Cournot Duopoly

LEARNING GOALS:

VI.B.1: Solve Cournot and Bertrand models.
VI.B.3: Show that collusion by firms is not sustainable in a Cournot model.
VII.B.1: Define the best response function and the Nash equilibrium.

Consider a market consisting of two firms that each have the following demand and cost functions:

$Q=14-p$, $C(q)=2q$

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Suppose Firm 2 exits the market. Model this in the graph by moving this firm’s choice of quantity to 0. Use the graph to determine the profit-maximizing quantity that the Firm 1 (now a monopoly) produces and the profit they earn. That is, drag their choice of quantity until profit is as high as you can make it. Verify this result using algebra.
Now consider a market where the firms need to account not only for their own output, but also how much the other firm produces. If Firm 1 knows that Firm 2 will produce two units of output, how much should Firm 1 produce to maximize their profit? Use the graph to determine your answer, then verify using algebra.
Suppose that Firm 2 now reacts to Firm 1 setting their level of output to the above quantity. What is the optimal output for Firm 2 given that Firm 1 produces the amount in (b)? Use the graph to determine your answer, then verify using algebra.
Continue this back-and-forth process a few more times using the graph. What outcome does it converge to? That is, at what quantities are both firms optimally responding to what the other firm is doing? What is the price, and the total profit?
Starting with each firm’s profit maximization problem, algebraically demonstrate why the quantities you found in (d) are the optimal levels of output for each firm given that they know each other’s best response functions.
A Nash equilibrium is a situation where a unilateral deviation by a single agent is not beneficial for that agent. Briefly explain why this scenario is a Nash equilibrium.
Use the graph to determine whether there is a combined level of output where both firms make more profit than they do in the Cournot equilibrium. What quantity did you find? Now algebraically determine the optimal level of total output. How does this compare with the optimal level of output you calculated for Firm 1 in (a) when they did not have to compete with Firm 2?
Explain why the situation in (g) is NOT a Nash equilibrium.