How to solve Cournot best response functions

The following question pertains to the Cournot model of oligopoly. This post shows a trick for solving these best response functions without using calculus, although a calculus based method is shown at the end of the post.

Suppose there are two firms in an industry and the inverse demand function for the industry is:
P = 45 – 2Q
Assume that the MC functions for the two firms are:
MC1 = 15
MC2 = 12
1. Solve for the equilibrium P, Q, q1, and q2 values, assuming there is no collusion between the two firms.

2. Solve for the equilibrium P, Q, q1, and q2 values, assuming there is collusion
between the two firms. Assume MC = 13.

To solve this problem we need to know two critical equations. 

The first is what the best response functions for Cournot model’s look like, and the other is what the collusion function looks like.  I was unable to find a textbook that specifically detailed the Cournot Model’s best response function while including a marginal cost, without going into a calculus derivation so I made my own, the demonstration is shown at the bottom of this post.

The best response function for each firm will be equal to:

Q1= (a-c-bQ2)/2b

Where Q1 and Q2 designate the quantities of output chosen by each firm, a and b are the intercept and slope from the demand function, (ie. P=a-bQ), and c represents marginal cost.  What we need to do now is plug in the values given in the question.  We do this for both Q1 and Q2.  This gives us the best response functions for each of the Cournot firms.  Then we must plug Q2 into Q1, and solve Q1 for a specific number (rather than variables).  Then we must plug this number into Q2's best response function to get a number for Q2.  These steps are shown below:

So now we know that according to the best response functions, firm one should produce 4.5, and firm two should produce 6.  This results in an equilibrium quantity of 10.5, and an equilibrium price of 24.  Also note that firm two has the higher quantity and the lower marginal cost, which makes intuitive sense.

Next we need to solve for equilibrium values when they collude.  In order to do this, we assume that the two companies behave as a monopoly.  In general, we would just close the higher marginal cost firm down (since it has a constant marginal cost), but since this problem tells us to assume a MC of 13, we will just do that.  Note that the following set-up is a monopoly problem, because 2 firms are behaving as one.

Remember that profit maximization for a monopoly occurs where marginal revenue is equal to marginal cost (a solution to this problem using calculus is shown at the end of the post).  It is also necessary to remember that marginal revenue has twice the slope of the demand function.  Since the slope of the demand curve is 2 (from P=45-2Q), we know that the marginal revenue curve is twice this, or 4.  So we can write the marginal revenue function as MR=45-4Q.  We then set MR=MC.

45-4Q=13

32=4Q

Q=8

So equilibrium quantity is 8, and equilibrium price is 29 (45-2*8).  How the two firms decide to divide up production is up to you, but to end up with an average marginal cost of 13, firm 1 would have to produce roughly 1/3 of the quantity (about 2.67).

These answers make sense, because the equilibrium quantity for the Cournot outcome is higher than the Monopoly outcome.  The price of the Cournot outcome is also lower than the Monopoly outcome, which fits theory, meaning that increased competition decreases prices (monopoly is the absence of competition).

Demonstration for why the above equation works for linear demand curves and constant marginal costs:

So now we have solved for the best response function of firm one (how much firm one should produce given firm 2’s output decision).

Calculus derivation for monopoly: