Suppose that, as in Section 11.3.2, each of two firms 1 and 2 provides two components A and B. The offerings of both firms are horizontally differentiated. A consumer of type (θ_A, θ_B) ∈ [0, 1]² derives a net surplus r – θ_A - (1 – θ_B) - p¹_A - p²_B if she buys component A from firm 1 at price p¹_A and component B from firm 2 at price p²_B. Correspondingly, for other systems of A and B. The gross surplus is assumed to be zero if the consumer does not buy a system. Consider only values of r such that the market is fully covered in equilibrium. Consumers of mass 1 are uniformly distributed on the unit square.

1. Suppose that firm 1 incurs a constant marginal cost of zero for each component  and that firm 2 incurs a marginal cost of c (with c < 3). Thus, firm 1 is more efficient. Determine the profit function of each firm when each firms sells each component separately. Determine equilibrium prices and equilibrium profits in the setting in which firms simultaenously set prices for both components.

2. Consider the same setting as before when both firms offer only a system (pure bundling). Determine equilibrium prices and equilibrium profits in the setting in which firms simultaneously set prices for their system.

3. Compare your results in (1) and (2). When is it more profitable for firm 1 to sell components in a bundle? When is it more profitable for firm 2 to sell components in a bundle?

4. Suppose now that firm 1 is more efficient producing component A and firm 2 is more efficient producing component B. In particular, suppose that, for component A, firm 1 incurs a constant marginal cost of zero and that firm 2 incurs a marginal cost of c, while the reverse holds for component B. Determine equilibrium prices and equilibrium profits under independent selling and pure bundling. Discuss the profitability of pure bundling in this setting.