Q.If duopoly behaviour is one that is described by Cournot, the market demand curve is given by the equation q = 200 – 4p, and both the firms have zero costs, find the quantity supplied by each firm in equilibrium and the equilibrium market price.

In a Cournot duopoly with zero costs, total output settles at two-thirds of the competitive output, shared equally. …

The market demand is $q = 200 - 4p$, which inverted is

and both firms have zero cost ($MC = 0$). Let the two firms supply $q_1$ and $q_2$, so total output is $q = q_1 + q_2$. In the Cournot model each firm chooses its own output taking the other's output as given, to maximise its profit

Setting the marginal profit to zero, $\dfrac{\partial \pi_1}{\partial q_1} = 50 - \dfrac{2q_1 + q_2}{4} = 0$, gives the reaction condition $2q_1 + q_2 = 200$. By symmetry the second firm gives $2q_2 + q_1 = 200$. Solving (with $q_1 = q_2 = q^*$):

$3q^* = 200 \;\Rightarrow\; q^* = \frac{200}{3} \approx 66.7 \text{ units per firm}.$ …