Consider a duopoly of two icecream vendors on a beach competing on prices (like in Bertrand duopoly) but we relax the assumption that products are identical.

We think of the beach as a line of length 1. There are 2 vendors, at each end of the beach. Each vendor produces icecream at constant marginal cost c. The vendors simultaneously choose the price of their icecream p1 and p2. Potential customers are uniformly distributed along the beach, with a total population normalized to 1. Each potential customer buys exactly one icecone either from vendor 1 or vendor 2, so the total demand is 1. Given its position on the beach (line y) (i.e., the customer is at distance y from vendor 1 and 1 y from vendor 2), a customer will buy its unit:

from firm 1 if p1 + ty² < p2 + t(1 y)²

from firm 2 if p1 + ty² > p2 + t(1 y)² (1)

and will toss a fair coin if these two quantities are equal.

Conditions (1) translate what the customers care about: they try to buy from the vendor with smallest price + t(distance)² . If we think of the line as representing geographical distance, t(distance)² is a transport cost of the ice melting.

However, other interpretations are possible. For instance, we can think of the line as representing some aspect of the product quality. Then, the proximity to a firm represents the preference and the term t(distance)² measures the inconvenience of having to move away from the preferred product. Parameter t represents how much the products are differentiated. If t = 0, the products are identical.

The demand resulting from customers choices to vendor i is denoted di(p1, p2). As usual, each vendor's objective is to maximize its profit:

u1(p1, p2) = p1d1(p1, p2) cd1(p1, p2),

u2(p1, p2) = p2d2(p1, p2) cd2(p1, p2).

1.Will either vendor i ever set a price pi < c? Why?

2.Suppose that vendor 2 sets a price p2. What is the maximum price p1 for which vendor 1 captures the entire market (i.e., has demand 1)?

3.Suppose that prices p1 and p2 are close enough that each vendor has positive demand. Find the position of the customer who is indifferent between buying from vendor 1 and buying from vendor 2. Compute the demand to vendor 1 d1(p1, p2).

4.Suppose that prices p1 and p2 are close enough that each firm has positive demand. Compute the best-response of vendor 1 to p2.

5.Draw the complete best response diagram (be careful while plotting BR1(p2) when p2 < c t and p2 > 3t + c and similarly for BR2).