Question

A county consists of two towns, Right and Left. Each has one-third of the population; the remaining one-third lives uniformly scattered on the one straight road that connects the two towns. Everyone is an imbiber. A and B own the only two beer licenses. Each drinker will go regularly to whichever bar is closest to him. The price of a drink in each bar is equal and determined by the government. The only variable that A and B have control over is their location.

(a) Find the Nash equilibrium locations of the two establishments.

(b) Comment whether Nash equilibrium locations are "socially optimal" ones?

Answer 1

(a)Nash equilibrium would be halfway between the towns. As we know nash equilibrium occurs when given the opponent's location,firm wouldn't deviate from its current position so as to maximize its profits. At halfway both of them would get access to an equal percentage of the population. Therefore they have equal market share. Any movement away from the centre given rival firm holds its position in the centre will result in a reduction in market share. Therefore the centre position is the Nash equilibrium.

b)It is not a socially optimal one because the majority of the population has to travel to the centre to reach the destination. Therefore 2/3 population(total population living in towns)has to travel halfway through the distance between two towns to reach the location A more socially optimal one would be to have the firms one each at each town. Therefore total distance travelled is reduced.2/3 the population would have to cover 0 distance.