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The detailed solution for exercise 9 is attached for your ready reference. Good luck ud83dudc4d  Given information:-(1)\begin{array}{l}Q=100-P \\\alpha \\P=100-Q\end{array}Cost function is identical\begin{aligned}C & =10 \theta+\frac{1}{2} \theta^{2} \\\therefore M C & =\frac{d C}{d \theta}\left[10 \theta+\frac{1}{2} \theta^{2}\right] \\& =10(1)+\frac{1}{\chi} \times 2 \times Q^{2-1} \\& =10+Q\end{aligned}(a) If they act as perfect competetors. then they are playing Bertrend competetion. In such a case.\begin{aligned}\operatorname{Price}(P) & =M C \\100-Q & =10+\theta \\100-10 & =2 Q \\90 & =20 \\Q & =\frac{90}{2}=45\end{aligned}Hence, each firm will produce \theta_{E}=Q_{D}=\frac{Q}{2}\therefore P=100-Q=100-143=\$ 55P Profits:-(2)\begin{aligned}\pi_{E} & =P \times \theta_{E}-\left[10 \theta_{E}+\frac{1}{2} \theta_{E}^{2}\right] \\& =55 \times 22.5-\left[10 \times 22.5+\frac{1}{2} \times(22.5)^{2}\right] \\& =1237.5-[225+253.125] \\& =1237.5-478.125 \\& =\$ 759.375\end{aligned}Hence, each firm will earn $759.375-(b) Cournot Duopoly\begin{array}{l}P=100-Q_{D} \quad \text { where } Q=Q_{E}+Q_{D} \\P=100-Q_{E}-Q_{D}\end{array}Everglow:-\text { Total Revenve } \begin{aligned}(T R) & =P \times Q_{E} \\& =\left(100-Q_{E}-Q_{D}\right) \times Q_{E} \\T R & \left.=100 Q_{E}-Q_{E}^{2}-Q_{D} \cdot Q_{E}\right] \\\therefore M R & =\frac{d T R}{d Q_{E}}\left[100 Q_{E}-Q_{E}^{2}-Q_{D} \cdot Q_{E}\right) \\& =10041-2 Q_{E}^{2-1}-Q_{D}(1) \\M R & =100-2 Q_{E}-Q_{D}\end{aligned}(3)Setting M R=M C_{E}\begin{array}{l}100-2 Q_{E}-Q_{D}=10+Q_{D} \\100-10-Q_{D}=3 Q_{E} \\90-Q_{D}=3 Q_{E} \\Q_{E}=\frac{90}{3}-\frac{Q_{D}}{3}\end{array}EverglowNow, since dimlit also faces the same demand with identical cost strudure. Hence, their reacton function will be :-Q_{D}=30-\frac{1}{3} Q^{3}Putting the value of Q_{D} in Q_{E}\begin{array}{l}Q E=30-\frac{1}{3} Q_{D} \\Q E=30-\frac{1}{3}\left(30-\frac{1}{3} Q E\right) \\Q_{E}=30-10+\frac{1}{9} Q_{E} \\\text { Q } E-\frac{1}{9} Q_{E}=20 \\\frac{9 Q E-Q E}{9}=20 \\8 Q E=9 \times 20 \\\text { QE }=\frac{9 \times 20}{8}=22.5 \\\end{array}(4)\begin{aligned}\therefore Q_{D} & =30-\frac{1}{3} Q_{E} \\& =3 \Delta-\frac{1}{3}(22.5) \\& =30-7.5 \\& =22.5\end{aligned}Sion, qu andietye pradoned\text { So, Priv: } \begin{aligned}P & =100-Q_{E}-Q_{D} \\& =100-22.5-22.5 \\& =100-45 \\& =\$ 55\end{aligned}Since, quantily produad and price is same as in part (a).Hence, each firm will eann a profit=\$ 759.375^{\circ}(c) When Everglow is playing stadcelberg, then they will inconporate Dimlit reaction function in thein total Revenve function to decide their quantity.Dintit Reaction function:\theta_{D}=30-\frac{1}{3} \theta_{E}Total Revenve function of Everglow:-(5)T_{R}=100 Q_{E}-Q_{E}^{2}-\theta_{D} \cdot Q_{E}Putting value of Q_{D} in the above equation\begin{aligned}T R & =100 Q_{E}-\theta_{E}^{2}-\theta_{E}\left(30-\frac{1}{3} Q_{E}\right) \\& =100 Q_{E}-Q_{E}^{2}-30 \theta_{E}+\frac{1}{3} Q_{E}^{2} \\& =70 Q_{E}-\frac{3 Q_{E}^{2}+\theta_{E}^{2}}{3} \\T R & =70 Q_{E}-\frac{-2 \theta_{E}^{2}}{3}\end{aligned}\therefore New T R function:T R_{E}=70 Q_{E}-\frac{2}{3} Q_{E}^{2}So,\begin{aligned}M R_{E} & =\frac{d T R_{E}}{d Q_{E}}\left[70 Q_{E}-\frac{2}{3} Q_{E}^{2}\right] \\& =70(1)-\frac{2}{3} \times 2 \times Q_{E}^{2-1} \\M R_{E} & =70-\frac{4}{3} \theta_{E}\end{aligned}\begin{array}{l}\text { Setting } M R_{E}=M C \\70-\frac{4}{3} Q_{E}=10+\theta_{E} \\70-10=Q_{E}+\frac{4}{3} Q_{E} \\b_{0}=\frac{7 Q_{E}}{3}\end{array}\begin{aligned}Q_{E} & \left.=\frac{60 \times 3}{7}=25.71\right] \\\therefore Q_{D} & =30-\frac{1}{3} Q_{E} \\& =30-\frac{1}{3}(25.71) \\& =30-8.57=21.43\end{aligned}(6)\text { So, Price : } \begin{aligned}P & =100-Q_{E}-Q_{D} \\& =100-25.71-21.43 \\& =100-47.14 \\& =\$ 52.86\end{aligned}Profits :Eviglow\begin{array}{l}\pi_{E}=P_{\times} \theta_{E}-\left[10 \theta_{E}+\frac{1}{2} \theta_{E}^{2}\right] \\A_{D}=P \times Q_{D}=\left[10 Q_{D}+\frac{1}{2} Q_{D}^{2}\right] \\=52.86 \times 25.71 \\-\left[10(25.71)+\frac{1}{2}(25.71)^{2}\right] \\=52.86 \times 21.43 \\-\left[10(21.43)+\frac{1}{2} \times(21.43)^{2}\right] \\=1132.7898-\left[\begin{array}{l}214.3 \\+229.62\end{array}\right] \\=1359.0306-587.6 \\=1132.7898-443.92 \\=\$ 771.4306 \\=\$ 688.8698 \\\end{array}(7)(d) when they collude they will ack like monopolist and each one will produce half of the total quantity produced (Q).\begin{array}{l}P=100-Q \\\text { Total Revenve }\left(T R_{M}\right)=P \times Q .(100-\theta) \times \theta \\\qquad T R_{M}=100 Q-Q^{2} \\\therefore M R_{M}=\frac{d T R_{M}\left[100 Q-Q^{2}\right]}{d Q}=100-2 Q \\M R_{M}=100\end{array}Setting M R_{M}=M C\begin{aligned}100-2 \theta & =10+Q \\100-10 & =3 Q \\90 & =3 Q \\Q & =\frac{90}{3}=30\end{aligned}Hence, each firm will produce Q_{E}=Q_{D}=\frac{Q}{2}\begin{aligned}\therefore P \text { ive : } P=100- & \theta=100-30 \\= & \$ 70\end{aligned}=\frac{30}{2}=15(8)Profits :-\begin{aligned}\pi_{E} & =70 \times 15-\left(10(15)+\frac{1}{2}(15)^{2}\right] \\& =1050-(150+112.5) \\& =1050-262.5 \\& =\$ 787.5 \\\pi_{D} & =70 \times 15-\left(10(15)+\frac{1}{2} \times(15)^{2}\right] \\& =1050-262.5 \\& =\$ 787.5\end{aligned} ...