Consider a (Salop’s) circular city. The length of its circumference (perimeter) is 1. The number of people in the city is also normalized to 1. Each citizen buys one unit of the good from the closest firm. Let τ denote the unit transportation cost to a consumer. For simplicity, it is assumed that the firms have zero fixed costs and zero variable costs.
(i) Assume that there are two firms, one at location 0, and the other one at location 1/2 on the circle. Write down the profit of each firm as a function of its price. Calculate the best response function of each firm. Calculate the Nash Equilibrium prices, (p1*, p2*). Calculate the equilibrium market shares and profit levels of the firms. Calculate the total consumers’ surplus. Calculate the average transportation cost for each customer/citizen.
(ii) Now, assume that there are three firms, at locations 0, 1/3, and 2/3. Calculate the best response of each firm to the prices of the other two firms. Calculate the symmetric Nash Equilibrium. Calculate the equilibrium market shares and profit levels of the firms. Calculate the total consumers’ surplus. Calculate the average transportation cost for each customer/citizen.
