Male Voice: In the last video we saw how
there could be an industry that has two firms,
a duopoly, and if those two firms coordinate
they could behave as a monopolist and they could
optimize their collective economic profit.
In the last video we saw that would happen
when they produced 50 units per period,
and they could split it, assuming these were
two identical firms, by each producing half of it.
In the case of the last video, it was 250 units
per firm. Then we saw that there was an incentive
to cheat; that by producing extra units,
from a market's point of view, the marginal
economic, or the economic profit on those
incremental units would be negative, so the whole
economic profit would shrink a little bit
as you produced units beyond that,
but the cheater would get a bigger chunk
of those units, or the bigger chunk of that
economic profit. The cheater could actually gain,
go from $250 per time period to $280,
and it would be all at the expense of the non-cheater,
and then some, who would lose even more
than what the cheater gained.
Obviously who was initially the non-cheater
has an incentive now to cheat, and they'll
both keep increasing, they'll both keep increasing
production so that if they wanted to keep
doing this one-upmanship. They both have
the incentive to keep going assuming that they
don't hold to their cartel agreement
until you get to a quantity where there's no
economic profit left. Right over here, the way
I've drawn it, the demand curve intersects
the average total cost curve right over here,
and there's no economic profit left.
We're producing a good quantity.
It looks like it's about 75 units combined;
75 units for the whole market.
But at this point, the market price
is equal to the average total cost,
and so there's no economic profit per unit
on average. What I want to do is think about this
in kind of a Game theoretic way.
Let's look at a bunch of states.
This is the optimal state that we are starting off in.
You can actually call it the Pareto optimal state,
named after Vilfredo Pareto.
All it means is that's the state where there's
no other state where you can make someone
better off without making the other person worse off.
Any of the states here, there are states,
for example, where blue is better off.
For example, in this state right over here
blue is better off, but green is worse off.
So that's why it's called Pareto optimality.
Now, what I want to think about is how these characters
will change their state due to their incentives.
Then we'll talk a little bit about Nash Equilibrium as well.
On this axis, up here, lets' say this is one
of the competitors. This is where they produce 25,
and let's say on the ultimate cheating
quantity of 75, and this is somewhat close to
the market, or that is the equilibrium quantity
if this was perfect competition,
they produce half of that, so this is them
producing 37.5 units. As we go from 25 to 37.5 units,
they are cheating more. This is more cheating
and over here, this was no cheating.
We can do the same thing for the blue player.
I'll write them as B. This is them producing 25.
This is them producing 37.5. As we go up and up
and up, they are cheating more.
This is a lot of cheating, or more cheating.
To think of it in a Game theoretical way,
this is the Pareto optimal state right over here.
It's optimal in many ways.
This is they've maximized the total economic
profit here. There's no other state that one
person would benefit without making the other worse.
Now, let's think about whether this is
a Nash equilibrium. Let's remind oursleves
what Nash equilibrium was.
This was a state where holding all the other
players constant, so in this case there's only one
other player, a player can't gain by changing strategy.
In this case, changing strategy is changing
your output. Let's see if that is true of this state
right over here. Well, let's hold A constant.
If A is constant, we're in this column right
over here. Is there something B can do,
is there change or strategy B can do,
that would allow B to gain?
Sure. B can increase production.
That's what we saw in the last video.
We would go from this bottom right state
to one right above it. Now B's economic profit
is 280, A's is 200. The pie has shrunk,
but B has got a larger chunk of it.
That was not a Nash equilibrium.
There is, holding all others constant,
there is a player that can gain by changing their strategy.
The Nash equilibrium definition, just to make sure,
they say it's a state where holding others
constant no player can gain by changing strategy.
We just showed that at least one player can gain
by changing strategy holding others constant.
The same would be true if we went the other way around.
If we held B constant at 25, A could gain
by changing his strategy, could go right over there.
This is not a Nash equilibrium.
Then regardless of what state we go to,
if we go to this state, it's still not
a Nash Equilibrium. If we hold A constant,
B could improve by increasing his production;
or if we hold B constant, then A can still
improve by cheating even more.
None of these are Nash equilibriums.
From any one of these states, if you hold A constant,
B could produce more; or if you hold B constant,
A could produce more and get some gain.
Over here, A's going from 130 to 160 and getting
some gain. You can imagine this keeps happening
incrementally. They keep producing more and more
and more. We kind of go there, then we go there,
then maybe we go there, then we go there.
Then maybe A cheats some more, then B
cheats some more, then A cheats a little bit more,
B cheats a little bit more, maybe a little bit more
past that, then A cheats a little bit more.
The whole time the whole economic profit pie,
which is the sum of A and B, is getting smaller
and smaller until finally A finally cheats
and they're at zero economic profit.
Now let's think about whether this is
a Nash equilibrium. Clearly, they won't want
to move backwards. If you hold A constant,
B would not want to move down. Then he would
lose economic profit. That doesn't work.
He doesn't gain by doing that. If you hold B
constant, A wouldn't want to move to the right.
A would also lose economic profit.
Now you might say what if they produced beyond 37.5?
Why can't they keep producing and go beyond there?
Holding A constant, if B were to produce more than 37.5
from this state right over here, then the total
pie will get negative and it doesn't matter
if B's getting a larger or smaller chunk of that pie.
B's chunk is going to be negative.
He's going to drive down the price even more.
You can see it over here.
If they increase quantity beyond
this market quantity of 75, 37.5 each,
if we go beyond that, the price that they would
be selling at, at that quantity over there,
is lower than the average total cost.
You're going to be, the total economic,
the average economic profit per unit
is going to be negative. There will be a total
of negative economic profit.
Neither of them will want to produce more
from this state either. All of a sudden
in this top-left
state, holding others constant;
if you hold A constant, B can't gain by
changing his strategy, and if you hold B constant
A can't gain by changing his strategy,
so we are, up here, in a Nash equilibrium.
This is a Nash equilibrium.
Like the prisoner's dilemma, it was not
the optimal state. The optimal state was here,
but because they both wanted to cheat,
they both wanted to do this one-upmanship,
they both broke their contracts,
they could end up in this state over here.
But this state is stable.
There's nothing holding the other party equal.
There's nothing that they could do to change, to optimize.
What they could do, and this is not what Nash
applies to, they could say okay,
we've been really ruining each others' business.
Let's go coordinate again and I'm going to decrease
production if you decrease production.
That is not, and they could maybe try to go back
to this state, and that does not mean that this is not
a Nash equilibrium because by coordinating again
we're not holding the others constant.
We're saying I'm changing my strategy while
you're changing your strategy. Maybe only
through another agreement they could go over here.
That still doesn't mean that this is not
a Nash equilibrium. This is a Nash equilibrium.
If there's no coordination, if you hold one player
constant, the other player cannot change their strategy,
or change their production, for a gain.