Home / Business / Economics / Chapter 12 Exercise 9 *9. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C; = 10Q: + {Q}(i = E,D) = Q = QE + QD a. Unable to recognize the potential for collusion, the two firms Chapter 12 Exercise 9 *9. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C; = 10Q: + {Q}(i = E,D) = Q = QE + QD a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of Qp, lp, and P? What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oli- gopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of Qe Qp, and P? What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of Qg, Qp, and P? What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of QQp, and P? What are each firm's profits? Chapter 12 Exercise 11 *11. Two firms compete by choosing price. Their demand functions are Q. = 20 - P1 + P2 and Q2 = 20 + P - P2 where P, and P, are the prices charged by each firm, respectively, and Q and are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.) b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Question Chapter 12 Exercise 9 9. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C; = 10Q: + {Q}(i = E,D) = Q = QE + QD a. Unable to recognize the potential for collusion, the two firms Chapter 12 Exercise 9 9. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C; = 10Q: + {Q}(i = E,D) = Q = QE + QD a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of Qp, lp, and P? What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oli- gopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of Qe Qp, and P? What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of Qg, Qp, and P? What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of QQp, and P? What are each firm's profits? Chapter 12 Exercise 11 *11. Two firms compete by choosing price. Their demand functions are Q. = 20 - P1 + P2 and Q2 = 20 + P - P2 where P, and P, are the prices charged by each firm, respectively, and Q and are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.) b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

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Transcribed Image Text: Chapter 12 Exercise 9 *9. Demand for light bulbs can be characterized by Q = 100 – P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C; = 10Q: + {Q}(i = E,D) = Q = QE + QD a. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of Qp, lp, and P? What are each firm's profits? b. Top management in both firms is replaced. Each new manager independently recognizes the oli- gopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of Qe Qp, and P? What are each firm's profits? c. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of Qg, Qp, and P? What are each firm's profits? d. If the managers of the two companies collude, what are the equilibrium values of QQp, and P? What are each firm's profits? Chapter 12 Exercise 11 *11. Two firms compete by choosing price. Their demand functions are Q. = 20 - P1 + P2 and Q2 = 20 + P - P2 where P, and P, are the prices charged by each firm, respectively, and Q and are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.) b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

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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} ...