Question

2. (15 pts) Consider a Stackelberg duopoly game of quantity competition in U.S. cigarettes between Philip Morris (PM, biggest brand is Marlboro) and RJ Reynolds (RJR, biggest brand is Camel). Philip Morris is the leader, and Reynolds is the follower. (We will obviously ignore state-minimum pricing and taxes for this question.) Market demand is described by the inverse demand function P 12 0.005Q. Each firm has a constant unit cost of production equal to 2. Prices are in S/pack, and quantities are in 10M packs/year. a) Solve for the Nash equilibrium outcomes of quantity, price, and firm variable profits b) Suppose Reynold's unit cost of production is cRjR < 2. What value would cRIR have so that in the Nash equilibrium the two firms, leader and follower, had the same market share (i.e., quantity sold)?

Answer 1

Solution:

Inverse market demand curve is: P = 12 - 0.005*Q where Q = q1 + q2, q1 is quantity produced and sold by Philip Morris and q2 is quantity produced and sold by RJ Reynolds

With constant unit cost of production = 2, the total cost for each firm = 2*qi; i= {1, 2}

a) Solving the Stackelberg duopoly game with PM as leader and RJ as the follower:

First, finding the best response curves:

Profit for PM, W1 = P*q1 - TC(q1)

W1 = (12 - 0.005*(q1 +q2))q1 - 2*q1

First finding the best response function of RJ firm:

For firm 2, profits, W2 = P*q2 - TC(q2)

W2 = (12 - 0.005*(q1 +q2))q2 - 2*q2

W2 = (10 - 0.005*q1)q2 - 0.005*q2²

Using first order condition, ,

0W2/012 = 0

= 10 - 0.005*q1 - 0.01*q2 = 0

So, best response function of firm 2, BR2: q2(q1*) = 1000 - 0.5q1* ... (*)

Now with PM (or firm 1) as the leader and RJ (or firm 2) as the follower, we have

Using BR2 as obtained above, BR2: q2(q1) = 1000 - 0.5q1

Substituting this value of quantity of follower, q2, in profit function of the leader firm, W1, we get

W1 = (12 - 0.005*(q1 + 1000 - 0.5q1))q1 - 2*q1

W1 = 5q1 - 0.0025*q1²

Now, optimizing using FOC for firm 1,

= 5 - 0.005*q1* = 0

or q1* = 5/0.005  = 1,000

Substituting this in (*), q2 = 1000 - 0.5*1000  = 5,00

So, Nash equilibrium quantity produced by PM is 10,000M packs/year and by RJ is 5,000M packs/year

Now the equilibrium price, P* = 12 - 0.005(1000 + 500) = $4.5 per pack

Firms' variable profits:

For PM, W1 = 4.5*1000 - 2*1000 = $2,500*10M per year

For RJ, W2 = 4.5*500 - 2*500 = $1,250*10M per year

b) For PM and RJ to have equal quantities in Nash equilibrium, that is q1* = q2*, under Stackelberg, we have to find the per unit cost of production for firm 2. Let's call it c, so now

Profit for firm 2 becomes: W2 = (12 - 0.005*(q1 +q2))q2 - c*q2

W2 = (12 - c - 0.005*q1)q2 - 0.005*q2²

Then, solving required FOC, best response function of firm 2 becomes:

12 - c - 0.005*q1 - 0.01*q2 = 0

q2(q1*) = (12 - c)*100 - 0.5q1* ... (**)

Note that profit function for firm 1 is still unchanged: W1 = (12 - 0.005*(q1 +q2))q1 - 2*q1

Again, profit for firm 1 becomes (on substitution):

W1 = (10 - 0.005*(q1 + (12 - c)*100 - 0.5q1))*q1

W1 = (4 - 0.5*c)*q1 - 0.0025*q1²

So, gives us 4 - 0.5*c - 0.005*q1 = 0

q1 = 800 - 100*c

Then, using (**), q2 = (12 - c)*100 - 0.5*(800 - 100*c)

q2 = 800 - 50*c

Since, q1 = q2

We have 800 - 100*c = 800 - 50*c

So, c = 0

Thus, with per unit cost of production equal to 0 for firm 2, can only two firms have same market share, that is produce equal quantity.