# 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.

### Cournot Duopoly Calculator

With this tool, you can calculate prices, qantities and profits for a simple Cournot Duopoly price demand function:

- p: price
- q: quantity
- a: reservation quantity