Price Competition in Differentiated Product Markets (25 points) Hoffman's and Sundae's both sell ice cream on Point Pleasant beach, a one mile stretch of beach in New Jersey. Hoffman's is located 0.25 miles from the North end of the beach and Sundae's is located 0.25 from the South end of the beach. There are 800 bathers uniformly distributed along the beach. All consumers value the ice cream at $8, but have a disutility of walking to a vendor of 2 dollars per mile. Each vendor has a marginal cost of $1 per cone. Assume everyone eats ice cream.

(a) (2.5 points) What is the utility of a consumer at location x (between 0 and 1) from purchasing ice cream from Hoffman's?

(b) (5 points) Derive an expression for the location of the consumer who gets the same utility from purchasing from Hoffman's as they do from Sundae's.

(c) (2.5 points) An expression for the demand for Hoffman's ice cream as a function of each firm's price.

(d) (5 points) Suppose Sundae's sets a price of pS = 2, what is the optimal price for Hoffman's in this situation?

(e) (5 points) Suppose pH = 2 and pS is the price you found in the previous question. If you are the consumer located exactly in the middle of the beach, which firm do you purchase from? Explain your answer.

(f) (2.5 points) Sundae's is considering moving closer to Hoffman's. What is one reason that might be good for Sundae's profits?

(g) (2.5 points) Sundae's is considering moving closer to Hoffman's. What is one reason that might be bad for Sundae's profits?