Question

Assume that there are 2 firms locating in Hotelling’s linear city of length 1. Each firm has a constant marginal cost of c. Consumers are uniformly distributed and have identical preferences represented by U=V-Pi-t(x*-xi)2 . where V-is consumer’s reservation value (consumer surplus from consuming her ideal product at zero price), Pi is the price of firm i’s product, x* is consumer’s location, 0≤x 1 and xi is firm i’s location. a) If both firms are required to locate in the middle, what price would they charge? b) If both firms are required to locate at two different ends of the city what price will they charge?

Answer 1

Ans a) – If both firms are required to locate in the middle, what price would they charge?

The linear city (Hotelling) –

- Linear city of length 1.

• Consumers are distributed uniformly along the city,

N = 1

• Quadratic transportation costs t per unit of length.

• They consume either 0 or 1 unit of the good

Minimal differentiation

• 2 shops are located at the same location X0

• P1and P2 are the prices charged by the 2 shops.

• Price of going to shop 1 for a consumer at x is

P1 + t(X0 − x)2

• Price of going to shop 2 for a consumer at x is

P2 + t(X0− x)2

• The consumers compare prices.... Bertrand competition

• Nash equilibrium in prices is

p∗i = p∗j = c

• and the equilibrium profits are

Π1= Π2 = 0

…………………………………..

Ans b) –

Maximum differentiation –

The linear city (Hotelling) –

- Linear city of length 1.

• Consumers are distributed uniformly along the city,

N = 1

• Quadratic transportation costs t per unit of length.

• They consume either 0 or 1 unit of the good

-----2 shops are located at the 2 ends of the city, shop 1 is at x = 0 and of shop 2 is at x = 1. c unit cost

• P1 and P2 are the prices charged by the 2 shops.

• Price of going to shop 1 for a consumer at x is P1+ tx2.

• Price of going to shop 2 for a cons. at x P2 + t(1 − x)2

• The utility of a consumer located at x is

Minimal differentiation –