In this lesson, we're continuing our path towards investigating how

a firm would optimize behavior.

In other words, maximize profits, in a world where the firm knew its costs, and

it was given some exogenous price, some signalistic price.

Taking that price and its cost, how does it maximize profits?

And since we're talking a lot about profits, in this particular video, what

I want to do is think about, how would you actually show profits on our graph?

So let's recall the type of graph that we've been dealing with here.

We'll draw our axes system.

And we have on the vertical axis, dollars and cents, and

on the horizontal axis, we have quantity.

And we know that profit maximization requires a very simple rule,

and that is, you want to set marginal revenue equal to marginal cost.

And in our particular instance that we're dealing with here where the price is

parametric to the firm, it's fixed,

then marginal revenue is going to just be equal to whatever that price is.

We call that P0, and so we'll call this MR0.

It's a horizontal line.

Any output level you choose, every extra one, you get exactly that price.

So suppose it's $3 a unit.

If you produce 100 you get $3 for the 100th unit, 101 you get $3 for

the 101st unit.

That's why it's shaped like this.

And then we want to put marginal cost in there, and

we understood marginal was cost is really U-shaped.

But we also knew that while it's U-shaped,

the downward sloping side is kind of a red herring.

It's not where we want to go, okay?

The marginal cost curve on the upward sloping side is where the firm

is going to maximize profits.

And that point occurs where marginal revenue equals marginal cost.

And we'll call this point, for now, q*.

And so the question is,

given the situation, can I depict profits?

That is, total revenue minus total cost.

Well, I actually have a term for total revenue already.

I can see that.

I can see that on this graph because I know total revenue is

equal to price, which is P0, times quantity, q*.

Well, if I were to go ahead and take, let's say, price,

that's this vertical height, times quantity, that's this horizontal distance.

Well, if you remember,

multiplying base times height will give you the area of that rectangle.

And so the area of this rectangle would be total revenue.

Now, how about total cost?

Well, we wonder about this.

Do we have anything on this picture that's going to help us out?

And you might be thinking, well, you know what?

I know marginal cost is the cost of the extra unit.

Each extra unit is depicted by the marginal cost.

So if I were to do that old calculus trick of integrating the area under that curve.

In other words, if I would just go about my business of perhaps adding up every one

of these vertical bars underneath it.

For each output level, how much was the extra cost associated with that?

Maybe that's a proxy for total cost.

Turns out that that's a trick that doesn't really work because,

as we know, marginal cost has absolutely no information about fixed cost in it.

Fixed cost and marginal cost are like apples and oranges.

They just don't get together.

And so in this case, all I can get by adding up all those areas under

the marginal cost curve would be the variable cost, but

nothing on the fixed cost.

So I got a problem.

I need something else.

And that something else is not going to be too hard to get.

Because what I'm going to do is bring back one of those little bubbles that comes up

over the cartoon character's head, get's this brilliant flash and says,

you know what?

I recall that we defined average total

cost as equal to total cost over output.

So if I were to go ahead and multiply both sides by q,

I get q times average total cost would be equal to total cost.

So now I see that if you just show me the average total cost curve,

I can actually figure out from that graph what it's going to look like.

So let's draw another graph.

We'll again put the marginal revenue on this graph, and we'll put marginal cost.

But I'm just not even going to show you that downward sloping side.

Before I had it as a dashed line because we know it's there, but

it's not relevant to the firm, so why clutter the graph?

We'll call this MC.

And now I remember from what we just did that profit

is equal to total revenue minus total cost.

But that's the same thing as saying profit is equal to price

times quantity minus average total cost times quantity.

We just did that in that little recall bubble there.

We discovered that total cost is just the same as quantity times average cost.

That means I can rewrite this as saying profit is equal to quantity times,

I'll just pull quantity out of the bracketed term,

price minus average total cost.

And that's good because what that tells us now is that profit is just written as

the number of units you produce times the per unit markup.

Price minus average cost is the per unit markup.

This is quantity times per unit markup.

You're going to have profits.

Well, I need an average cost curve.

What's my average cost curve look like?

Well, we remember the average cost curve is U-shaped, and

we remember that it has its minimum on the marginal cost curve.

So that's going to look something like this.