La Durazno is the only resort hotel on a small desert island off the coast of South America. It faces two market segments: bargain travelers and high-end travelers. The demand curve for bargain travelers is given by Q_¹ = 400 - 2P_1. The demand curve for high-end travelers is given by Q₂ = 500 - P₂. In each equation, Q denotes the number of travelers of each type who stay at the hotel each day, and P denotes the price of one room per day. The marginal cost of serving an additional traveler of either type is $20 per traveler per day.

1. Under the assumption that there is a positive demand from each type of traveler, what is the equation of the overall market demand curve facing the resort?
2. What is the profit-maximizing price under the assumption that the resort must set a uniform price for all travelers? For the purpose of this problem, you may assume that at the profit-maximizing price, both types of travelers are served. Under the uniform price, what fraction of customers are bargain travelers, and what fraction are high end?
3. Suppose that the resort can engage in third-degree price discrimination based on whether a traveler is a high-end traveler or a bargain traveler. What is the profit-maximizing price in each segment? Under price discrimination, what fraction of customers are bargain travelers and what fraction are high end?
4. The management of La Durazno is probably unable to determine, just from looking at a customer, whether he or she is a high-end or bargain traveler. How might La Durazno screen its customers (i.e., cause them to self-identify type through their choices) so that it can charge the profit-maximizing discriminatory prices you derived in question (3)?