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# Monopolist optimizing price: Total revenue

Video transcript

What I want to start thinking
about in this video is, given that we do have a
monopoly on something, and in this example,
in this video, we're going to have a
monopoly on oranges. Given that we have a
monopoly on oranges and a demand curve for
oranges in the market, how do we maximize our profit? And to answer that
question, we're going to think about
our total revenue for different quantities. And from that we'll get
the marginal revenue for different quantities. And then we can compare that
to our marginal cost curve. And that should give
us a pretty good sense of what quantity we should
produce to optimize things. So let's just figure
out total revenue first. So obviously, if
we produce nothing, if we produces 0 quantity,
we'll have nothing to sell. Total revenue is
price times quantity. Your price is 6 but
your quantity is 0. So your total revenue is going
to be 0 if you produce nothing. If you produce 1 unit--
and this over here is actually 1,000
pounds per day. And we'll call a unit
1,000 pounds per day. If you produce 1 unit,
then your total revenue is 1 unit times $5 per pound. So it'll be $5 times, actually
1,000, so it'll be $5,000. And you can also view it as
the area right over here. You have the height is price
and the width is quantity. But we can plot that, 5 times 1. If you produce 1 unit,
you're going to get $5,000. So this right over here
is in thousands of dollars and this right over here
is in thousands of pounds. Just to make sure that we're
consistent with this right over here. Let's keep going. So that was this point, or
when we produce 1,000 pounds, we get $5,000. If we produce 2,000
pounds, now we're talking about our price
is going to be $4. Or if we could say
our price is $4 we can sell 2,000 pounds,
given this demand curve. And our total revenue
is going to be the area of this
rectangle right over here. Height is price,
width is quantity. 4 times 2 is 8. So if I produce
2,000 pounds then I will get a total
revenue of $8,000. So this is 7 and 1/2,
8 is going to put us something right about there. And then we can keep going. If I produce, or if the
price is $3 per pound, I can sell 3,000 pounds. My total revenue is this
rectangle right over here, $3 times 3 is $9,000. So if I produce
3,000 pounds, I can get a total revenue of $9,000. So right about there. And let's keep going. If I produce, or if the
price, is $2 per pound, I can sell 4,000 pounds. My total revenue is $2
times 4, which is $8,000. So if I produce
4,000 pounds I can get a total revenue of $8,000. It should be even with that
one right over there, just like that. And then if the price is $1 per
pound I can sell 5,000 pounds. My total revenue is going
to be $1 times 5, or $5,000. So it's going to be
even with this here. So if I produce 5,000 units
I can get $5,000 of revenue. And if the price
is 0, the market will demand 6,000 pounds
per day if it's free. But I'm not going to generate
any revenue because I'm going to be giving
it away for free. So I will not be generating
any revenue in this situation. So our total revenue curve,
it looks like-- and if you've taken algebra you
would recognize this as a downward facing
parabola-- our total revenue looks like this. It's easier for me to draw
a curve with a dotted line. Our total revenue looks
something like that. And you can even
solve it algebraically to show that it is this
downward facing parabola. The formula right over
here of the demand curve, its y-intercept is 6. So if I wanted to write price
as a function of quantity we have price is equal
to 6 minus quantity. Or if you wanted to write in
the traditional slope intercept form, or mx plus b form-- and
if that doesn't make any sense you might want to review some
of our algebra playlist-- you could write it as p is
equal to negative q plus 6. Obviously these are
the same exact thing. You have a y-intercept of
six and you have a negative 1 slope. If you increase quantity by
1, you decrease price by 1. Or another way to think about
it, if you decrease price by 1 you increase quantity by 1. So that's why you have
a negative 1 slope. So this is price is a
function of quantity. What is total revenue? Well, total revenue is equal
to price times quantity. But we can write price as
a function of quantity. We did it just now. This is what it is. So we can rewrite it, or we
could even write it like this, we can rewrite the
price part as-- so this is going to be equal to negative
q plus 6 times quantity. And this is equal
to total revenue. And then if you
multiply this out, you get total
revenue is equal to q times q is negative q
squared plus 6 plus 6q. So you might recognize this. This is clearly a quadratic. Since you have a negative out
front before the second degree term right over here,
before the q squared, it is a downward
opening parabola. So it makes complete sense. Now, I'm going to leave
you there in this video. Because I'm trying
to make an effort not to make my videos too long. But in the next video
what we're going to think about is, what
is the marginal revenue we get for each of
these quantities? And just as a review,
marginal revenue is equal to change in
total revenue divided by change in quantity. Or another way to think about
it, the marginal revenue at any one of these
quantities is the slope of the line tangent
to that point. And you really have to do
a little bit of calculus in order to actually calculate
slopes of tangent lines. But we'll approximate it
with a little bit of algebra. But what we essentially want
to do is figure out the slope. So if we wanted to figure out
the marginal revenue when we're selling 1,000 pounds--
so exactly how much more total revenue do we get if
we just barely increase, if we just started selling
another millionth of a pound of oranges-- what's
going to happen? And so what we do
is we're essentially trying to figure out the
slope of the tangent line at any point. And you can see that. Because the change
in total revenue is this and change in
quantity is that there. So we're trying to find
the instantaneous slope at that point, or
you could think of it as the slope of
the tangent line. And we'll continue doing
that in the next video.