Cournot–Bertrand comparison in a mixed oligopoly

Abstract

We revisit the classic discussion comparing price and quantity competition, but in a mixed oligopoly in which one state-owned public firm competes against private firms. It has been shown that in a mixed duopoly, price competition yields a larger profit for the private firm. This implies that firms face weaker competition under price competition, which contrasts sharply with the case of a private oligopoly. Here, we adopt a standard differentiated oligopoly with a linear demand. We find that regardless of the number of firms, price competition yields higher welfare. However, the profit ranking depends on the number of private firms. We find that if the number of private firms is greater than or equal to five, it is possible that quantity competition yields a larger profit for each private firm. We also endogenize the price-quantity choice. Here, we find that Bertrand competition can fail to be an equilibrium, unless there is only one private firm.

Appendices

Appendix A

$$\begin{aligned} H_1\equiv & {} \delta (1-\delta )(a_i^2-2\delta a_0a_i+\delta a_0^2)n^2+((3-\delta )a_i^2-2\delta (3-\delta )a_0a_i\\&+\delta (\delta ^2-3\delta +4)a_0^2)n+a_0^2(2-\delta )^2\\ H_2\equiv & {} \delta ^4(a_i-2\delta a_0a_i+\delta a_0^2)n^5+(1-\delta )\delta ^3(6a_i^2-12\delta a_0a_i+7\delta a_0^2)n^4 \\&-(1-\delta )\delta ^2((11\delta -12)a_i^2+(-22\delta ^2+24 \delta )a_0a_i+(18\delta ^2-19\delta )a_0^2)n^3\\&+(1-\delta )\delta (6\delta ^2-17\delta +10)a_i^2+ (-12\delta ^3+34\delta ^2-20\delta )a_0a_i\\&+(18\delta ^3-44\delta ^2+25\delta )a_0)n^2-(1-\delta )^3((\delta -3) a_i^2+(-2\delta ^2+6\delta )a_0a_i\\&+(7\delta ^2-16\delta )a_0^2)n+(1-\delta )^3(2-\delta )^2a_0^2\\ H_3\equiv & {} 4(a_j-a_i)^2\delta ^5m^5-2\delta ^4(13a_j^2\delta -22a_i a_j\delta +13a_i^2\delta -11a_j^2+18a_i a_j\\&-11a_i^2)m^4+\delta ^3(65a_j^2\delta ^2-86a_i a_j\delta ^2 +69a_i^2\delta ^2-110a_j^2\delta +140a_i a_j\delta \\&-118a_i^2\delta +46a_j^2-56a_i a_j+50a_i^2)m^3 +(1-\delta )\delta ^2(75a_j^2\delta ^2-64a_i a_j\delta ^2\\&+83a_i^2\delta ^2-116a_j^2\delta +92a_i a_j\delta -132a_i^2 \delta +44a_j^2-32a_i a_j+52a_i^2)m^2\\&+(1-\delta )^2\delta (40a_j^2\delta ^2-16a_i a_j\delta ^2 +45a_i^2\delta ^2-56a_j^2\delta +20a_i a_j\delta -66a_i^2\delta \\&+19a_j^2-6a_i a_j+24a_i^2)m+(1-\delta )^3(8a_j^2\delta ^2 +9a_i^2\delta ^2-10a_j^2\delta \\&-12a_i^2\delta +3a_j^2+4a_i^2)\\ H_4\equiv & {} 2(a_j-a_i)^2(2-\delta )\delta ^2m^2+\delta (3a_j^2 \delta ^2-2a_i a_j\delta ^2+3a_i^2\delta ^2-10a_j^2\delta +8a_i a_j\delta \\&-10a_i^2\delta +7a_j^2-6a_i a_j+8a_i^2)m +(1-\delta )(a_j^2\delta ^2+a_i^2\delta ^2-4a_j^2\delta \\&-4a_i^2\delta +3a_j^2+4a_i^2) \end{aligned}$$

$$\begin{aligned} H_5\equiv & {} -(1-\delta )\delta ^4n^4+(1-\delta )(3\delta -2) \delta ^3n^3+(3\delta ^3-7\delta ^2+2\delta +3)\delta ^2n^2 \\&+(1-\delta )(4-\delta )(2-\delta ^2)\delta n+(1-\delta )^2 (2-\delta )(2+\delta )\\ H_6\equiv & {} \delta ^3(\delta ^2-3\delta +3)n^3+ \delta ^2(1-\delta )(\delta ^2-6\delta +11)n^2+ \delta (1-\delta )(3\delta ^2-14\delta +12)n\\&-(1-\delta )(2-\delta )(3\delta -2)\\ H_{7}\equiv & {} 2\delta ^3c_0n^3-\delta ^2(2c_i(7\delta -8)c_0 -2\alpha (1-\delta ))n^2+\delta ((3\delta -2)c_i\\&+(5\delta ^2-19\delta +10)c_0+(3\delta ^2-5\delta +2)\alpha )n -(\delta (\delta -2)c_i\\&-(2\delta ^3+7\delta -2-12 \delta +4)c_0+\alpha \delta (\delta ^2-3\delta +2))\\ H_{8}\equiv & {} 2\delta ^3c_in^3-\delta ^2((9\delta -6)c_i-2 \delta c_0-2(1-\delta )\alpha )n^2+\delta ((8\delta ^2-16\delta +6) c_i\\&-\delta (5\delta -4)c_0+(5\delta ^2-9\delta +4)\alpha )n +(\delta ^3+7\delta ^2-7\delta +2)c_i\\&+\delta (2\delta ^2-5\delta +2)c_0 -(2\delta ^3-7\delta ^2+7\delta -2)\alpha \\ H_{9}\equiv & {} -(\delta ^3n^3-\delta ^3n^2+3\delta ^2n-3\delta n- \delta ^3-\delta ^2+4\delta -2)(4\delta ^5n^5-26\delta ^5n^4\\&+24\delta ^4n^4+57\delta ^5n^3-118\delta ^4n^3+56\delta ^3n^3 -49\delta ^5n^2+186\delta ^4n^2-198\delta ^3n^2\\&+64\delta ^2n^2 +12\delta ^5n-103\delta ^4n+201\delta ^3n-146\delta ^2n+36\delta n+ \delta ^5+13\delta ^4\\&-54\delta ^3+72\delta ^2-40\delta +8)\\ H_{10}\equiv & {} (2\delta ^3n^3-9\delta ^3n^2+8\delta ^2n^2+8\delta ^3n-21\delta ^2n+10\delta n+\delta ^3+9\delta ^2-12\delta +4)^2\\ H_{11}\equiv & {} \delta c_i^2n^2-2\delta ^2c_0c_in^2+2\alpha \delta ^2c_in^2-2\alpha \delta c_in^2+\delta ^2c_0^2n^2- \alpha ^2\delta ^2n^2+\alpha ^2\delta n^2\\&+2\delta c_i^2n+3c_i^2n-4\delta ^2c_0c_in-6\delta c_0c_in +4\alpha \delta ^2c_in+2\alpha \delta c_in-6\alpha c_in\\&+\delta ^2c_0^2n+4\delta c_0^2n+2\alpha \delta ^2c_0n-2\alpha \delta c_0n-3\alpha ^2\delta ^2n+3\alpha ^2n-\delta ^3c_0^2-3 \delta ^2c_0^2\\&+4c_0^2+2\alpha \delta ^3c_0+6\alpha \delta ^2c_0-8\alpha c_0-\alpha ^2\delta ^3-3\alpha ^2\delta ^2+4\alpha ^2 \end{aligned}$$

$$\begin{aligned} H_{12}\equiv & {} 8\delta ^5m^5-16\delta ^4m^5+10\delta ^3m^5 -32\delta ^5m^4+76\delta ^4m^4-70\delta ^3m^4+25\delta ^2m^4\\&+50\delta ^5m^3-132\delta ^4m^3+138\delta ^3m^3-74\delta ^2m^3 +18\delta m^3-38\delta ^5m^2+113\delta ^4m^2\\&-126\delta ^3m^2+69\delta ^2m^2-22\delta m^2+4m^2+14\delta ^5m -48\delta ^4m+60\delta ^3m\\&-32\delta ^2m+6\delta m-2\delta ^5 +8\delta ^4-12\delta ^3+8\delta ^2-2\delta \\ H_{13}\equiv & {} 4 \delta ^2 m^4+8 \delta ^4 m^3-32 \delta ^3 m^3 +18 \delta ^2 m^3+2 \delta m^3-12 \delta ^4 m^2+64 \delta ^3 m^2\\&-77 \delta ^2 m^2+26 \delta m^2+6 \delta ^4 m-40 \delta ^3 m +68 \delta ^2 m-42 \delta m+8 m-\delta ^4+8 \delta ^3\\&-17\delta ^2+14 \delta -4 \end{aligned}$$

Appendix B

Proof of Proposition 1

From (4) and (2), we have

$$\begin{aligned}&\pi _i^{B}-\pi _i^{C}\\&\quad =\frac{\delta ^2n(a_i-\delta a_0)^2H_5}{\beta (1+n\delta )(1-\delta )(2-\delta +n\delta (1-\delta ))^2(\delta ^2n^2+3\delta (1- \delta )n+(1-\delta )(2-\delta ))^2}. \end{aligned}$$

Here, $\pi _i^{B}-\pi _i^{C}$ is positive (res. negative, zero) if $H_5(n,\delta )$ is positive (res. negative, zero). Figure 1 describes the region for $H_5(n,\delta )<0$. This shows (i) and (ii).¹¹

We have $\lim _{n\rightarrow \infty }H_5(n,\delta )=-\infty .$ This implies (iii). Q.E.D.

Proof of Proposition 2

From (3) and (1), we have

$$\begin{aligned}&SW^{B}-SW^{C} \\&\quad =\frac{\delta ^2n^2 (a_1-\delta a_0)^2H_6}{2\beta (1+n\delta )(1-\delta )(2- \delta +n\delta (1-\delta ))^2(\delta ^2n^2+3\delta (1-\delta )n+(1-\delta )(2-\delta ))^2}. \end{aligned}$$

Here, $SW^{B}-SW^{C}$ is positive (res. negative, zero) if $H_6(n,\delta )$ is positive (res. negative, zero). We now show that $H_6(1,\delta )>0$ and that $H_6(n,\delta )$ is increasing in n for $n \ge 1$.

Substituting $n=1$ into $H_6(n,\delta )$, we have $H_6(1,\delta )= (2-\delta ^2)^2>0$. We show that $H_6(n,\delta )$ is increasing in n for $n \ge 1$ if $\delta \in (0,1)$. We have that

$$\begin{aligned} \frac{\partial H_6(n,\delta )}{\partial n}= & {} 3\delta ^3(\delta ^2-3\delta +3)n^2+2\delta ^2(1-\delta ) (\delta ^2-6\delta +11)n\\&+\delta (1-\delta )(3\delta ^2-14\delta +12). \end{aligned}$$

This is increasing in n. Substituting in $n=1$, we have

$$\begin{aligned} \frac{\partial H_6(n,\delta )}{\partial n}|_{n=1}= \delta ^4(2+\delta )+4\delta (1-\delta )(3+2\delta ) >0. \end{aligned}$$

Thus, $\frac{\partial H_6(n,\delta )}{\partial n }>0$ for $n \ge 1$. Q.E.D.

Proof of Proposition 3

We have already discussed the equilibrium profit of each private firm when all firms choose the price contract ($\pi _i^B$). We show that given the contracts of other firms, a private firm has an incentive to choose the quantity contract.

Consider the subgame in which one private firm chooses the quantity contract and all the other firms choose the price contract. In equilibrium, the public firm names the following price:

$$\begin{aligned} p_0=\frac{H_{7}}{2\delta ^3 n^3+\delta ^2(8-9\delta )n^2+\delta (\delta -2)(8\delta -5)n+(2-3\delta )^2}, \end{aligned}$$

the private firms that choose the price contract name the following price:

$$\begin{aligned} p_i= & {} \frac{H_{8}}{2\delta ^3 n^3+\delta ^2(8-9\delta )n^2+\delta (\delta -2)(8\delta -5)n+(2-3\delta )^2},\\&\quad (i=1,2,\ldots ,n-1), \end{aligned}$$

and the private firm that chooses the quantity contract selects the following quantity:

$$\begin{aligned} q_n=\frac{(a_i-\delta a_0)(1+\delta (n-2))(2+\delta (2n-3))}{\beta (2\delta ^3(1-\delta )n^3+\delta ^2(9\delta ^2-17\delta +8)n^2-\delta (8\delta ^3-29\delta ^2+31\delta -10)n-(\delta ^4+8\delta ^3-21\delta ^2+16\delta -4))}. \end{aligned}$$

The private firm that chooses the quantity contract obtains the following profit:

$$\begin{aligned}&\pi ^{p,\ldots ,p,q}\nonumber \\&\quad =\frac{(a_i-\delta a_0)^2(1+n\delta )((n-2\delta )+1)^2((2n-3)\delta +2)^2}{\beta (1-\delta )(\delta (n-1)+1)(2\delta ^3n^3+\delta ^2(8-9\delta )n^2+\delta (8\delta ^2-21\delta +10)n+(\delta ^3+9\delta ^2-12\delta +4))^2}. \end{aligned}$$

(9)

From (4) and (9), we have that

$$\begin{aligned}&\pi _i^{B}-\pi ^{p,\ldots ,p,q} \\&\quad = \frac{\delta ^2(a_i-\delta a_0)^2H_{9}}{\beta (1-\delta )(\delta n+1)(\delta n-\delta +1) (\delta ^2n^2-3\delta ^2n+3\delta n+\delta ^2-3\delta +2)^2 H_{10}}. \end{aligned}$$

Here, $\pi _i^{B}-\pi ^{p,\ldots ,p,q}$ is positive (res. negative, zero) if $H_{9}(n,\delta )$ is positive (res. negative, zero). Figure 8 describes the region for $H_9(n,\delta )<0$. This shows (i).¹²

Fig. 8

The region for the Bertrand model fails to be an equilibrium

$\lim _{n \rightarrow \infty }H_{9}(n,\delta )=-\infty .$ This implies (ii).

We have already discussed the equilibrium welfare when all firms choose the quantity contract ($SW^{C}$). We show that given the contracts of all private firms, the public firm has an incentive to choose the price contract, regardless of $\delta$.

Consider the subgame in which the public firm chooses the price contract and all private firms choose the quantity contract. In equilibrium, the public firm names the following price:

$$\begin{aligned} p_0=m_0, \end{aligned}$$

and all private firms selects the following quantity:

$$\begin{aligned} q_i=\frac{a_i-\delta a_0}{\beta (1-\delta )(2+\delta (1+n))} \quad (i=1,2,\ldots ,n). \end{aligned}$$

Substituting these equilibrium price and quantity into the payoff function, we have the following welfare:

$$\begin{aligned} SW^{p,q,\ldots ,q}=\frac{H_{11}}{2\beta (1-\delta )(\delta n+\delta +2)^2}. \end{aligned}$$

(10)

From (1) and (10), we have that

$$\begin{aligned} SW^{p,q,\ldots ,q}-SW^{C} = \frac{n\delta ^2(a_i-\delta a_0)^2(\delta n (2-\delta ^2)+\delta (2-\delta )+4)}{2\beta (1-\delta )(\delta n+\delta +2)^2(\delta ^2n-\delta n+\delta -2)^2}>0. \end{aligned}$$

This implies (iii). Q.E.D.

Proof of Proposition 4

Let $\pi _j^{B}$ and $\pi _j^{C}$ be the profit of each private firm under Bertrand and Cournot competition, respectively. From (5) and (7), we have that $\pi _j^{B}-\pi _j^{C}$ is

$$\begin{aligned} \frac{\delta ^2(\delta m(a_i-a_j)-(1-\delta )a_j)^2H_{12}}{\beta (1-\delta )(2\delta m-\delta +1)(\delta m(2\delta -3)- (1-\delta )(2-\delta ))^2(2\delta ^2m(m-3)+5\delta m+(1-\delta )(2-3\delta ))^2}. \end{aligned}$$

Here, $\pi _j^{B}-\pi _j^{C}$ is positive (res. negative, zero) if $H_{12}(m,\delta )$ is positive (res. negative, zero). We show that $H_{12}(1,\delta )$ is positive, and $H_{12}(m,\delta )$ is increasing in m for $m\ge 1$. Substituting $m=1$ into $H_{12}(m,\delta )$, we have $H_{12}(1,\delta )=\delta ^4-4\delta ^2+4>0$. We have

$$\begin{aligned} \frac{\partial {H_{12}(m,\delta )}}{\partial {m}}= & {} \delta ^3 (10m^3-6m^2-2m-2)m+4\delta ^2(1-\delta )(13\delta ^2+6(1-\delta )\\&+19(1-\delta )^2)m^3+(25(1-\delta )^4+5\delta ^3(1-\delta )^3 +4\delta ^2(1-\delta )^4\\&+(1-\delta )^5+(1-\delta )^6+\delta ^2f_2(\delta ))m^2 +(2(1-\delta )^4+17\delta ^3(1-\delta )^4\\&+21\delta ^3(1-\delta )^3+15\delta ^2(1-\delta )^5+2(1-\delta )^7+\delta ^3f_3(\delta ))m\\&+2\delta (7\delta ^4-24\delta ^3+30\delta ^2-16\delta +3), \end{aligned}$$

where

$$\begin{aligned} f_2(\delta )\equiv & {} 8\delta ^3+21\delta ^2-57\delta +29,\\ f_3(\delta )\equiv & {} 21\delta ^3+52\delta ^2-41\delta +11. \end{aligned}$$

Substituting $m=1$ into this, we have

$$\begin{aligned} \frac{\partial {H_{12}(m,\delta )}}{\partial {m}}|_{m=1}=2(2-\delta )(\delta (4-3\delta )+2)>0. \end{aligned}$$

Here, $\frac{\partial H_{12}(m,\delta )}{\partial m}$ is increasing in m for $m\ge 1$ if $f_2(\delta )>0$ and $f_3(\delta )>0$.

First, we show that $f_2(\delta )>0$. We have that

$$\begin{aligned} \frac{d f_2(\delta )}{d \delta }=24\delta ^2+42\delta -57. \end{aligned}$$

Solving $\frac{d f_2(\delta )}{d \delta }=0$ leads to the following solutions:

$$\begin{aligned} \delta =\frac{-7+\sqrt{201}}{8},\quad \delta =\frac{-7-\sqrt{201}}{8}. \end{aligned}$$

Thus, $f_2(\delta )$ is minimized when $\delta =\frac{-7+\sqrt{201}}{8}$ for $\delta \in (0,1)$. Since

$$\begin{aligned} f_2\left( \frac{-7+\sqrt{201}}{8} \right) =\frac{2867-201^{3/2}}{8}>0. \end{aligned}$$

$f_2(\delta ) >0$ for $\delta \in (0,1)$.

Next, we show that $f_3(\delta )>0$. We have that

$$\begin{aligned} \frac{d f_3(\delta )}{d \delta }=-63\delta ^2+104\delta -41. \end{aligned}$$

Solving $\frac{d f_3(\delta )}{d \delta }=0$ leads to the following solutions:

$$\begin{aligned} \delta =\frac{41}{63},\quad \delta =1. \end{aligned}$$

Thus, $f_3(\delta )$ is minimized when $\delta =\frac{41}{63}$ for $\delta \in (0,1)$. Since

$$\begin{aligned} f_3 \left( \frac{41}{63} \right)= & {} \frac{6583}{11907}>0, \end{aligned}$$

$f_3(\delta ) >0$ for $\delta \in (0,1)$.

Therefore, $\frac{\partial {H_{12}(m,\delta )}}{\partial {m}}>0$ for $m \ge 1$.

We now compare the welfare. From (6) and (8), we have that $SW^{B}-SW^{C}$ is

$$\begin{aligned} \frac{\delta ^2m(\delta m-\delta +1)(\delta m(a_i-a_j)-(1-\delta )a_j)^2H_{13}}{2\beta (1-\delta )(2\delta m-\delta +1)(\delta m(2\delta -3)-(1-\delta )(2-\delta ))^2(\delta m(2\delta m-6\delta +5)+(\delta -1)(3\delta -2))^2}. \end{aligned}$$

Here, $SW^{B}-SW^{C}$ is positive (res. negative, zero) if $H_{13}(m,\delta )$ is positive (res. negative, zero). We show that $H_{13}(1,\delta )$ is positive, and $H_{13}(m,\delta )$ is increasing in m for $m>1$. Substituting $m=1$ into $H_{13}(m,\delta )$ we have $H_{13}(1,\delta )=\delta ^4-4\delta ^2+4>0$. We show that $H_{13}(m,\delta )$ is increasing in m for $m \ge 1$ if $\delta \in (0,1)$.

We have

$$\begin{aligned} \frac{\partial {H_{13}(m,\delta )}}{\partial {m}}= & {} 4\delta ^2(4m^2-3m-1)m+6\delta (1-\delta )(4\delta (1-\delta )+8\delta +1)m^2+2\delta f_4(\delta ) m \\&+6\delta ^4-40\delta ^3+68\delta ^2-42\delta +8, \end{aligned}$$

where

$$\begin{aligned} f_4(\delta )\equiv & {} -12\delta ^3+64\delta ^2-75\delta +26. \end{aligned}$$

Substituting in $m=1$, we have

$$\begin{aligned} \frac{\partial {H_{13}(m,\delta )}}{\partial {m}}\Big |_{m=1}=6\delta ^4-8\delta ^3-16\delta ^2+16\delta +8>0. \end{aligned}$$

Here, $\frac{\partial {H_{13}(m,\delta )}}{\partial {m}}$ is increasing in m for $m\ge 1$ if $f_4(\delta )>0$. We show $f_4(\delta )$ is positive if $\delta \in (0,1)$. We have that

$$\begin{aligned} \frac{d f_4(\delta )}{d \delta }=-36\delta ^2+128\delta -75. \end{aligned}$$

Solving $\frac{d f_4(\delta )}{d \delta }=0$ leads to the following solutions:

$$\begin{aligned} \delta =\frac{32+\sqrt{349}}{18},\quad \delta =\frac{32-\sqrt{349}}{18}. \end{aligned}$$

Here, $f_4(\delta )$ is minimized when $\delta =\frac{32-\sqrt{349}}{18}$. Because

$$\begin{aligned} f_4 \left( \frac{32-\sqrt{349}}{18} \right) =\frac{6686-349^{3/2}}{243}>0, \end{aligned}$$

$f_4(\delta ) >0$ for $\delta \in (0,1)$. Thus, $\frac{\partial {H_{13}(m,\delta )}}{\partial {m}}>0$ for $m \ge 1$. Q.E.D.

Proof of Proposition 5

We have already discussed the equilibrium profit of each private firm when all firms choose the price contract. Let $\pi _j^B$ denote this profit. We show that given the contracts of other firms, a private firm has an incentive to deviate and chooses the quantity contract. We consider the subgame in which one private firm (firm 3) chooses the quantity contract and all the other firms (firm 0, firm 1, and firm 2) choose the price contract. In equilibrium , the public firms (firm 0 and firm 1) name the following price:

$$\begin{aligned} p_i=\frac{(5\delta ^3+7\delta ^2+2\delta )m_j+(-13\delta ^2 -16\delta -4)m_i+5\alpha \delta ^3-3\alpha \delta ^2-2\alpha \delta }{10\delta ^3-9\delta ^2-16\delta -4}\quad (i=0,1), \end{aligned}$$

the private firm that chooses the price contract (firm 2) names the following price:

$$\begin{aligned} p_2=\frac{(10\delta ^3-4\delta ^2-9\delta -2)m_j+(-10\delta ^2-4\delta )m_i+5\alpha \delta ^2-3\alpha \delta -2\alpha }{10\delta ^3-9\delta ^2-16\delta -4}, \end{aligned}$$

and the private firm that chooses the quantity contract (firm 3) selects the following quantity:

$$\begin{aligned} q_3=\frac{-(3\delta ^2+5\delta +2)m_j-(-6\delta ^2-4\delta )m_i-3\alpha \delta ^2+\alpha \delta +2\alpha }{10\beta \delta ^4-19\beta \delta ^3-7\beta \delta ^2+12\beta \delta +4\beta }. \end{aligned}$$

The private firm that chooses the quantity contract (firm 3) obtains the following profit:

$$\begin{aligned} \pi ^{p,p,p,q}=\frac{(1+3\delta )(2+3\delta )^2(\delta (a_j-a_i)+(1-\delta )a_i)^2}{\beta (1-\delta )(1+2\delta )(10\delta ^3-9\delta ^2-16\delta -4)^2}. \end{aligned}$$

(11)

Substituting $m=2$ into (5) we have that $\pi _j^{B}$ for $m=2$ is

$$\begin{aligned} \pi _j^{B}=\frac{(1+\delta )^2(1+2\delta )(\delta (a_j-a_i) +(1-\delta )a_i)^2}{\beta (1-\delta )(1+3\delta )(\delta ^2-5 \delta -2)^2}\quad (j=2,3). \end{aligned}$$

(12)

From (12) and (11), we have that $\pi _j^{B}-\pi ^{p,p,p,q}$ is

$$\begin{aligned} \frac{\delta ^2f_5(\delta )(20\delta ^3(1-\delta ^2)+21\delta ^3(1-\delta )+44\delta ^3+126\delta ^2+56\delta +8)(\delta (a_j-a_i)+(1-\delta )a_i)^2}{\beta (1-\delta )(1+2\delta )(1+3\delta )(\delta ^2-5\delta -2)^2(10 \delta ^3-9\delta ^2-16\delta -4)^2}, \end{aligned}$$

where $f_5(\delta )=-20\delta ^3-3\delta ^2+13\delta +4$.

Here, $\pi _j^{B}-\pi ^{p,p,p,q}$ is positive (res. negative, zero) if $f_5(\delta )$ is positive (res. negative, zero). $f_5(\delta )<0$ if $\delta > \delta ^* \simeq 0.867.$ Q.E.D.