Two independent ice cream vendors own stands at either end of a 1 mile long beach. Everyday there are 100 beach-goers who come to the beach and distribute themselves uniformly along the water. Every beach-goer one wants exactly one ice cream during the day, and values the ice cream from both stands at $6. All of the beach-goers would rather be sunbathing or in the water, so they have a disutility to walking on the beach of $1 per mile. Both firms have a marginal cost of zero.

1. Each individual is also referenced by a location x on the beach between 0 and 1. What are the utilities of purchasing from Early's and Cali for the person at location .75, given that Early's names price pe and Cali names price pc? What are the utilities for each individual as a function of their location on the beach, x?
2. What is the demand for Early's Ice Cream and Cali Creamery given that the firms charge prices pe and pc, respectively?
3. What is Cali Creamery's best response function when Early's posts a price of pe?
4. What is the equilibrium prices both firms charge, if both firms have to determine their prices early in the morning without knowing the price of the other firm is going to charge.