There are two neighboring towns, Alfa and Beta. Firm A is located in Alfa, firm B in Beta. The two firms produce and sell identical goods, with fixed marginal costs of c. The two firms simultaneously decide on prices, $p_A$ and $p_B$ respectively. An inhabitant of Alfa can visit and shop at firm A at no cost. Driving to Beta costs $e\ge0$. Thus, an inhabitant of Alfa can observe $p_A$ at no costs. However, to observe $p_B$ costs e. The situation is symmetrical for someone who lives in Beta. The population in the two towns are identical. Hence, the demand curve is the same in both towns. Let $p_M$ denote the monopoly price. Assume monopoly profit is single-peaked (with its peak at $p_M$ ). Assume that a consumer who is indifferent between buying from A and B will elect to purchase from the firm in his home town.
(i) Derive all possible equilibrium prices in any symmetric pure-strategy Subgame Perfect Equilibria if $e=0$. Explain your answer.
(ii) Derive all possible equilibrium prices in any symmetric pure-strategy Subgame Perfect Equilibria if $e > 0$. Prove and explain your answer. Explain how and why the equilibrium changes with a small change in e.
what I think:
For first part (i)
I have three cases:
First case:
Both $p_A = p_B >0$. In this case, no firm want to deviate. Thus, I will calculate SPNE points by using cournot duopoly model.
Second case:
Both $p_A = p_B =0$. In this case, one of the firms wants to deviate by increasing its price a little bit. Thus, I have SPNE points at $q^*=0$ by using cournot duopoly model. So there’s no SPNE.
Third case:
$p_A \not= p_B >0$ in this case, again, one of the firms wants to deviate in order to increase its profit. So there is no SPNE.
I did the first part in this way. But I don’t know to what extent it is true. And if it is true, I could not proceed it..
For second part (ii)
Under symmetric equilibrium $$p_A+e=p_B+e=c$$. I think that I can show this equilibrium by using proof by contradiction.
And the most important point is that I could not do the second part (ii) as well. Please explain how I can solve it.
Any help is appreciated. Thank you for your helps in advance.