Cournot Duopoly

The Cournot Duopoly is another example of how selfish (uncoordinated) behavior can lead to a worse outcome than unselfish behavior.

Suppose we have to sellers A and B that face a price-demand-function:

p: price

p = 1000 – (q_A + q_B)

Whereas q_A is the quantity offered by A and q_B is the quantity offered by B.

For simplicity, we neglect cost. Each seller wants to maximize his profit (=turnover).

The profit is calculated as:

G_A: profit for A

G_A = p * q_A= [1000 – (q_A + q_B)] * q_A

To find the maximum, we calculate the first derivation and set it zero:

dG_A/dq_A = 1000 – 2q_A – q_B = 0

This yields:

q_A = (1000 – q_B)/2

For B, we do the same and get:

q_B = (1000 – q_A)/2

In the end, we get the quantities:

q_A = 333.33

and

q_B = 333.33

The profit for each seller is:

G_A = [1000 – (333.33 + 333.33)] * 333,33 = 111,111.1112

G_B = [1000 – (333.33 + 333.33)] * 333,33 = 111,111.1112

The total profit is 222,222.222.

If we replace the 1000 in our price demand function by a parameter a, we get as quantities

q_A = 1/3*a

q_B = 1/3*a

Is this the best result we can expect? If we assume that both sellers coordinate their quantities (like acting as one company), we get the following situation:

G_C = p * Q

G_C: profit in the coordination case

Q: total quantity

G_C = (1000 – Q)*Q

dG_C/dQ = 1000 – 2Q = 0

Q = 500

We can see that the optimal quantity here is 500 (compared to 333.33 + 333.33 in the selfish case).

The total profit is 250,000.

This is more than in the selfish case.