3. (16 points) Consider a Salop circle model. There are consumers uniformly spread along the circumference of a circle with length 1. The consumer gets a utility of consumption equal to U(t, t) = 2 - 2|t - | where t is the product purchased and T is the customer's most preferred product. Each customer has an outside option of buying from none of the firms and consuming a substitute instead for a total surplus (utility minus cost) of 1. There is a fixed cost of entering the industry equal to .25. a. (8 points) Suppose there are four firms in the industry. Where do they locate? What price do they charge? b. (8 points) Suppose there are eight firms in the industry. Where do they locate? What price do they charge? c. (Extra credit: 8 points) Which scenario (part a or part b) has higher social welfare? (Just guessing isn't going to get you any points. )- Consider how a single firm would price the good - Charging a price of p the firms will get all customers that live close enough - Customers will buy if u - clt - t*| - p z u Rearranging: u-u-P - 2 It -t* | C Thus, the quantity sold is q = 2- U-u-P CThis is the demand function - The inverse demand can be found by rearranging: p = u -u NIn q Revenue: Rev = pq - Marginal Revenue: MR = u -u - cq - Optimal quantity: q* = C - Optimal price: p u-u = 2- You now solve for the optimal quantity and price Inverse demand: p =p +cd -cq Marginal revenue: MR = p + cd - 2cq pted - Optimal quantity: q* = 2c pted - Optimal price: p* = 2c