Cournot Nash Equilibrium Between Two Firms Suppose we have two firms with specialized, but similar products. Suppose market demand for the two products is:
$$p_1(q_1,q_2)=a-bq_1-dq_2$$
$$p_2(q_1,q_2)=a-bq_2-dq_1$$
where $d \in (-b,b)$. Suppose that both firms have cost $c(q)=q$
What does $d$ mean intuitively?
Is the Cournot Nash Equilibrium for this
$$q_1=\frac{2ba - ad + dc'(q_2)-c'(q_1)2b}  {1 - d^2}$$
$$q_2=\frac{2ba - ad + dc'(q_1)-c'(q_2)2b}  {1 - d^2}$$
Thanks a ton.


What does d mean intuitively?

To answer this question, think about the "vanilla" Cournot competition case, where products $p_1$ and $p_2$ are identical; they're perfect substitutes.  In this case, increases in production from your competitor (i.e. $q_2$) displaces your own production, so $d = b$ and
$p_1(q_1,q_2) = a - b(q_1+q_2)$.
On the other hand, if an increase in production of $q_2$ increases demand for your own product $q_1$, then these products are compliments.  Be careful about stating they are perfect compliments, because without looking at consumer indifference curves, we can't determine this.
In this case, $d$ is negative, and is bounded by $-b$.
In short, $d$ is a measure of the degree to which these two goods are complements or substitutes.  Another approach would be to take the derivative of demand with respect to production of the other good, like this: 
$\frac{\partial p_1}{\partial q_2} = -d$.
If $d>0$, $\frac{\partial p_1}{\partial q_2} <0$ and $q_2$ is a complement to $q_1$.  Likewise, if $d<0$, $\frac{\partial p_1}{\partial q_2} >0$ and $q_2$ is a substitute for $q_1$.  Because of the symmetry of the problem, both will either be complements or substitutes. However, in the real world this is not always the case.

What is the Cournot-Nash equilibrium?

The Cournot-Nash equilibrium is the output {$q_1,q_2$} from which neither firm can profitably deviate.  To answer this, you need to find the best response function for each firm by solving for the optimal output, given the production of the other firm.  This is accomplished by equating Marginal Revenue = Marginal Cost. Note that the marginal cost of production is zero; i.e. $c'(q_1) = c'(q_2)=0$.
$BR_1(q_2) = \frac{a-dq_2}{2b}$ and $BR_2(q_1) = \frac{a-dq_1}{2b}$.
The Cournot-Nash equilibrium is located where these two Best Response functions intersect.  Solving the system of two equations and two unknowns, I get:
$q_1^* = q_2^* = \frac{a(\frac{1}{2}-\frac{d}{4b})}{b-\frac{d^2}{4b}}$.
