*}}{\partial v^{*}}\overset{s}{=}\,\, \left[ \,2\,v^{*}-1\,\right] \Delta _{1}-2\left[ \,v^{*}-1\,\right] \Delta _{1}\,\,=\Delta _{1}>0\,\text {.}\, \end{aligned}$$

Equation (6) implies that for $\,i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&\frac{\partial w_{i}^{0*}}{\partial v^{0*}}=\,\,-\,\frac{-\,b\,\Delta _{j}\left[ \,2\,v^{0*}-1\, \right] -2\left[ \,\Delta _{i}-b\,v^{0*}\,\Delta _{j}\,\right] }{\left[ \,2\,v^{0*}-1\,\right] ^{2}}=\frac{2\,\Delta _{i}-b\,\Delta _{j}\,}{\left[ \,2\,v^{0*}-1\,\right] ^{2}}>0 \end{aligned}$$

(58)

$$\begin{aligned}&\Rightarrow \frac{dw_{1}^{0*}}{d\Delta _{2}} =\frac{\partial w_{1}^{0*}}{\partial \Delta _{2}}+\frac{ \partial w_{1}^{0*}}{\partial v^{0*}}\,\,\frac{\partial v^{0*}}{ \partial \Delta _{2}} \nonumber \\&\quad =\frac{b\,v^{0*}}{2\,v^{0*}-1}+\frac{ 2\,\Delta _{1}-b\,\Delta _{2}}{\left[ \,2\,v^{0*}-1\,\right] ^{2}} \left\{ \,-\,\frac{2}{H}\,v^{0*}\left[ \,v^{0*}-1\, \right] \left[ \,2\,v^{0*}-1\,\right] \left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \,\right\} \text {.} \end{aligned}$$

(59)

The last equality in (59) reflects (55). (59) implies that $\frac{dw_{1}^{0*}}{ d\Delta _{2}}<0$ for b sufficiently close to 0. $\square$

Proof of Proposition 3

Equation (6) implies:

$$\begin{aligned} \frac{\partial w_{1}^{*}}{\partial \Delta _{1}}=\frac{ v^{*}-1}{2\,v^{*}-1}\text { and }\frac{\partial w_{1}^{*}}{ \partial v^{*}}=\Delta _{1}\left[ \,\frac{2\,v^{*}-1-2\left( \,v^{*}-1\,\right) \,}{\left( \,2\,v^{*}-1\,\right) ^{2}} \,\right] \,\,=\frac{\Delta _{1}}{\left[ \,2\,v^{*}-1\,\right] ^{2}}\,\text {.} \nonumber \\ \end{aligned}$$

(60)

Equations (55) and (60) imply:

$$\begin{aligned}&\frac{dw_{1}^{*}}{d\Delta _{1}}=\frac{\partial w_{1}^{*}}{\partial \Delta _{1}}+\frac{\partial w_{1}^{*}}{\partial v^{*}}\,\,\frac{\partial v^{*}}{\partial \Delta _{1}} \nonumber \\&\quad =\frac{v^{*}-1}{2\,v^{*}-1}-\frac{ \Delta _{1}}{\left[ \,2\,v^{*}-1\,\right] ^{2}}\,\,\frac{2}{H} \,\,v^{*}\left[ \,v^{*}-1\,\right] \left[ \,2\,v^{*}-1\,\right] \left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \nonumber \\&\quad =\frac{\left[ \,v^{*}-1\,\right] G_{1}(v^{*})}{H\left[ \,2\,v^{*}-1\,\right] } \end{aligned}$$

(61)

where

$$\begin{aligned}&G_{1}(v)\equiv H-2\,v\,\Delta _{1}\left[ \,\Delta _{1}\,-b\,\Delta _{2}\,\right] \,\,=\,\,\left[ \,\Delta _{1}\, \right] ^{2}+\left[ \,\Delta _{2}\,\right] ^{2}-2\,b\,\Delta _{1}\,\Delta _{2}-2\,v\,\Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \nonumber \\&\quad =\,\,-\,\left[ \,2\,v-1\,\right] \left[ \,\Delta _{1}\,\right] ^{2}+\left[ \,\Delta _{2}\,\right] ^{2}+2\,b\left[ \,v-1\,\right] \Delta _{1}\,\Delta _{2}\, . \end{aligned}$$

(62)

The first equality in (62) reflects (34). ( 62) implies that when $\Delta _{1}\ge \Delta _{2}$ and $v>1$:

$$\begin{aligned}&G_{1}(v)\le \,\,\left[ \,\Delta _{1}\,\right] ^{2}\left[ \,1-\left( \,2\,v-1\,\right) +2\,b\left( \,v-1\,\right) \,\right] \nonumber \\&\overset{s}{=}\,\,2-\,2\,v+2\,b\left[ \,v-1\,\right] \,\,=\,\,-\,2\left[ \,v-1\,\right] +2\,b\left[ \,v-1\,\right] \,\,=\,\,-\,2 \left[ \,1-b\,\right] \left[ \,v-1\,\right] \text {.} \end{aligned}$$

(63)

Equations (61) and (63) imply that when $v^{*}>1$:

$$\begin{aligned} \frac{dw_{1}^{*}}{d\Delta _{1}}<0\text { if }\,\Delta _{1}\,\ge \,\Delta _{2}\,\text {.} \end{aligned}$$

(64)

Equation (62) implies:

$$\begin{aligned}&G_{1}(v)=\Delta _{1}^{2}\,+\,\Delta _{2}^{2}\,-\,b\,\Delta _{1}\,\Delta _{2}\,-\,b\,\Delta _{1}\,\Delta _{2}\,-\,2\,v\,\Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad =\Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\, \right] \,+\,\Delta _{2}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \,-\,2\,v\,\Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \ \ \ \ \ \ \ \ \nonumber \\&\quad =-\,\left[ \,2\,v-1\,\right] \Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \,+\,\Delta _{2}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad >-\,\left[ \,2\,v-1\,\right] \Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \,+\,b\,\Delta _{1}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad =\left[ \,b\left( \,\Delta _{2}-b\,\Delta _{1}\,\right) \,-\,\left( \,2\,v-1\,\right) \left( \,\Delta _{1}-b\,\Delta _{2}\,\right) \,\right] \Delta _{1}\,\text {.}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(65)

Observe that:

$$\begin{aligned}&b\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] \,>\,\left[ \,2\,v-1\,\right] \left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \Leftrightarrow b\,\Delta _{2}-b^{2}\,\Delta _{1}>\,\left[ \,2\,v-1\,\right] \Delta _{1}-b\left[ \,2\,v-1\,\right] \Delta _{2} \nonumber \\&\quad \Leftrightarrow 2\,v\,b\,\Delta _{2}>\,\, \left[ \,2\,v-1+b^{2}\,\right] \Delta _{1}\ \ \Leftrightarrow \ \ \Delta _{2} >\,\left[ \,\frac{2\,v-1+b^{2}}{2\,v\,b}\,\right] \Delta _{1}\,\text { .}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(66)

Also observe that

$$\begin{aligned}&\frac{2\,v-1+b^{2}}{2\,v\,b}\,>\,b\text { when }v\,>\,1 \nonumber \\&\text {because }\frac{2\,v-1+b^{2}}{2\,v\,b}>b \Leftrightarrow 2\,v-1+b^{2}>2\,v\,b^{2} \nonumber \\&\Leftrightarrow 2\,v-1>b^{2}\left[ \,2\,v-1\,\right] \Leftrightarrow 1\,>\,b^{2}\,\text {.}\, \end{aligned}$$

(67)

Equations (61), (65), (66), (67), and Assumption 1 ensure that when $v^{*}>1$, $\,\frac{ dw_{1}^{*}}{d\Delta _{1}}>0$ under the condition specified in the Proposition. $\square$

Proof of Proposition 4

Equations (6) and (45) imply:

$$\begin{aligned} \frac{dw_{1}^{*}}{db}=\frac{\partial w_{1}^{*}}{ \partial b}+\frac{\partial w_{1}^{*}}{\partial v^{*}}\,\,\frac{ \partial v^{*}}{\partial b}=\frac{\partial w_{1}^{*} }{\partial v^{*}}\,\,\frac{\partial v^{*}}{\partial b}=0\text { if }\,\Delta _{1}\,=\,\Delta _{2}\text {.} \end{aligned}$$

(68)

Equations (6) and (45) imply that if $\,\Delta _{1}\,=\,\Delta _{2}$ , then:

$$\begin{aligned} \frac{dw_{2}^{\bullet *}}{db}=\frac{\partial w_{2}^{\bullet *}}{\partial b}+\frac{\partial w_{2}^{\bullet *}}{ \partial v^{\bullet *}}\,\,\frac{\partial v^{\bullet *}}{\partial b} =\frac{\partial w_{2}^{\bullet *}}{\partial b}= \frac{v^{\bullet *}\,\Delta _{1}\,}{2\,v^{\bullet *}-1}>0\, . \end{aligned}$$

(69)

Equations (7) and (45) imply that for $i\in \{1,2\}$:

$$\begin{aligned} \frac{dp_{i}^{\bullet *}}{db}=\frac{\partial p_{i}^{\bullet *}}{\partial b}+\frac{\partial p_{i}^{\bullet *}}{ \partial v^{\bullet *}}\,\,\frac{dv^{\bullet *}}{db}= \frac{\partial p_{i}^{\bullet *}}{\partial v^{\bullet *}}\,\,\frac{ dv^{\bullet *}}{db}=0\text { if }\,\Delta _{1}\,=\,\Delta _{2}\,\text {.} \end{aligned}$$

Equation (8) implies that for $i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned} \frac{\partial X_{i}^{\bullet *}}{\partial b}=\,\,\left[ \frac{ v^{\bullet *}}{2\,v^{\bullet *}-1}\right] \frac{\left[ \,1-b\,\right] \left[ \,-\Delta _{j}\,\right] +\Delta _{i}-b\,\Delta _{j}}{\left[ \,1-b\, \right] ^{2}}=\frac{v^{\bullet *}\left[ \,\Delta _{i}-\Delta _{j}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] ^{2}}\,\text {.} \end{aligned}$$

(70)

Equations (8), (45), and (70) imply that if $\Delta _{1}=\Delta _{2}$, then for $i\in \{1,2\}$:

$$\begin{aligned} \frac{dX_{i}^{\bullet *}}{db}=\frac{\partial X_{i}^{0*}}{\partial b}+\frac{\partial X_{i}^{0*}}{\partial v^{0*}}\,\,\frac{dv^{0*}}{db}=0\, . \end{aligned}$$

$\square$

Proof of Proposition 5

Equations (45), (60), and (68) imply:

$$\begin{aligned} \frac{dw_{1}^{*}}{db}=\frac{\Delta _{1}}{ \left[ \,2\,v^{*}-1\,\right] ^{2}}\,\,\frac{\partial v^{*}}{\partial b}<0\text { if }\,\Delta _{1}\,\ne \,\Delta _{2}\,\text {.} \end{aligned}$$

Equations (6), (45), and (58) imply:

$$\begin{aligned}&\frac{dw_{2}^{\bullet *}}{db}=\frac{ \partial w_{2}^{\bullet *}}{\partial b}+\frac{\partial w_{2}^{\bullet *}}{\partial v^{\bullet *}}\,\,\frac{\partial v^{\bullet *}}{ \partial b} \nonumber \\&\quad =\frac{v^{\bullet *}\,\Delta _{1}}{2\,v^{\bullet *}-1\,}-\,\frac{2\,\Delta _{2}\,-b\,\Delta _{1}}{\left[ \,2\,v^{\bullet *}-1\,\right] ^{2}}\,\,\frac{v^{\bullet *}\left[ \,v^{\bullet *}-1\,\right] \left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,\Delta _{2}-\Delta _{1}\,\right] ^{2}}{H\left[ \,1-b\,\right] } \nonumber \\&\quad =\frac{v^{\bullet *}\,G_{b}}{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] H\,}\overset{ s}{=} G_{b}(v^{\bullet *}) \end{aligned}$$

(71)

$$\begin{aligned}&\text {where }G_{b}(v)\equiv \,\,\Delta _{1}\left[ \,1-b\,\right] H-\left[ \,v-1\,\right] \left[ \,2\,\Delta _{2}\,-b\,\Delta _{1}\,\right] \left[ \,\Delta _{2}-\Delta _{1}\,\right] ^{2}. \end{aligned}$$

(72)

Equations (34) and (72) imply:

$$\begin{aligned}&G_{b}(v)\equiv \,\,\Delta _{2}\,\Delta _{1}\left[ \,1-b\,\right] \left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] +\,\Delta _{1}^{2}\left[ \,1-b\, \right] \left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] \nonumber \\&\qquad -\,\left[ \,v-1\,\right] \left[ \,2\,\Delta _{2}-b\,\Delta _{1}\,\right] \left[ \,\Delta _{2}-\Delta _{1}\,\right] ^{2} \nonumber \\&\quad =\,\,\left[ \,1-b\,\right] \left[ \,\Delta _{2}^{2}\,\Delta _{1}-2\,b\,\Delta _{2}\,\Delta _{1}^{2}+\Delta _{1}^{3}\, \right] -\,\left[ \,v-1\,\right] \left[ \,2\,\Delta _{2}-b\,\Delta _{1}\, \right] \left[ \,\Delta _{2}-\Delta _{1}\,\right] ^{2} \nonumber \\&\quad =\,\,\Delta _{2}^{2}\,\Delta _{1}-2\,b\,\Delta _{2}\,\Delta _{1}^{2}+\Delta _{1}^{3}-b\,\Delta _{2}^{2}\,\Delta _{1}+2\,b^{2}\Delta _{2}\,\Delta _{1}^{2}-b\,\Delta _{1}^{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\qquad -\,\,2\,v\,\Delta _{2}^{3}-2\,v\,\Delta _{2}\,\Delta _{1}^{2}-2\,b\,v\,\Delta _{2}\,\Delta _{1}^{2}+4\,v\,\Delta _{2}^{2}\,\Delta _{1}\,+b\,v\,\Delta _{2}^{2}\,\Delta _{1}+b\,v\,\Delta _{1}^{3} \nonumber \\&\qquad +\,\,2\,\Delta _{2}^{3}+2\,\Delta _{2}\,\Delta _{1}^{2}+2\,b\,\Delta _{2}\,\Delta _{1}^{2}-4\,\Delta _{2}^{2}\,\Delta _{1}\,-b\,\Delta _{2}^{2}\,\Delta _{1}-b\,\Delta _{1}^{3}\ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad =\,\,\left[ \,-\,2\,b+4\,v+b\,v-3\,\right] \Delta _{2}^{2}\,\Delta _{1}+2\left[ \,1+b^{2}-v\left( \,1+b\,\right) \,\right] \Delta _{2}\,\Delta _{1}^{2}\ \ \ \ \ \ \ \ \ \ \nonumber \\&\qquad -\,\,2\left[ \,v-1\,\right] \Delta _{2}^{3}+\left[ \,1-2\,b+b\,v\,\right] \Delta _{1}^{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\quad =\,\,\Delta _{2}^{2}\,\left\{ \,\left[ \,4\,v-2\,b+b\,v-3\, \right] \Delta _{1}-2\left[ \,v-1\,\right] \Delta _{2}\,\right\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nonumber \\&\qquad +\,\,\Delta _{1}^{2}\,\left\{ \,\left[ \,1-2\,b+b\,v\,\right] \Delta _{1}-2 \left[ \,v\left( \,1+b\,\right) -1-b^{2}\,\right] \Delta _{2}\,\right\} .\ \ \ \ \ \ \ \ \ \end{aligned}$$

(73)

Equation (73) implies that if

$$\begin{aligned} \Delta _{2}>\max \,\left\{ \,\frac{4\,v-2\,b+b\,v-3}{2\left[ \,v-1\,\right] }\,\Delta _{1},\frac{1-2\,b+b\,v}{2\left[ \,v\left( 1+b\right) -1-b^{2}\,\right] }\,\Delta _{1}\,\right\} \text {,} \end{aligned}$$

(74)

then $G_{b}(v)<0$, and so $\frac{dw_{2}^{0*}}{db}<0$, from (71).

Equation (74) can be restated as:

$$\begin{aligned} \Delta _{2}>\frac{\left[ \,4+b\,\right] v-2\,b-3}{2\left[ \,v-1\,\right] }\,\,\Delta _{1}\,\text {.} \end{aligned}$$

(75)

To see why, observe that:

$$\begin{aligned}&\frac{\left[ \,4+b\,\right] v-2\,b-3}{2\left[ \,v-1\,\right] }>\frac{1-2\,b+b\,v}{2\left[ \,v\left( \,1\,+\,b\,\right) \,-\left( \,1\,+\,b^{2}\right) \,\right] } \nonumber \\&\quad \Leftrightarrow \,\ \left[ \,v-1+b\left( v-b\right) \,\right] \left[ \,4\,v-2\,b+b\,v-\,3\,\right] \,>\,\left[ \,v-1\,\right] \left[ 1-2\,b+b\,v\,\right] \nonumber \\&\quad \Leftrightarrow \,\ 4\left[ \,v\,-1\,\right] ^{2}+b\left[ \,v-b\,\right] \left[ \,4\,v-2\,b+b\,v-3\,\right] \,\,>0\,\text {.} \end{aligned}$$

(76)

The inequality in (76) holds because $f(v,b)\equiv 4\,v-2\,b+b\,v-3>0$. To see why, observe that $\frac{\partial f(v,b)}{\partial v}=4+b>0$ and $\frac{ \partial f(v,b)}{\partial b}=v-2$. Therefore, $\frac{\partial f(v,b)}{ \partial b}<0$ for $v\in [\,1,2)$ whereas $\frac{ \partial f(v,b)}{\partial b}\ge 0$ for $v\ge 2$. Consequently: (i) $\min f(v,b)=f(1,1)=0\,$ for $v\in [\,1,2)$; and (ii) $\min f(v,b)=f(2,0)=5>0$ for $v\ge 2$. Therefore, $f(v,b)\ge \min f(v,b)\ge 0$ for $v>1$ and $b\in [\,0,1\,]$.

Further observe that

$$\begin{aligned}&\frac{4\,v-2\,b+b\,v-3\,}{2\left[ \,v-1\,\right] }>b\ \ \Leftrightarrow 4\,v-2\,b+b\,v-3>2\,b\left[ \,v-1\, \right] \nonumber \\&\Leftrightarrow 4\,v-b\,v-3>0\ \ \Leftrightarrow v\left[ \,4-b\,\right] -3>0\,.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{aligned}$$

(77)

The inequality in (77) holds when $v>1$ because $\,4-b\,>\,3$. Finding (ii) in the Proposition follows from (74), (75), ( 77), and Assumption 1.

Equation (7) implies that for $i\in \{1,2\}$:

$$\begin{aligned} \frac{dp_{i}^{\bullet *}}{db}=\frac{ \partial p_{i}^{\bullet *}}{\partial v^{\bullet *}}\,\,\frac{ dv^{\bullet *}}{db}=\Delta _{i}\left[ \,\frac{ 2\,v^{\bullet *}-1-2\left( v^{\bullet *}-1\right) }{\left( 2\,v^{\bullet *}-1\right) ^{2}}\,\right] \frac{dv^{\bullet *}}{db} =\frac{\Delta _{i}}{\left[ \,2\,v^{\bullet *}-1\,\right] ^{2}}\,\,\frac{dv^{\bullet *}}{db}\,\text {.} \end{aligned}$$

(78)

Equations (45) and (78) imply that $\frac{ dp_{i}^{\bullet *}}{db}<0$ if $\,v^{\bullet *}>1$ and $\Delta _{1}\ne \Delta _{2}$.

Equation (8) implies that for $i,j\in \{1,2\}$ ($j\ne i$):

$$\begin{aligned}&\frac{\partial X_{i}^{\bullet *}}{\partial b}=\frac{ v^{\bullet *}\left[ \,-\,\Delta _{j}\left( 1-b\right) +\Delta _{i}-b\,\Delta _{j}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] ^{2}}=\frac{v^{\bullet *}\left[ \,\Delta _{i}-\Delta _{j}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] ^{2}}\,\text {, and } \nonumber \\&\frac{\partial X_{i}^{\bullet *}}{\partial v^{\bullet *}} =\,\,\left[ \,\frac{\Delta _{i}-b\,\Delta _{j}}{1-b}\,\right] \left[ \,\frac{ 2\,v^{\bullet *}-1-2\,v^{\bullet *}}{\left( 2\,v^{\bullet *}-1\right) ^{2}}\,\right] \,\,=\,\,-\,\frac{\Delta _{i}-b\,\Delta _{j}}{ \left[ \,1-b\,\right] \left[ \,2\,v^{\bullet *}-1\,\right] ^{2}}\,\text { . } \end{aligned}$$

(79)

Equations (8), (45), and (79) imply:

$$\begin{aligned}&\frac{dX_{i}^{\bullet *}}{db}=\frac{\partial X_{i}^{\bullet *}}{\partial b}+\frac{\partial X_{i}^{\bullet *}}{ \partial v^{\bullet *}}\,\,\frac{dv^{\bullet *}}{db} \nonumber \\&\quad =\frac{v^{\bullet *}\left[ \,\Delta _{i}-\Delta _{j}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\, \right] ^{2}}+\frac{v^{\bullet *}\left[ \,v^{\bullet *}-1\,\right] \left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,\Delta _{i}-\Delta _{j}\, \right] ^{2}\left[ \,\Delta _{i}-b\,\Delta _{j}\,\right] }{H\left[ \,2\,v^{\bullet *}-1\,\right] ^{2}\left[ \,1-b\,\right] ^{2}} \nonumber \\&\quad =\frac{v^{\bullet *}\left[ \,\Delta _{i}-\Delta _{j}\,\right] \left\{ \,H+\left[ \,v^{\bullet *}-1\,\right] \left[ \,\Delta _{i}+\Delta _{j}\,\right] \left[ \,\Delta _{i}-b\,\Delta _{j}\, \right] \,\right\} }{H\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\, \right] ^{2}} > reqless \ 0\ \ \Leftrightarrow \ \ \Delta _{i}\ > reqless \ \Delta _{j}\,\text {.} \end{aligned}$$

(80)

Equation (80) implies that the output of the most efficient supplier increases and the output of the least efficient supplier declines as b increases.

Equation (8) implies:

$$\begin{aligned}&X_{1}^{\bullet *}+X_{2}^{\bullet *}=\frac{v^{\bullet *}\left[ \,\Delta _{1}-b\,\Delta _{2}+\Delta _{2}-b\,\Delta _{1}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] }= \frac{v^{\bullet *}\left[ \,\Delta _{1}+\Delta _{2}\,\right] \left[ \,1-b\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] }=\frac{v^{\bullet *}\left[ \,\Delta _{1}+\Delta _{2}\,\right] }{2\,v^{\bullet *}-1} \nonumber \\&\Rightarrow \frac{d\left( X_{1}^{\bullet *}+X_{2}^{\bullet *}\right) }{db}=\frac{\partial \left( X_{1}^{\bullet *}+X_{2}^{\bullet *}\right) }{\partial v^{\bullet *}}\,\,\frac{dv^{\bullet *}}{db}\overset{s}{=}\,\,-\,\frac{1 }{\left[ \,2\,v^{\bullet *}-1\,\right] ^{2}}\,\,\frac{dv^{\bullet *} }{db}\overset{s}{=}\,\,-\,\frac{dv^{\bullet *}}{db}\,\text {. } \end{aligned}$$

(81)

Equations (45) and (81) imply that $\frac{d\left( X_{1}^{\bullet *}+X_{2}^{\bullet *}\right) }{db}>0$ when $v^{\bullet *}>1$ and $\Delta _{1}\ne \Delta _{2}$. $\square$

Proof of Lemma 3

If $F=0$, the conclusion follows from finding (i) in Lemma 1.

If $F>0$, then finding (ii) in Lemma 1 implies that $v>1$ for $v\in \{v^{*},v^{0*}\}$. Therefore, $\lambda >0$ at the solution to [RP] because $\gamma \in \left[ \,0,1\,\right]$ and $v=\gamma +\lambda$ . Consequently, (4), (7), (8), and (17) imply:

$$\begin{aligned}&\left[ \,p_{1}^{\bullet *}-c_{u}-c_{1}\,\right] X_{1}^{\bullet *}+\left[ \,p_{2}^{\bullet *}-c_{u}-c_{2}\,\right] X_{2}^{\bullet *}=F \nonumber \\&\quad \Rightarrow \frac{\left[ \,v^{\bullet *}-1\,\right] \Delta _{1}}{2\,v^{\bullet *}-1}\,\,\frac{v^{\bullet *} \left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] \left[ \,1-b\,\right] }+\frac{\left[ \,v^{\bullet *}-1\, \right] \Delta _{2}}{2\,v^{\bullet *}-1}\,\,\frac{v^{\bullet *}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] }{\left[ \,2\,v^{\bullet *}-1\, \right] \left[ \,1-b\,\right] }=F \nonumber \\&\quad \Rightarrow \frac{v^{\bullet *}\left[ \,v^{\bullet *}-1\,\right] }{\left[ \,2\,v^{\bullet *}-1\,\right] ^{2}}= \frac{F\left[ \,1-b\,\right] }{\Delta _{1}\left[ \,\Delta _{1}-b\,\Delta _{2}\,\right] +\Delta _{2}\left[ \,\Delta _{2}-b\,\Delta _{1}\,\right] }\, . \end{aligned}$$

(82)

Observe that:

$$\begin{aligned}&\frac{\partial }{\partial v}\left( \frac{v\left[ \,v-1\, \right] }{\left[ \,2\,v-1\,\right] ^{2}}\right) \,\overset{s}{=}\,\,\left[ \,2\,v-1\,\right] ^{2}\left[ \,2\,v-1\,\right] -v\left[ \,v-1\,\right] 4 \left[ \,2\,v-1\,\right] \nonumber \\&\overset{s}{=}4\,v^{2}-4\,v+1-4\,v^{2}+4\,v =1>0\,\text {.} \end{aligned}$$

(83)

(83) implies there is a unique v that satisfies (82). Therefore, $v^{*}=v^{0*}$. $\square$

Proof of Proposition 6

The proposition follows from (7) and (8) because $v^{*}=v^{0*}$, from Lemma 3. $\square$

Proof of Proposition 7

Equation (6) implies that for $v=v^{*}= v^{0*}$:

$$\begin{aligned} w_{1}^{*}-w_{1}^{0*}=\frac{\left[ \,v-1\,\right] \Delta _{1}}{2\,v-1}+\frac{\Delta _{1}-b\,v\,\Delta _{2}}{2\,v-1\,}= \frac{\,v\left[ \,\Delta _{1}-b\,\Delta _{2}\right] }{2\,v-1} >0\text {. } \end{aligned}$$

$\square$

Proof of Proposition 8

$v^{*}=v^{0*}$, from Lemma 3. Therefore, the conclusion follows directly from (6). $\square$