Question

Problem 2. Two independent ice cream vendors own stands at either end of a 4 mile long beach. Everyday there are 400 beach-goers who come to the beach and distribute themselves uniformly along the water. Every beach-goer wants exactly one ice cream during the day, and values the ice cream from both stands at $10. All of the beach-goers would rather be sunbathing or in the water, so they have a disutility to walking on the beach of $0.25 per mile. Early's Ice Cream, the firm at location 0, is an early riser and always posts her price first. Cali Creamery, at location 4, is more laid back and posts her price just before the beach opens (the beach requires all prices be posted by the time the beach opens). Both stands get their ice cream from the same place and pay $1 per unit. 1. Each individual is also referenced by a location r on the beach between 0 and 4. What are the utilities of purchasing from Early's and Cali for the person at location = 2.50, given that Early's names price pe and Cali names price pe? What are the utilities for each individual as a function of their location on the beach, r? 2. What is the demand for Early's Ice Cream and Cali Creamery given the firms name prices pe and pe? 3. What is Cali Creamery's profit maximizing pricing strategy (e.g. complete contingent) given Early posts first? 4. What is the Sub-game Perfect Nash Equilibrium outcome for this market? Report prices, quantities, and profits for each firm. 5. Early's owner feels that her hard work is not paying off, she hires you as a business consultant. She's annoyed that Cali is always undercutting her price and is considering waiting to post so that Cali will not learn her price before naming her own. She wants you to predict how waiting to post a price will affect profits. What will Early's profits be under this new regime? What advice do you give her?