2. The Circle Line (30 points): The circle line is a new metro line that runs around the perimeter of a circular town called Discadel. The town's perimeter is equal to 1. The introduction of the new circle line has attracted food retailers that want to open grocery stores around it. A unit mass of consumers are uniformly distributed around the circle line. Assume that the utility of food shopping is high enough such that all consumers do their food shopping at the new stores. Further assume that consumers have transportation cost given by ed, where t is the unit transportation cost and d is the distance to the supermarket. Initially, n retailers decide to obtain a license to open a store at a fixed cost F. Once the stores open they compete by choosing prices p simultaneously. The cost to the retailer of each shopping trip is c = 5. Assume that the food retailers' stores are equidistant from one another. (a) Find the equilibrium values of the following as functions of t, n and F: i. (5 points) The price (p*) ii. (5 points) The number of shopping trips (q*) iii. (5 points) Profits per firm (**). (b) (3 points) Suppose F = 2 and 4 retailers entered into the market in equilibrium. We also know that each of those stores are making zero profits. Determine the corresponding value of t. (c) (4 points) How many retailers, n, enter into the market if t = 50 and F = 1.5. Assume that that n must be an integer number such as 1, 2,3, .. (d) Consider a social planner who minimizes the sum of transportation costs and total fixed costs. Consider t = 64 and F = 4. i. (5 points) Determine the socially optimal number of retailers in the market. ii (3 points) With respect to the equilibrium number of retailers, does the social planner prefer more or less differentiation in the market? Explain your