Firm 1 has a single retail location at the left hand extreme point of a mile-lon

Question

Firm 1 has a single retail location at the left hand extreme point of a mile-long Main Street while firm 2 is located at the right hand extreme. There are 100 consumers who are evenly distributed over this market. The value of the product sold is V=$10 to any consumer. The marginal cost of production is $1.40 to firm 1 and $3.20 to firm 2. The transportation cost is $1 per mile.

1. What is the Nash equilibrium price charged by firm 1?

2. What is the Nash equilibrium price charged by firm 2?

3. In Nash equilibrium, Firm 1 makes a profit of ?

Explanation / Answer

Firm 1's demand curve: Q1=12-2(P1)+(P2)
Firm 2's demand curve: Q2=12-2(P2)+(P1)
Where (P1) and (P2) are the price that Firms 1 and 2 charge and Q1 and Q2 are the resulting quantities that they sell.
We assume that both firms set their prices at the same time and that each firm takes its competitor's price as fixed.

Q1=12-2*1.40+3.20=12.4

Q2=12-2*3.20+1.40=7

totalQ1=12.4*100=1240

toatalQ2=7*100=700

Profit function for firm 1 is (P1Q1)-10 = 12(P1)-2(P1)^2+(P1P2)-10

Profit function for firm 2 is (P2Q2)-10 = 12(P2)-2(P2)^2+(P1P2)-10

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