So, the marginal revenue is a horizontal line.

So, if I ask you,

"Hey, remember the rule?

You want to have marginal revenue equals marginal cost to maximize profit.

Where is that?" Well, the first thing you're going to note is,

"Hey Larry, there's two of them, there's an alpha and there's a beta".

We'll call this q sub alpha.

We'll call this q sub beta.

Marginal revenue equals marginal costs.

Which one of these maximizes profit?

Well, it turns out, first,

you should be starting to get a little nervous.

"Hey Larry, I thought you told us

marginal revenue equals marginal cost is going to do it."

It is, but there's two of them. So, which one's right?

Well, just for a moment,

imagine you were sitting here at alpha,

and you were to increase output by just one unit,

what would happen to your profit side?

Well, the extra revenue from that unit,

that's the definition of marginal revenue.

The extra revenue for an extra unit of output would be here.

The extra cost for that unit would be here.

Marginal cost is the extra cost for that unit of production.

Marginal revenue is the extra revenue for that unit of production.

Well, since the extra revenue exceeds the extra cost,

your profits are definitely growing.

Alpha can't be right because there's a point to the right where it's better.

In fact, if you were to continue to step out here,

you'd continue to make revenue on the margin in excess of cost on the margin.

Positive revenue above cost until you get right to beta.

Once you're at beta, if you go one more unit to the right, it's a bad deal.

That extra unit has extra revenue less than what it costs you to produce.

You don't want to make that unit.

That unit is going to get you,

it's going to tear down part of your profits,

because you're getting money back for that unit,

that's the marginal revenue,

that's smaller than the cost that you incurred to produce that unit.

So, what you're going to want to do is go out to beta.

Now you're going to say to me, "Well Larry,

how come you told us marginal revenue equals marginal cost maximizes profits?"

Actually, I was a little bit sloppy when I did that,

because we know when we take this simple calculus rule of the first-order condition,

what we're actually finding is an extreme condition or it could be a minimum.

In fact, let me draw an extra graph here.

Imagine this curve.

Well, setting the slope of that function equal to zero would be,

that's the first derivative, it would define local,

a local maximum there,

a local minimum here.

It would find a valley. It would find the peak.

It would find a valley. It would find a peak.

So, setting the first derivative equal to zero,

finds an extreme point but you have to do

that infamous second derivative test to find out whether you found a maximum or minimum.

Now, I'm not interested in having you work out

any second derivative let alone first derivative test of this thing,

I just want you to know that what really happened here is that on that earlier picture,

this point would be associated with beta that was done.

Marginal revenue equals marginal cost on the higher output side on the right.

This point probably about right here,

would be associated with alpha.

That's your maximum loss side,

or put it differently, minimum profit.

So, if you set the slopes of those two curves equal,

you're going to find the biggest gap between them.

One of those is a very attractive big gap, revenue exceeds cost.

The other is a very unattractive, cost exceed revenue.

In fact, cost exceed revenue at the biggest point possible.

That's a terrible position to be in.

So, alpha minimizes profits, beta maximizes profits.

So, going forward, on a graph like this,

we're just going to ignore this first one

and only focus on the intersection on the right side,

but we're still not quite done with this.

I want to draw one more picture.

I am going to label this as q sub wrong.

One of the things that students sometimes fall into is they'll say, "Hey,

I think q sub wrong is right because that's the largest gap between revenue and cost."

It's the largest gap between marginal revenue and marginal cost,

but you're not interested in maximizing that.

You're not interested in maximizing marginal,

you're interested in maximizing the total number of

coins that's in your cash register at the end of the day.

This is a nice point.

Maybe you could find that output level q sub wrong,

that particular jar of mayonnaise and hang a gold star on it, and say,

"Your production actually maximize

the difference between marginal revenue and marginal cost."

It means nothing to your shareholders. It means nothing.

You want to maximize total profit and clearly,

if you could just keep increasing output passes point,

this extra jar of mayonnaise or whatever the product is,

would give extra coins in the cash register of this high and extra cost of this.

That's certainly a positive growth to your profit.

Boom! You got another one.

You're going to get all this. Boom! It's just

going to keep growing as you go to the right.

Each additional one of these,

right out to the beta, is going to make that profit pi get bigger and bigger.

Once you get the beta, don't go any farther to the right because the extra,

the higher costs would be higher on the margin than revenue,

and the profit would be shrinking again.

So, we know when we hit marginal revenue equals marginal cost.

We want to hit marginal revenue and marginal cost,

where marginal cost is on the upwards sloping side.