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What I want to start thinking
about in this video is,

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given that we do have a
monopoly on something,

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and in this example,
in this video,

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we're going to have a
monopoly on oranges.

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Given that we have a
monopoly on oranges

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and a demand curve for
oranges in the market,

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how do we maximize our profit?

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And to answer that
question, we're

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going to think about
our total revenue

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for different quantities.

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And from that we'll get
the marginal revenue

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for different quantities.

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And then we can compare that
to our marginal cost curve.

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And that should give
us a pretty good sense

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of what quantity we should
produce to optimize things.

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So let's just figure
out total revenue first.

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So obviously, if
we produce nothing,

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if we produces 0 quantity,
we'll have nothing to sell.

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Total revenue is
price times quantity.

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Your price is 6 but
your quantity is 0.

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So your total revenue is going
to be 0 if you produce nothing.

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If you produce 1 unit--
and this over here

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is actually 1,000
pounds per day.

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And we'll call a unit
1,000 pounds per day.

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If you produce 1 unit,
then your total revenue

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is 1 unit times $5 per pound.

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So it'll be $5 times, actually
1,000, so it'll be $5,000.

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And you can also view it as
the area right over here.

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You have the height is price
and the width is quantity.

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But we can plot that, 5 times 1.

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If you produce 1 unit,
you're going to get $5,000.

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So this right over here
is in thousands of dollars

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and this right over here
is in thousands of pounds.

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Just to make sure that we're
consistent with this right

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over here.

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Let's keep going.

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So that was this point, or
when we produce 1,000 pounds,

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we get $5,000.

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If we produce 2,000
pounds, now we're

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talking about our price
is going to be $4.

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Or if we could say
our price is $4

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we can sell 2,000 pounds,
given this demand curve.

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And our total revenue
is going to be

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the area of this
rectangle right over here.

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Height is price,
width is quantity.

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4 times 2 is 8.

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So if I produce
2,000 pounds then

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I will get a total
revenue of $8,000.

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So this is 7 and 1/2,
8 is going to put

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us something right about there.

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And then we can keep going.

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If I produce, or if the
price is $3 per pound,

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I can sell 3,000 pounds.

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My total revenue is this
rectangle right over here,

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$3 times 3 is $9,000.

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So if I produce
3,000 pounds, I can

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get a total revenue of $9,000.

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So right about there.

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And let's keep going.

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If I produce, or if the
price, is $2 per pound,

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I can sell 4,000 pounds.

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My total revenue is $2
times 4, which is $8,000.

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So if I produce
4,000 pounds I can

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get a total revenue of $8,000.

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It should be even with that
one right over there, just

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like that.

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And then if the price is $1 per
pound I can sell 5,000 pounds.

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My total revenue is going
to be $1 times 5, or $5,000.

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So it's going to be
even with this here.

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So if I produce 5,000 units
I can get $5,000 of revenue.

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And if the price
is 0, the market

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will demand 6,000 pounds
per day if it's free.

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But I'm not going to generate
any revenue because I'm

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going to be giving
it away for free.

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So I will not be generating
any revenue in this situation.

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So our total revenue curve,
it looks like-- and if you've

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taken algebra you
would recognize this

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as a downward facing
parabola-- our total revenue

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looks like this.

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It's easier for me to draw
a curve with a dotted line.

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Our total revenue looks
something like that.

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And you can even
solve it algebraically

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to show that it is this
downward facing parabola.

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The formula right over
here of the demand curve,

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its y-intercept is 6.

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So if I wanted to write price
as a function of quantity

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we have price is equal
to 6 minus quantity.

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Or if you wanted to write in
the traditional slope intercept

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form, or mx plus b form-- and
if that doesn't make any sense

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you might want to review some
of our algebra playlist--

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you could write it as p is
equal to negative q plus 6.

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Obviously these are
the same exact thing.

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You have a y-intercept of
six and you have a negative 1

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slope.

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If you increase quantity by
1, you decrease price by 1.

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Or another way to think about
it, if you decrease price by 1

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you increase quantity by 1.

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So that's why you have
a negative 1 slope.

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So this is price is a
function of quantity.

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What is total revenue?

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Well, total revenue is equal
to price times quantity.

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But we can write price as
a function of quantity.

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We did it just now.

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This is what it is.

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So we can rewrite it, or we
could even write it like this,

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we can rewrite the
price part as-- so this

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is going to be equal to negative
q plus 6 times quantity.

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And this is equal
to total revenue.

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And then if you
multiply this out,

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you get total
revenue is equal to q

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times q is negative q
squared plus 6 plus 6q.

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So you might recognize this.

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This is clearly a quadratic.

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Since you have a negative out
front before the second degree

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term right over here,
before the q squared,

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it is a downward
opening parabola.

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So it makes complete sense.

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Now, I'm going to leave
you there in this video.

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Because I'm trying
to make an effort

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not to make my videos too long.

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But in the next video
what we're going

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to think about is, what
is the marginal revenue we

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get for each of
these quantities?

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And just as a review,
marginal revenue

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is equal to change in
total revenue divided

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by change in quantity.

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Or another way to think about
it, the marginal revenue

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at any one of these
quantities is the slope

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of the line tangent
to that point.

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And you really have to do
a little bit of calculus

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in order to actually calculate
slopes of tangent lines.

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But we'll approximate it
with a little bit of algebra.

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But what we essentially want
to do is figure out the slope.

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So if we wanted to figure out
the marginal revenue when we're

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selling 1,000 pounds--
so exactly how much more

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total revenue do we get if
we just barely increase,

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if we just started selling
another millionth of a pound

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of oranges-- what's
going to happen?

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And so what we do
is we're essentially

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trying to figure out the
slope of the tangent line

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at any point.

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And you can see that.

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Because the change
in total revenue

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is this and change in
quantity is that there.

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So we're trying to find
the instantaneous slope

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at that point, or
you could think of it

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as the slope of
the tangent line.

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And we'll continue doing
that in the next video.