Stackelberg Duopoly

This game considers a market with two sellers A and B.

The linear price-demand function is:

p = 1000 – q_A – q_B

For simplicity, we assume zero costs.

Both sellers plan their quantities q_A and q_B sequentially. Seller A starts his consideration by assuming that B offers q_B = 0. Hence, A can act like a monopoly.

His profit G is

G_A = q_A * p

G_A = q_A * (1000 – q_A – 0)

Finding the maximum by the first derivative:

dG_A/d q_A = 1000-2*q_A

As a result, we get for A:

q_A^*= 500

G_A= 500*(1000-500-0) = 250,000

Now, seller B enters the market. He finds the following situation:

q_A= 500

With this information, he optimizes his quantity q_B.

G_B = q_B *p

G_B=q_B(1000-500-q_B) = 500q_B-q_B^2

First derivative to find maximum:

dG_B/q_B = 500-2q_B

As result we get for B:

q_B^*=250

G_B=65,500

But this changes the situation for A.

His profit drops to

G_A = (1000 – q_A – q_B)*q_A

G_A = (1000 – 500 – 250) * 250

G_A = 125,000

The total profit (125,000 + 65,500 = 190,000) is below the monopoly profit.

Further optimization

In this situation, it is beneficial for A to reconsider his quantity q_A by taking into account q_B = 250. Recalculating the optimal quantity for A yields q_A^new = 375. The profit is 140,625. This is higher than the profit with q_A = 500 (and q_B = 250).

Now, B can recalculate his optimal quantity and adapt it accordingly. Then A would recalculate his quantity again and so forth.

The following table shows the new optimal quantities in every round:

qA	500
qB	250
qA	375
qB	312.5
qA	343.75
qB	328.125
qA	335.9375
qB	332.03125
qA	333.984375
qB	333.007813
qA	333.496094
qB	333.251953
qA	333.374023
qB	333.312988
qA	333.343506
qB	333.328247

We can see that the quantities are approaching 1000/3 = 333.33 for each seller. In total this is 2/3*1000 which would be the result of the Cournot Duopoly.