 # Monopoly Part 5 Finding the profit maximizing Q

a monopolist actually does. So this is going to be
a series of lectures, and I’m going to cover
how a monopolist maximizes profits. And before we really
get into this, we’re going to need to go ahead and get two pieces
of information. One is we need
to use demand data to determine total revenue
at different levels of quantity and the marginal revenue
of different levels of Q. So I’m going to go ahead and do a little example
here for you guys. So what do we mean by all that? Up to now, we have often
looked at the demand curve as a function of the price. We’ve often looked at quantity
demanded as a function of price. Now we’re going to go
the other way around and we’re going to look at price
as a function of quantity. Then we’re going
to use that information to determine total revenue, and then we’re going
to use that information to determine marginal revenue. So let’s suppose
that if we have our monopolist and they decide they want
to sell one unit, they can charge \$11 for it. Then if they produce one unit,
their total revenue is \$11. And as they went
from zero to one, their revenue went
from zero to 11. So the marginal revenue
of that first unit is one. And let’s suppose
that as they go to two units, they have to cut their price to 10 bucks to be able
to sell the additional unit. So if they produce two units, they’re going to have \$20
of total revenue, two times ten. And the marginal revenue
here then is the change in revenue over the change
in the number of units. So marginal
revenue equals change in total revenue
over change in quantity, which in this case is nine. And then if we
have three units produced and that makes us
have to cut the price to \$9, then now our
total revenue is 27. And we can see that the marginal
revenue is seven. And one more here,
if producing four means that we have to cut our price
to eight to be able to solve them all, then the marginal revenue
of this fourth one is \$5. And you can see that in all
cases, actually, the marginal revenue is less
than the price, and that sometimes
gets a little bit tricky for people
to understand why that is. And the reason is not only
did we sell the second one for \$10, we also cut the price
on the first one down to \$10. So that’s why we have
the marginal revenue even lower. Likewise, when we go to three, not only did we sell the third
one for \$9 instead of \$10, we also cut price
on the previous two. So that’s why our
marginal revenue is \$9 minus \$2 gets us to \$7 and so on
and so forth down here. You can see that if we
kept on going this way, we would eventually
get into a situation where we would
have negative marginal revenue because eventually the amount
that we would gain on price from selling one more unit
would be less than the loss, the lower amount that we would
get for the previous units. So as we’re going
to maximize our profit, we need to find out what our total revenue is
for all of our different levels of Q. Once we’ve got that, we’re going to go ahead and match that with
some marginal cost information. So I’m going to go ahead
and set up another example here. Now that you already understand
the concept of marginal revenue, I can go a little
bit faster here. Let’s suppose
that when they sell zero units, that’s consistent
with setting a price of \$6. And obviously you get zero total
revenue if you sell zero units. Marginal revenue
doesn’t make sense here because there’s
no previous level to compare to. Total costs of zero units
would be our fixed costs. And let’s suppose
our fixed costs are \$4. Marginal cost doesn’t
make any sense here because there’s no previous
level of costs to compare it to. If we have quantity of one, let’s suppose that if we want
to sell one unit, we have to cut our price to \$5. That gives us
a total revenue of \$5. Marginal revenue is how much
we changed our revenue, and we changed it by \$5. And let’s suppose that producing this first unit caused us
to incur \$2.50 of costs. So marginal cost was \$2.50. And you can see that we
made ourselves better off by producing this first unit
\$2.50 to costs. So our marginal profit
on that is \$5 minus \$2.50 gives us \$2.50
of marginal profit. If selling two units means
that we have to cut our price to \$4, then our total revenue
of two units is \$8. The marginal revenue
of this second unit is \$3. And let’s suppose that this one has a lower
marginal cost than the first one because we have some increasing
returns for awhile. Let’s suppose that we know
that it has a \$1.50 of marginal costs. Notice I can go ahead
and solve this around. Before I did this one
minus this one gets you there. And so total costs of one
minus total costs of zero gets me the marginal cost
of the first unit. We can also turn that around and go total cost of one
plus marginal cost of the second gets us
to total cost of the second. So I think that is \$8. \$6.50 plus \$1.50 is \$8. And this firm is now
breaking even. Just to make things
a little less boring, I’m going to go ahead and make this demand curve start
to change its slope. So to sell that third unit, I have to cut my price
but not by a full dollar here. And so now my total
revenue is three times \$3.50 or \$10.50 and the marginal
revenue here is \$2.50. And let’s suppose this one has
a marginal cost of \$1 so that we have total
cost of \$9. And notice this third
unit was still profitable to produce because it
added \$2.50 to our revenue and \$1 to our costs. And now we’re going to go ahead
and say that \$3 if we want to sell four units,
\$12 of revenue. We have a marginal
revenue here of \$1.50. And let’s say we now are going
to have marginal costs of \$2 for this fourth unit so that our
total costs go to \$11. And you can see
now that this fourth unit was not very profitable for us. It only brought
in \$1.50 of revenue, but it had a marginal
cost of \$2, so its marginal profit was \$.50. So here is
our profit maximizing quantity. Again, just
like with perfect competition, to maximize profits,
we want to produce all the units that have marginal revenue
greater than marginal cost. So the first
unit passes that test. The second unit, \$8 is more than \$1.50. Yeah,
that passes the test. The third unit, \$2.50 is more
than \$1 so it passes the test. Third and fourth unit does not
pass the test. So that’s how we’re going
to maximize profits– produce all the units of Q that have marginal revenue
greater than marginal costs, just like with perfect
competition.

## One comment

1. sangjin lee says:

Thank you. that helped a lot!