Cournot Duopoly with Tariff

Inverse Demand Function

p(z)=a-bz=a-b(x+y)

Foreign firms

π*=x(p(z)-t)-C*(x)=x(a-b(x+y)-t)-cx

FOC ⇒ Reaction Function of Foreign Firms

x=r*(y,t)=(a-c-t)/(2b)-(1/2)y

Home firms

π=y(p(z)-t)-C(y)=y(a-b(x+y)-t)-cx

FOC ⇒ Reaction Function of Home Firms

y=r(x)=(a-c)/(2b)-(1/2)y

Nash Equilibrium

x=(a-c-2t)/(3b)

y=(a-c+t)/(3b)

t↑⇒ x↓, y↑ (C→D)

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

Bertrand Model

Model
Two firms with the same marginal cost (MC=c), producing a homogeneous product and competing in prices.

homogeneous product → consumers purchase from cheapest firms

Bertrand's theorem
Let (p1*,p2*) be a Nash Equilibrium. Then, p1*=p2*=c.

If each firm has a different marginal cost c1, c2 (c1<c2), the Nash equilibria are (p1, p2) such that c1≤p1=p2≤c2.

http://en.wikipedia.org/wiki/Bertrand_competition

Dominance and Nash Equilibria

Definition

Strategy s is strictly dominant if strategy s strictly dominates every other possible strategy.

Strategy s is weakly dominant if strategy s dominates all other strategies, but some are only weakly dominated.

Strategy s is strictly dominated if some other strategy exists that strictly dominates s.

Strategy s is weakly dominated if some other strategy exists that weakly dominates s.


Important!

1. A strictly dominant strategy must be played in Nash Equilibria.

2. Strictly dominated strategies cannot be played in Nash Equilibria.

3. Weakly dominated strategies may be played in Nash Equilibria.

4. If, after completing iterated elimination of strongly dominated strategies, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.

5. If, after completing iterated elimination of weakly dominated strategies, there is only one strategy for each player remaining, that strategy set is also a Nash equilibrium. (The Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium.)

 http://en.wikipedia.org/wiki/Strategic_dominance

Cartesian Product of Convex/ Compact

1. Any Cartesian product of convex sets is convex.

meboo.convexoptimization.com/BOOK/convexgeometry.pdf

2. Any Cartesian product of compact sets is compact.

www.ms.unimelb.edu.au/~rubin/math127/compact.pdf