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PROFESSOR: OK, so what we're going to do today is continue our discussion of oligopoly which we started last time.

I want to return to the example we were using last time to discuss the example of Cournot competition. We started last time by talking about general game theory and the prisoner's dilemma and the concept of Nash equilibrium.

Then we turned to a specific example of a Cournot competition, the way we can specifically model two firms competing in a market.

And so if you recall the example from last time, we had demand curve of the form p equals 339 minus q where this is thousands of passengers per month. And we had a marginal cost of $147. And we have flat marginal cost, so average cost equals marginal cost equals 147, equals average cost. So flat marginal cost.

And what we said, so therefore if the firm was a monopolist, if American Airlines say was a monopolist, they would set marginal revenue equal to marginal cost which would be 339 minus 2q equals 147. And so the monopolist was going to set a quantity of 96. That was the monopolist quantity with the price. So the quantity of monopolist is 96 and the price of monopolist is $243. So that was the monopoly case.

Something you should be very facile with for tomorrow night is solving those kinds of monopoly problems. And then we said, OK, but what if, in fact, it recognizes that it's not a monopoly. There's another firm in the market. United is in the market as well.

Well in that case it has to consider its residual demand.

So in that case we said the qa, its residual demand, for American was equal to the total market q minus the quantity absorbed by United, qu. And we showed last time what this leads to graphically.

And so we re-handed out figure 16-3 just to review.

The notion was that each firm develops a best response curve.

Each firm says, based on what the other firm's going to do, here is my best level of production. Here's my profit maximizing level of production.

And each firm having developed a curve, there's an equilibrium where those curves intersect because at that point both firms are happy.

You've achieved your Nash equilibrium because at that point both firms are satisfied with the strategy they're playing given what the other firm's playing. So that point of intersection, both firms' best response are consistent with each other. And that's we developed last time graphically.

Intuitively, I think it follows from the same logic, the idea is we're playing this game and the game will only have a stable outcome if we're both satisfied with the outcome. If we're not both satisfied we'll continue to change our behavior and it won't be stable.

And so what I want to do now is talk about how we solve for this mathematically. So let's just think about the mathematics of solving for Cournot equilibrium. So what's the mathematics now? What's American Airlines's residual demand function?

Well their price, p sub a, is equal to the total demand, 339, minus what they supply minus what United supplies. So the price in the market, they don't have separate prices, the price in the market is going to be 339 minus what each of them supplies.

So now American can't control q sub u, American has to take that as an outside given factor. That's what they're responding to. So American says, I need to optimize this with respect to what I can control which is q sub a.

So how does it do it? Well, it computes marginal revenue. Well marginal revenue is p times q sub a.

I want to return to the example we were using last time to discuss the example of Cournot competition. We started last time by talking about general game theory and the prisoner's dilemma and the concept of Nash equilibrium.

Then we turned to a specific example of a Cournot competition, the way we can specifically model two firms competing in a market.

And so if you recall the example from last time, we had demand curve of the form p equals 339 minus q where this is thousands of passengers per month. And we had a marginal cost of $147. And we have flat marginal cost, so average cost equals marginal cost equals 147, equals average cost. So flat marginal cost.

And what we said, so therefore if the firm was a monopolist, if American Airlines say was a monopolist, they would set marginal revenue equal to marginal cost which would be 339 minus 2q equals 147. And so the monopolist was going to set a quantity of 96. That was the monopolist quantity with the price. So the quantity of monopolist is 96 and the price of monopolist is $243. So that was the monopoly case.

Something you should be very facile with for tomorrow night is solving those kinds of monopoly problems. And then we said, OK, but what if, in fact, it recognizes that it's not a monopoly. There's another firm in the market. United is in the market as well.

Well in that case it has to consider its residual demand.

So in that case we said the qa, its residual demand, for American was equal to the total market q minus the quantity absorbed by United, qu. And we showed last time what this leads to graphically.

And so we re-handed out figure 16-3 just to review.

The notion was that each firm develops a best response curve.

Each firm says, based on what the other firm's going to do, here is my best level of production. Here's my profit maximizing level of production.

And each firm having developed a curve, there's an equilibrium where those curves intersect because at that point both firms are happy.

You've achieved your Nash equilibrium because at that point both firms are satisfied with the strategy they're playing given what the other firm's playing. So that point of intersection, both firms' best response are consistent with each other. And that's we developed last time graphically.

Intuitively, I think it follows from the same logic, the idea is we're playing this game and the game will only have a stable outcome if we're both satisfied with the outcome. If we're not both satisfied we'll continue to change our behavior and it won't be stable.

And so what I want to do now is talk about how we solve for this mathematically. So let's just think about the mathematics of solving for Cournot equilibrium. So what's the mathematics now? What's American Airlines's residual demand function?

Well their price, p sub a, is equal to the total demand, 339, minus what they supply minus what United supplies. So the price in the market, they don't have separate prices, the price in the market is going to be 339 minus what each of them supplies.

So now American can't control q sub u, American has to take that as an outside given factor. That's what they're responding to. So American says, I need to optimize this with respect to what I can control which is q sub a.

So how does it do it? Well, it computes marginal revenue. Well marginal revenue is p times q sub a.

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