Question

2- Consider Hotelling’s model (a street of length 1, consumers uniformly distributed along the street). Suppose that each consumer has a quadratic transportation cost equal to 2d², where d is the distance traveled. Suppose that there are two firms, one located in 0.25 and the other one in 1. Assume production costs are equal to c=5m.u. for one product unit.

a) Find the indifferent consumer’ location.

b) Determine the demand functions for the two firms.

c) Determine the best response functions.

d) If the two firms compete in prices and settle at a Nash equilibrium, which are the equilibrium prices? Determine the corresponding total profits.

Answer 1

$$x=0.75-x$$

$$2x=0.75$$

$$x=0.375$$

$$Q_{D1}=(0.25+0.375)\times 2d^2=1.25d^2$$

$$Q_{D2}=(1-0.625)\times 2d^2=0.75d^2$$

$$C_1=5\times1.25d^2=6.25d^2$$

$$C_2=5\times0.75d^2=3.75d^2$$

$$\pi=TR-TC=(1.25d^2+0.75d^2)\times (p -5)$$

$$\pi=2d^2(p-5)$$