# 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.

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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:

$q_1^* = q_2^* = \frac{a(\frac{1}{2}-\frac{d}{4b})}{b-\frac{d^2}{4b}}$.

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Thank you very much! This helps a ton, and I realize why I was wrong when I calculated the Cournot-Nash Equilbrium. – Kitsune Cavalry Feb 24 '12 at 3:32
Great, glad to help. Please approve my answer by clicking the check mark. – baha-kev Feb 24 '12 at 16:28

In response to your first question, the function $p_1(q_1,q_2)=a-bq_1-dq_2$ can be thought of as both

• the highest price firm 1 can charge and
• the willingness-to-pay (WTP) of the marginal buyer (the person who bought $q_1^{st}$ unit) of good 1 when each firm $i=1,2$ is choosing to sell $q_i$ units.

This is because...if firm 1 wanted a higher price than $p_1(q_1,q_2)$, the marginal consumer would refuse (because $p_1(q_1,q_2)$ is that buyer's WTP), and so selling $q_1$ units would no longer be feasible.

Okay, now we know that it is a formula for the marginal WTP for good 1 conditional on a pair of choices $(q_1,q_2)$. Now think about what it looks like when $q_2$ is fixed, at say $q_2=5$. Then we have a function for the price of good 1 in terms of its quantity, $$p_1=c-bq_1,$$ where $c=a-5d$ is a constant from firm 1's perspective. This expression is called firm 1's residual demand curve. Given a value for $q_2$, this is a curve in the $(p_1,q_1)$ plane just like the demand curve a monopolist faces. This point of view is useful (to me) when solving for equilibrium in a Cournot market.

Anyhow, this is a long way of saying that $d$ shifts this demand curve left and right. As baha said, for $d>0$ as $q_2$ increases, demand for $q_1$ falls, so the goods are substitutes; and for $d<0$, they are complements.

Footnote: I think $d$ is bounded in absolute value by $b$ because, intuitively, we don't believe that the "own-price" effect can be larger than the cross-price effect. Also, it may be that $|d|\geq b$ would prevent the market from having an equilibrium (?)...consider it an exercise.

@baha: c(q) = q, so the marginal cost is 1, not zero. I don't have the rep to comment on your post directly.

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