Consider Hotelling's location model. In the town of Sandwich, Massachusetts, two sandwich vendors, Giorgo's Panini's (P) and la Donia Torta (7), need to decide where to locate their shop on Main street. The length of Main Street is one and there is a mass one of consumers who are distributed uniformly along Main street. Denote the location choice of shop P as p and that of shop Tas t. All consumers have an equal high willingness to pay for sandwiches from both vendors. However, they dislike travelling from their home to a shop's location. The cost of doing so is linear: if a consumer is located at point x on the interval, then the travel cost to sandwich shop P is the absolute value of the distance: [x p], and equivalently the travel cost to sandwich shop T is |x t]. The mayor of Sandwich, a tremendous sandwich lover, fixed the sandwich price at $5. The shops can produce a sandwich at a cost of $3 and choose their location to maximize profits. - 3.1. Given location choices p and t, find the demand for each shop (the percentage of consumers that buys at shop P and at shop T). How does a shop's demand depend on its location and that of the other shop. 3.2. Show that (p*, t*) = (1, 1) is a Nash equilibrium. 3.3. Show that (p*, t*) = (, ) is the unique Nash equilibrium.