Since all firms are identical they have the same unit cost and

Nash equilibrium has the property that every firm produces the same amount, okay?

So let's say q star is the Nash equilibrium output of each firm and

the definition of Nash equilibrium says that this should be mutual best reply.

So if other firms are producing q star it's your best reply to

produce the same amount, q star.

So let's write down the best reply, relationship, by means of equation.

So you are producing q star, that should be a best reply to other people

producing q star, and we have already calculated the best reply function.

It's half of this number here, a minus c over

b minus total output of the remaining firms, okay,

so since there are n firms, the number of other firms is n-1,

and each of those n-1 firms are producing q star of Nash equilibrium,

so therefore this is what we denoted by large Q,

the total quantity of other firms, and this is your best reply, and

at the Nash equilibrium everybody's best responding each other, okay?

So this is a simple equation.

One equation and you have one known q star, so you can solve for q star.

So, let's do that.

So let's multiply both sides by 2.

So 2 times q star is equal to a minus c divided by

b minus n minus 1 and q star, okay, so

you move this term from right to left,

and what you have here is N plus 1 times q star,

equals to this number here, a minus c divided by b, okay.

So therefore q star is equal to 1 over n

plus 1 times this number here, a minus c, divided by b, okay.

So therefore, the total output at

the Nash equilibrium was N firms is N times this number, so

N over N plus 1 times this number here, a minus c divided by b, okay?

So let me just rewrite this part, N over N plus 1,

so N is equal to N plus 1 minus 1.

That's the numerator.

The denominator is N time, N plus 1, so

this is equal to, say, 1 minus, 1 over N plus 1.

Okay, so if you have N firms, total production is this type

this quantity times this number here, so by using the power of math we

have identified that the total output when N firms present in the market,

and let's compare this result with perfectly competitive market.

Again, this is a picture of perfectly competitive market, and in the perfect

competition, total output is this yellow number here, and on the other hand oh,

in contrast, if you have N firms, we have calculated the total output, and

it's all, it's equal to this number, okay, and as you can see,

as N increases, this part here, 1 over N plus 1, decreases sharply.

So everything is quickly brought into this yellow number.

So let me show you several cases.

If N is equal to 1, the quantity is small and

the price is higher, but an, as N increases, the total

output quickly converges to competitive equilibrium total output, 'kay?

So Nash Equilibrium sets the following things.

If there are few firms in the market, price is very high and quantity is small,

but as the number of firms increase, price goes down and eventually it converges

to competitive, perfect competitive market equilibrium, okay?

So, large number of firms actually implies perfect competition.

This is the prediction by Nash equilibrium.

So by using the concept of Nash equilibrium,

and by using the power of math to find Nash equilibrium,

we have derived the Law of Market Competition in economics.