Suppose there are 2 firms in a vertically differentiated market. Consumers buy either one unit of any of the two goods or they do not purchase in the market. If they do not purchase in the market their indirect utility is 0. If they purchase good i their indirect utility is θ s_i - p_i, where θ is the preference parameter of a consumer, s_i is the quality of good i and pi is the price of good i. Assume that there exist consumers of mass 1 whose preference parameter is uniformly distributed on [0, 1].

1. Determine the demand function of each firm depending on prices and qualities.

2. Suppose that qualities are given, s₁ < s₂, and that firms face constant marginal costs of production c, which are independent of quality. Determine equilibrium prices and profits in the game in which firms simultaneously set prices.

3. Suppose firms set qualities from an interval [0, s̅]. Which qualities will result in subgame perfect equilibrium in the game in which firms set qualities simultaneously at stage 1 and set prices simultaneously at stage 2?