Two firms produce and sell differentiated products that are substitutes for each

Question

Two firms produce and sell differentiated products that are substitutes for each other. Their demand curves are Firm 1: Q1 = 40-3P1+ P2 Firm 2: Q240- 3P2+ P 1 Both firms have constant marginal costs of $2.20 per unit. Both firms set their own price and take their competitors 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 S 9.32 and produce 21.36 units of output. (Enter your responses rounded to two decimal places.) Each firm will earn a profit of $ 152.08. (Enter your response rounded to two decimal places.)

Explanation / Answer

Consider the given problem here there are 2 firm and their demand curve curves and the “MC” are also given in the question.

So, here the 1st firm’s payoff function is given below.

=> 1 = P1*Q1 – MC1*Q1 = P1*(40 – 3*P1 + P2) – 2.2*(40 – 3*P1 + P2).

=> 1 = 40*P1 – 3*P1^2 + P2*P1 – 2.2*(40 – 3*P1 + P2). So, here the 1st firm will try to maximize “1” with respect to “P1” given that “P2” will remain same.

=> 1/ P1 = 40 - 6*P1 + P2 – 2.2*(- 3) = 40 + 6.6 + P2 – 6*P1 = 46.6 + P2 – 6*P1.

=> 1/ P1 = 46.6 + P2 – 6*P1 = 0, be the “Reaction Function” of the 1st firm.

Similarly the 2nd firm’s payoff function is given below.

=> 2 = P2*Q2 – MC2*Q2 = P2*(40 – 3*P2 + P1) – 2.2*(40 – 3*P2 + P1).

=> 2 = 40*P2 – 3*P2^2 + P2*P1 – 2.2*(40 – 3*P2 + P1). So, here the 1st firm will try to maximize “1” with respect to “P2” given that “P1” will remain same.

=> 2/ P2 = 40 - 6*P2 + P1 – 2.2*(- 3) = 40 + 6.6 + P1 – 6*P2 = 46.6 + P1 – 6*P2.

=> 2/ P2 = 46.6 + P1 – 6*P2 = 0, be the “Reaction Function” of the 2nd firm.

Now, the intersection of “Reaction Function of Firm1” and “Reaction Function of Firm2” will determine the optimum price of the both firms here.

=> 46.6 + P2 – 6*P1 = 0 ………………….(1)

=> 46.6 + P1 – 6*P2 = 0 ………………….(2)

We can see that both tese equations are identical to each other => at the optimum “P1=P2”.

=> 46.6 + P2 – 6*P1 = 0, => 46.6 = 5*P1, => P1 = 46.6/5 = 9.32.

So, “P1=P2=9.32”, => Q1 = 40 – 3*P1 + P2 = 40 – 2*P1 = 40 – 2*9.32 = 21.36.

So, the each firm will produce “Q1=Q2=21.36”.

So, the profit of the 1st firm is “1 = P1*Q1 – MC1*Q1 = (9.32-2.2)*21.36 = 152.08.

So, here each firm will produce “Q1=Q2=21.36” will charge “P1=P2=9.32” and will earn “1=2=152.08” as a profit.

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