Cournot Model

• In this article, we are going to look at cournot models, where firms compete in an oligopoly.

Modeling the situaiton

• Imagine that we have two firms producing homogeneous products.
• Each firm can choose it's quantity, $q_i \in [0, \infty), i = 1, 2$
• Each unit of production, the firm has to pay c, the production cost. Assume the production cost remains constant.
• The price is determined by the market, by the formula below $p(Q) = a - Q$
• Note that the price decreases as quantity increases. Q indicates the total amout of quantity.
• Each firm wants to maximize it's own profit.

Normal form of Cournot Model

• for those who do not know normal form, it's just the matematical descripion of player's options and utility function (the payoff).
• $n=2$
• $S_i = [0, \infty), i = 1, 2$
• $u_i(q_i, q_j) = q_i (a - q_i-q_j) - cq_i$

Finding the best response function

• Let us be in the side of firm 1. We want to maximize our profit, by altering our production.
• Note that our utility function, $u_1(q_1, q_2) = q_1(a - q_1 - q_2) - cq_1$
• is a parabola upside down, in respect to $q_1$
• Thus, we take the derivative of our utility function in respect to our quantity. $\frac{\partial u_1}{\partial q_1} = a - c - q_2 - 2q_1 = 0$
• Thus, our best response is, $q^*_1 = \frac{a - c- q_2}{2}$
• Note that the situation is symmetric, we also have our best response function for firm 2, $q^*_2 = \frac{a - c- q_1}{2}$

Finding Nash equilibrium

• Recall that in nash equilibrium, both firms would have to have no incentive to unilaterally devicate from current strategy profile.
• That means, both firms are best responding to each other.
• Thus, both equations must hold. $q^*_1 = \frac{a - c- q^*_2}{2}$ $q^*_2 = \frac{a - c- q^*_2}{2}$
• Solving this equation, we obtain the nash equilibrium, which is, $q^*_1 = q^*_2 = \frac{a - c}{3}$