There are two firms, noted A and B, producing a homogeneous good; their marginal cost of production is normalized to zero. There is a unit mass of consumers who have a unit demand for the good; their reservation price for the good is equal to r > 0. In period 1, consumers bought one unit of the good from one or the other firm; in particular, a mass αA of consumers bought from firm A, and a mass αB from B, withαA +αB = 1. We concentrate here on the period 2 game: the two firms set simultaneously their price (noted pA and pB) and consumers decide from which firm to buy, given the two prices and given the presence of a switching cost z < r, which they have to incur if they switch firms from period 1 to period 2. You are asked to characterize the Nash equilibrium in the price competition game of period 2. In particular, answer the following three questions. 

1. Given r and z, for which pairs (αA, αB) does a symmetric pure-strategy Nash equilibrium exist in this game? 

2. For which values of r and z will there be no symmetric pure-strategy Nash equilibrium in this game for any pair (αA, αB)? 

3. Give the economic interpretation of your results.