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:
[latex display=”true”]P\quad =\quad A\quad -\quad BQ[/latex]Cost function:
[latex display=”true”]P\quad=\quad{Cq }_{ i }[/latex]We know in a Cournot model that at equilibrium, all firms will produce exactly the same amount [latex]{ q }_{ i }[/latex]. Therefore, the sum of all those individual yet [su_tooltip style=”yellow” position=”top” title=”definition” content=”That means all of outputs for each firm are exactly the same!”]homogeneous[/su_tooltip] outputs will equal [latex]{ Q }^{ * }[/latex]:
[latex display=”true”]{ 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})[/latex]Armed with this knowledge, let’s expand our inverse demand function by substituting [latex]Q[/latex] with [latex]({q}_{1}\quad+\quad{q}_{2}\quad+\quad{q}_{3}\quad+\quad\cdots\quad+\quad{q}_{N})[/latex]:
[latex display=”true”]P\quad =\quad A\quad -\quad B({q}_{1}\quad+\quad{q}_{2}\quad+\quad{q}_{3}\quad+\quad\cdots\quad+\quad{q}_{N})[/latex]Let’s factor out one [latex]{q}_{i}[/latex] since we’ll be solving for one.
Hmm, [latex]{q}_{1}[/latex] seems like a good choice.
[latex display=”true”]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 })[/latex]Now, to simplify a little, let us pause and consider [latex]({ q }_{ 2 }\quad +\quad { q }_{ 3 }\quad +\quad { q }_{ 4 }\quad +\quad \cdots \quad +\quad { q }_{ N })[/latex]. What exactly is this?
It’s actually [latex]{Q}_{N}[/latex] with one of the [su_tooltip position=”bottom” content=”i is 1 in our case”][latex]{q}_{i}[/latex][/su_tooltip] taken out. This means we can write that as:
[latex display=”true”]({ 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 }[/latex]This allows us to rewrite and greatly simplify our demand function:
[latex display=”true”]P\quad =\quad A\quad -\quad B{ q }_{ 1 }\quad -\quad B{ Q }_{ n-1 }[/latex]Marginal Revenue & Costs
To get [latex]{q}_{1}^{*}[/latex] we must equate marginal revenue to marginal cost ([latex]{MR}_{1} = {MC}_{1}[/latex]) and solve for [latex]{q}_{1}[/latex].
Let’s find our marginal revenues and costs: the derivatives of our demand and cost functions with respect to [latex]{q}_{1}[/latex].
We know that marginal cost is simply the derivative of the cost function we are given with respect to the [latex]{q}_{i}[/latex] we need.
[latex display=”true”]{MC}_{1}\quad =\quad \frac { \delta C{q}_{1} }{ \delta {q}_{1} }\quad=\quad C[/latex] [latex display=”true”]{ 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 } }[/latex]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 [latex]{q}_{i}[/latex] in question):
[latex display=”true”]{ MR }_{ 1 }\quad =\quad A\quad -\quad 2B{q}_{1}\quad -\quad B{ Q }_{ n-1 }[/latex]Profit Maximization
Now that we have both marginal cost and revenue, we can now profit maximize and find [latex]{q}^{*}_{1}[/latex].
[latex display=”true”]{ MR }_{ 1 }\quad =\quad {MC}_{1}[/latex] [latex display=”true”]A\quad -\quad 2B{q}_{1}\quad -\quad B{ Q }_{ n-1 }\quad =\quad C[/latex]We know [latex]{ Q }_{ n-1 } = Q – { q }_{ 1 }[/latex] so let’s substitute it in our equation above:
[latex display=”true”]A\quad -\quad 2B{q}_{1}\quad -\quad B(Q – { q }_{ 1 })\quad =\quad C[/latex] [latex display=”true”]A\quad -\quad 2B{ q }_{ 1 }\quad -\quad BQ\quad+\quad B{ q }_{ 1 }\quad =\quad C[/latex] [latex display=”true”]A\quad -\quad B{ q }_{ 1 }\quad -\quad BQ\quad =\quad C[/latex]Solving for [latex]{q}_{1}[/latex]:
[latex display=”true”]{ q }^{ * }_{ 1 }\quad =\frac { \quad A\quad -\quad B Q \quad -\quad C }{ B } [/latex]Now, we know that [latex]\sum _{ i=1 }^{ N }{ {q}_{i}} = n{q}_{i}[/latex]. Let’s substitute this for [latex]Q[/latex].
[latex display=”true”]{ q }^{ * }_{ 1 }\quad =\frac { \quad A\quad -\quad nB{q}_{i} \quad -\quad C }{ B }[/latex] [latex display=”true”]{ q }^{ * }_{ 1 }\quad =\quad \frac { A }{ B } \quad -\quad n{ q }_{ i }\quad -\quad \frac { C }{ B }[/latex] [latex display=”true”]{ q }^{ * }_{ 1 }\quad +\quad n{ q }_{ i }\quad =\quad \frac { A }{ B } \quad -\quad \frac { C }{ B }[/latex] [latex display=”true”](1\quad +\quad n){ q }^{ * }_{ 1 }\quad =\quad \frac { A }{ B } \quad -\quad \frac { C }{ B }[/latex] [latex display=”true”]{ q }^{ * }_{ 1 }\quad =\quad (\frac { A }{ B } \quad -\quad \frac { C }{ B } )(\frac { 1 }{ (1\quad +\quad n) })[/latex] [latex display=”true”]{ q }^{ * }_{ 1 }\quad =\quad (\frac { A\quad-\quad C}{B} )(\frac { 1 }{ (1\quad +\quad n) } )[/latex]And again, since we know [latex]n{ q }^{ * }_{ 1 }=Q[/latex]:
[latex display=”true”]{Q}^{*}\quad =\quad (n)(\frac { A\quad -\quad C }{ B } )(\frac { 1 }{ (1\quad +\quad n) } )[/latex]Now that we have total market output at equilibrium, we can deduce what equilibrium price will be in the market.
[latex display=”true”]{P}^{*}\quad =\quad A\quad -\quad B((n)(\frac { A\quad -\quad C }{ B } )(\frac { 1 }{ (1\quad +\quad n) } ))[/latex]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.
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