Cournot Oligopoly with N Firms

I recently took ECON 325 with Basyl Golovedstkyy (SFU) for my summer semester. It’s a fairly comprehensive course focusing on Industrial Organization modelling how firms can behave when they have market power but aren’t completely in a monopolistic setting.

One of the problems I had an issue on personally on the final exam was this model of a Cournot oligopoly with a variable amount (n) of firms. Being so frustrated with myself, I decided to go over it again slowly now that my semester is over because I know that it certainly will pop up in the future.

If you’re looking for a clear explanation of the model, look no further! Now, let’s begin.

A Cournot-Style Oligopoly

Model Setup

Let’s keep things fairly general.

Market inverse demand curve:

P\quad =\quad A\quad -\quad BQ

Cost function: 

P\quad=\quad{Cq }_{ i }

We know in a Cournot model that at equilibrium, all firms will produce exactly the same amount { q }_{ i }. Therefore, the sum of all those individual yet homogeneous outputs will equal { Q }^{ * }:

{ Q }^{ * }\quad=\quad\sum _{i = 1 }^{ N }{ { q }_{ i } }\quad=\quad({q}_{1}\quad+\quad{q}_{2}\quad+\quad{q}_{3}\quad+\quad\cdots\quad+\quad{q}_{N})

Armed with this knowledge, let’s expand our inverse demand function by substituting Q with ({q}_{1}\quad+\quad{q}_{2}\quad+\quad{q}_{3}\quad+\quad\cdots\quad+\quad{q}_{N}):

P\quad =\quad A\quad -\quad B({q}_{1}\quad+\quad{q}_{2}\quad+\quad{q}_{3}\quad+\quad\cdots\quad+\quad{q}_{N})

Let’s factor out one {q}_{i} since we’ll be solving for one.

Hmm, {q}_{1} seems like a good choice.

P\quad =\quad A\quad -\quad B({ q }_{ 1 })\quad-\quad B({ q }_{ 2 }\quad +\quad { q }_{ 3 }\quad +\quad { q }_{ 4 }\quad +\quad \cdots \quad +\quad { q }_{ N })

Now, to simplify a little, let us pause and consider ({ q }_{ 2 }\quad +\quad { q }_{ 3 }\quad +\quad { q }_{ 4 }\quad +\quad \cdots \quad +\quad { q }_{ N }). What exactly is this?

It’s actually {Q}_{N} with one of the {q}_{i} taken out. This means we can write that as:

({ q }_{ 2 }\quad +\quad { q }_{ 3 }\quad +\quad { q }_{ 4 }\quad +\quad \cdots \quad +\quad { q }_{ N })\quad =\quad \sum _{i = 1 }^{ N }{ { q }_{ i } }\quad -\quad { q }_{ 1 }\quad =\quad Q\quad -\quad { q }_{ 1 }\quad=\quad { Q }_{ n-1 }

This allows us to rewrite and greatly simplify our demand function:

P\quad =\quad A\quad -\quad B{ q }_{ 1 }\quad -\quad B{ Q }_{ n-1 }

Marginal Revenue & Costs

To get {q}_{1}^{*} we must equate marginal revenue to marginal cost ({MR}_{1} = {MC}_{1}) and solve for {q}_{1}.

Let’s find our marginal revenues and costs: the derivatives of our demand and cost functions with respect to {q}_{1}.

We know that marginal cost is simply the derivative of the cost function we are given with respect to the {q}_{i} we need.

{MC}_{1}\quad =\quad \frac { \delta C{q}_{1} }{ \delta {q}_{1} }\quad=\quad C { MR }_{ 1 }\quad =\quad \frac { \delta PQ }{ \delta { q }_{ 1 } } \quad =\quad \frac { \delta (A\quad -\quad B{ q }_{ 1 }\quad -\quad B{ Q }_{ n-1 }){ q }_{ 1 } }{ \delta { q }_{ 1 } } \quad =\quad\frac { \delta (A{ q }_{ 1 }\quad -\quad B{ q }^{ 2 }_{ 1 }\quad -\quad B{ q }_{ 1 }{ Q }_{ n-1 }) }{ \delta { q }_{ 1 } }

Taking the derivative and simplifying further (we can also use our knowledge that marginal revenue is simply the inverse demand with double the slope for the {q}_{i} in question):

{ MR }_{ 1 }\quad =\quad A\quad -\quad 2B{q}_{1}\quad -\quad B{ Q }_{ n-1 }

Profit Maximization

Now that we have both marginal cost and revenue, we can now profit maximize and find {q}^{*}_{1}.

{ MR }_{ 1 }\quad =\quad {MC}_{1} A\quad -\quad 2B{q}_{1}\quad -\quad B{ Q }_{ n-1 }\quad =\quad C

We know { Q }_{ n-1 } = Q - { q }_{ 1 } so let’s substitute it in our equation above:

A\quad -\quad 2B{q}_{1}\quad -\quad B(Q - { q }_{ 1 })\quad =\quad C A\quad -\quad 2B{ q }_{ 1 }\quad -\quad BQ\quad+\quad B{ q }_{ 1 }\quad =\quad C A\quad -\quad B{ q }_{ 1 }\quad -\quad BQ\quad =\quad C

Solving for {q}_{1}:

{ q }^{ * }_{ 1 }\quad =\frac { \quad A\quad -\quad B Q \quad -\quad C }{ B }

Now, we know that \sum _{ i=1 }^{ N }{ {q}_{i}} = n{q}_{i}. Let’s substitute this for Q.

{ q }^{ * }_{ 1 }\quad =\frac { \quad A\quad -\quad nB{q}_{i} \quad -\quad C }{ B } { q }^{ * }_{ 1 }\quad =\quad \frac { A }{ B } \quad -\quad n{ q }_{ i }\quad -\quad \frac { C }{ B } { q }^{ * }_{ 1 }\quad +\quad n{ q }_{ i }\quad =\quad \frac { A }{ B } \quad -\quad \frac { C }{ B } (1\quad +\quad n){ q }^{ * }_{ 1 }\quad =\quad \frac { A }{ B } \quad -\quad \frac { C }{ B } { q }^{ * }_{ 1 }\quad =\quad (\frac { A }{ B } \quad -\quad \frac { C }{ B } )(\frac { 1 }{ (1\quad +\quad n) }) { q }^{ * }_{ 1 }\quad =\quad (\frac { A\quad-\quad C}{B} )(\frac { 1 }{ (1\quad +\quad n) } )

And again, since we know n{ q }^{ * }_{ 1 }=Q:

{Q}^{*}\quad =\quad (n)(\frac { A\quad -\quad C }{ B } )(\frac { 1 }{ (1\quad +\quad n) } )

Now that we have total market output at equilibrium, we can deduce what equilibrium price will be in the market.

{P}^{*}\quad =\quad A\quad -\quad B((n)(\frac { A\quad -\quad C }{ B } )(\frac { 1 }{ (1\quad +\quad n) } ))

From hereon in, you can go ahead and find the revenues or loses, equilibrium profits for each firm or the total profit the industry makes.

If this helped, please leave a comment below.

If you really feel like I’ve saved you with this, you can send me some of your hard earned money with PayPal or Kofi. That would super be cool!

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