Question

Two firms produce and sell differentiated products that are substitutes for each other. Their demand curves are Firm​ 1: Upper Q 1Q1 ​= 40minus−3Upper P 1P1​+ Upper P 2P2 Firm​ 2: Upper Q 2Q2 ​= 40minus−3Upper P 2P2​+ Upper P 1P1 Both firms have constant marginal costs of ​$4.604.60 per unit. Both firms set their own price and take their​ competitor's price as fixed. Use the Nash equilibrium concept to determine the equilibrium set of prices. Since the firms are​ identical, they will set the same prices and produce the same quantities. In​ equilibrium, each firm will charge a price of ​$nothing and produce nothing units of output. ​(Enter your responses rounded to two decimal​ places.) Each firm will earn a profit of ​$nothing. ​(Enter your response rounded to two decimal​ places.)

Answer 1

Answer:

$P_1=P_2=9.86\\Q_1=Q_2=20.28\\\pi_1=\pi_2=137.09$

Explanation:

$Q_1 = 40 - 3p_1 +p_2, MC_1 = 3.10$

$TR_1 = P_1 \times Q_1 = 40P_1 - 3P_1^2 + P_1P_2$

$TC_1 = MC_1 \times Q_1 = 3.10 (40 - 3P_1 + P_2)$

$= 124 - 9.3P_1 + 3.10P_2$

$Profit, \pi_1 = TR_1 - TC_1$

$= 40P_1 - 3P_2^2 + P_1P_2 - 124 + 9.3P_1 - 3.10P_2$

$49.3P_1 - 3.10P_2 + P_1P_2 - 3P_1^2 - 124$

$\frac{d\pi}{dP_1} = 0$     when profit is maximized

$= 49.3 + P_2 - 6P_1 = 0$

$6P_1 - P_2 = 49.3$        [Response Function for firm 1]  ...........(1)

again

$B_2 = 40 - 3P_2 + P_1 , MC_2 = 3.10$

$TR_2 = P_2 \times B_2 = 40P_2 - 3P_2^2 + P_1P_2$

$TC_2 = MC_2 \times B_2 = 3.10(40 - 3P_2 + P_1)$

$= 124 - 9.3P_2 + 3.10P_1$

$\pi_2 =TR_2 - TC_2 = 40P_2 - 3P_2^2 +P_1P_2 - 124 + 9.3P_2 - 3.10P_1$

$= 49.3P_2 - 3.10P_1 + P_1P_2 - 3P_2^2$

check additional details in the attached file