Consider a quantity-setting oligopoly (Cournot model) where demand is linear and each firm's marginal cost is a constant C(i) for each firm i. The demand function is P = a − b(qi + q−i), where the subindex i denotes firm i's own production while subindex −i denotes the combined production of all other firms in the market. In the simple case of a duopoly, we have only q1 and q2.
1. Suppose we're in a duopoly market. On a graph with q1 on the x-axis and q2 on the y-axis, draw each firm's best-response function. Label the Nash Equilibrium of the game played by the firms.
Note: Remember that best response in a Cournot game specifies the production of the firm given the production of all other firms in the market - in a duopoly, this would be a function of firm 1's production given how much firm 2 produces: q1(q2). Explain your answer by arguing what happens with this function.2. Show how the equilibrium changes as Firm 2's marginal cost decreases.