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Studying for a test? Prepare with these 2 lessons on Game theory and Nash equilibrium.
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Video transcript
What I want to do in this video is analyze why it makes sense for two companies that make up a duopoly to coordinate. To get into an agreement, which may or may not be legal-- probably would be illegal-- and restrict quantity. But also think about why there's a strong incentive for either or both of the parties to cheat their agreement and produce more quantity than they agreed to produce. So let's say that both of our players in our duopoly-- and this would actually apply to an oligopoly generally, but the analysis would be a little bit more difficult if we had more than two players-- but let's say each player has an identical-- they're identical companies. And they both have a marginal cost curve that looks something like that. So they both have an individual marginal cost curve that looks like that. And they both have an average total cost curve that looks something like this. And they are identical. So I'll just draw it once. This is the marginal cost and average total cost for both firms. Now let's think about what it would look like for the market. Well, one way to think about it-- pick an arbitrary marginal cost. So for one firm, what can they produce, or what quantity will they be at that marginal cost? Well, they'll be at this quantity for that marginal cost. But if you have two firms that are just like that, they could have twice as much quantity to be at that point in marginal cost. So two firms will be over there. And if you picked this marginal cost, one firm would produce that quantity to be right at that marginal cost, for that next incremental good. But two firms could produce two, especially if they have the exact same cost structure. So what you're going to have is you're essentially adding this curve to itself in the horizontal direction. So if you look at the marginal cost curve for both firms together, you're essentially going to get a curve that is twice is fat as the marginal cost curve for one firm. So it will look something like this. And I'll do it in yellow. So it will look something like that. So that is the marginal cost for the market, where the market in this example is both of these firms. And that will also be true for the average total cost. If at this price-- or actually, I should say, if the average total cost is up here for one firm-- that means that they are producing this quantity. But two firms together could produce twice the quantity of that average total cost. So two firms would produce twice. And so what you're going to have is an average total cost curve that is twice as fat as the average total cost curve for one firm, if you talk about the market. So the market's average total cost curve is going to look something like this. It's going to be twice as fat. It's the exact same logic. It's going to look something like that. So that is the average total cost curve for the market. So, so far, the convention that I've ended up using is orange for an individual firm, and then this dotted yellow line for the market as a whole. Now let's think about what a good equilibrium-- or what the right price should be if they were able to coordinate together. If they were to essentially combine their firms and almost behave like a monopoly. And to think about that, we're going to have to draw a demand curve. So let me draw the market demand curve. Let's say the market demand curve looks something like that. It's really big, so it's hard for me. And we'll assume that this is a line. So it's not-- well, that's pretty good. So this is the market demand curve. So if both of these firms operated together, if they-- I drew the market demand curve. I also want to draw the market marginal revenue curve. Now remember, we're going to assume that both of these firms are acting together. If they perfectly coordinate, they can join their capacities and act essentially like a monopoly. So if they did act like a monopoly, their marginal revenue curve would be twice the slope of this market demand curve. So it would hit the horizontal axis right over there. And so it would look something like this. So this right over here is the market marginal revenue curve. So if they were to behave like a monopoly, you could view this dotted line as their marginal cost curve. This would be their average total cost. And now this is their marginal revenue. If they were to behave as a monopoly, what would be the optimal quantity? Well, it would be right there, right where marginal revenue is equal to marginal cost. Before that, they would keep wanting to produce because marginal revenue is higher than marginal cost. And then after that, they don't want to produce, because marginal cost is higher than marginal revenue, and they're going to take economic losses on each of those incremental units. And so this is the quantity that they would produce. And the price they would get for that-- they just have to go to the market demand curve-- they would get this price right over here. Let's say they would get that price right over there. And the actual-- their average total cost per unit-- once again, we have to go to the market here. It's this dotted line right over here. That is their average total cost per unit. So their average economic profit per unit is going to be their revenue per unit, minus their average total cost per unit. So this height is their economic profit per unit. And if we multiply that times the total number of units, you would get their total economic profit if they coordinate perfectly, essentially behaving like a monopoly. And let's just say for argument that this height right over here-- let's say that that is 10. And let's say that this quantity that they would want to produce as a monopolist is 50. So what is the total economic profit here? Well, their total economic profit is 500. Total economic profit if they coordinate is 500. And so they see this, and they say, look, why don't we agree to each produce exactly half of this, and we would split the economic profit. And to see that, let's just say one firm says, OK. They both decide that they're going to produce 25. They're going to get this price for it up here, which was the market price. They're going to get that price for it, and their costs are right here. Now we're going on each individual firm. And that makes sense, because this cost is just twice as far away as this cost. And the dotted line yellow average total cost for the market is just a fatter version, twice as fat as the orange line. And so each firm will make this much economic profit per unit, times 25 units. And so each firm would make this orange area in terms of economic profit, or half of the entire 500, or 250 per firm. Now let's think about why there is an incentive for one or both of the firms to cheat. Let's say one firm in particular-- so the other firm holds at 25 units. But the other firm says, hey, I like this price. I'm already making economic profit. Let me produce 10 more units. So the other firm says, I'm not going to produce 25. I'm going to produce 35 units. And if that guy produces 35 units, and the other firm in the market-- the other duopolist, I guess we could say it-- continues to produce at 25, then the total market production is now going to be 60. Now what is the total economic profit? So we can go up the demand curve right over there. That's the new price. That right over there is the new price. The cost per unit is this right over here, and then the number of units that they're producing is 60. So the new economic profit is this area, in this bluish purplish color that I just drew. And even visually this is true-- looks like the demand curve and the average total cost curve have gotten closer together. So let's say that this height right over here is 8. And it's going to be $8 of economic profit per unit, times 60 units. So if they cheat-- let's talk about the cheating circumstance. This was coordinate, now let's think about if they cheat. Now we have 60 units for the whole market times $8 of economic profit per unit. You're going to have total economic profit of 480. Your total economic profit went down. And that makes sense, because now as a market, you're producing beyond the point where marginal revenue is equal to marginal cost. Now marginal cost as a market is higher than marginal revenue. And so all of this is essentially, you're creating economic loss because each of these incremental units as a market-- The cost is higher than the revenue, and you have an economic loss. And so that's why your total economic profit as a market went down from 500 to 480. But how much is this character going to be making? The one that decided to cheat? Well, he now has 35 units. He's producing 35 units, and he's getting an economic profit of $8 per unit. So he gets this entire area right over here. So let's multiply 35 times 8. I'll do it right over here. 35 times 8. 5 times 8 is 40, 3 times 8 is 24, plus 4 is 280. So now the cheating firm, Cheat, has $280 of economic profit in this period. And then the honest firm, or the fair firm-- what they're both doing might be illegal by even attempting to coordinate-- the non-cheater, I guess I could call them-- the non-cheater will have the rest. The non-cheater is going to have the balance of the economic profit. And the total economic profit was 480. The cheater's getting 280. The non-cheater is only going to get 200. So the cheater definitely benefited by increasing quantity past that optimal one. He went from 250 to 280. So it made sense for him. It reduced the total economic profit, and it really hurt the non-cheat right over there.