Bertrand Duopoly with Imperfect Substitute Goods

Demand Function of Firm 1

q1=a-bp1+dp2

Profit Function of Firm 1

π1=p1(a-bp1+dp2)-c(a-bp1+dp2)

FOC ⇒ Reaction Function of Firm 1

∂π1/∂p1=0 ⇒ p1=r1(p2)=(a+cb)/2b+(d/2b)p2

Demand Function for Firm 2

q2=a-bp2+dp1

Profit Function of Firm2

π2=p2(a-bp2+dp1)-c(a-bp2+dp1)

FOC ⇒ Reaction Function of Firm 2

∂π2/∂p2=0 ⇒ p2=r2(p1)=(a+cb)/2b+(d/2b)p1


Nash Equilibrium

p1*=p2*=(a+cb)/(2b-d)





p1=p, R*=r1, p2=q, R=r2, x=q1, y=q2

If p1 and p2 increase along the reaction function of firm 1, then the production of firm 1 will increase. In addition, if p1 and p2 increase along the reaction function of firm 2, then the production of firm 2 will increase.

Proof
Reaction function of firm1 ⇒ dp1/dp2=d/2b
Demand function of firm1 ⇒ dq1/dp1=-b+d(dp2/dp1)
From these equations we have
dq1/dp1=b>0 (or dq1/dp2=d/2>0)

http://www.econ.kobe-u.ac.jp/~myojo/io/io11.pdf