Why parties to cartels cheat | Game theory and Nash equilibrium | Microeconomics | Khan Academy

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.

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  1. It's more likely that when one firm cheats, the other will as well; so each firm will face a highly inelastic demand curve below the original cooperative price and quantity of $10 and 25 units each. If the competitor matches a price decrease, neither firm will gain much market share and total revenues and profits will decrease for both firms. The incentive, therefore, is to cooperate and maintain the original price. This is what the kinked demand curve model illustrates.

  2. I believe there are some errors.
    In the case of non-equal share of the market (one party is cheating), average total cost is not just 2 times fatter (for example one produces 0 as an extreme case).

    Also for cheating scenario, average total profit of 8$/unit is for 60 units total, and company can get 8$/unit if it produce (lets say) 30 units. The cheating party is producing 35 units, ([email protected] < [email protected]), thus getting more than 8$/unit, while non-cheating is getting less then 8$/unit.

  3. However, when a firm changes their quantity to 35, does the price change instantaneously? Because if not, one could argue that for a short period of time they'd still be selling it for 10$ and so one firm would still be doing 250$ and the other would be doing 300$. Also, do the marginal cost and the average total cost curves change over time? If you can predict those changes, won't it affect your decision?

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