Question

Consider a simplified model of horizontal product differentiation. In class we noted that with quadratic “transportation costs” the demand curve facing an individual firm was decreasing in its own price and increasing in the price of its rival. Suppose an industry has two firms with constant and symmetric marginal costs c1 = c2 = 6. Suppose given levels of product differentiation and transportation costs, demand for firm 1 and firm 2 is:

q1 = 60 − 2p1 + p2

q2 = 60 − p2 + p1/2.

Suppose these firms compete by setting prices. Find each firm’s best response to the price of the other firm, and then find the Nash equilibrium to the game.1

Answer 1

q1 = 60 − 2p1 + p2

q2 = 60 − p2 + p1/2

marginal costs: c1 = c2 = 6

Profit(Firm 1)= (p1-MC)q1=(p1-6)( 60 − 2p1 + p2)= 60p1-2p12+p2p1-360+12p1-6p2

Differentiate with respect to p1:

dProfit/dp1= 60-4p1+p2+12=0

p1= 72+p2 / 4 Best response curve of firm 1

Profit (Firm 2)= (p2-MC)q2= (p2-6)(60 − p2 + p1/2)= 60p2-p22+p1p2/2-360+6p2-3p1

Differentiate with respect to p2:

dprofit/dp2= 60-2p2+p1/2+6=0

66+p1/2= 2p2

132+p1 / 4= p2    Best response curve of firm 2

Solve their BRS for their nash equilibrium:

p1= 14.8

p2= 36.7

q1= 60-2(14.8)+36.7= 67.1

q2= 60-36.7+14.8/2= 30.7