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GREG HUTKO: Today we're going
to do Fall 2010, P Set Six,
Problem Number Four.
And for this problem we're going
to shift away from what
we've usually been talking
about, where we're dealing
with a straight equilibrium,
setting the demand curve equal
to the supply curve.
And now we're going to think
about what happens when the
supplier has market power.
When they're the only competitor
in the market, and
they can decide how much
quantity they want to produce.
And they don't have to worry
about other producers coming
in and producing.
Problem Number Four--
I'll read through Part A--
states, "A monopolist
firm faces the
following cost curve.
The cost equal Q squared plus
15, where Q is the output
produced, the demand for its
product is given by P equals
24 minus Q. We need to calculate
the non-price
discriminating consumer surplus,
the producer surplus,
and the deadweight loss
associated with the monopoly."
Now, what this problem's really
going to look like-- we
can think about it starting
with this graph.
Is, instead of producing output
to the point of the
equilibrium right here, the
supplier can actually make
more money by saying I'm only
going to produce to a point
right here, so this is what
we're looking for.
We're wondering how much is the
supplier actually going to
constrict the supply
in the market.
And when they constrict this
supply, what happens is this
small triangle right here
becomes the deadweight loss.
This is potential surplus that
would have existed when this
whole big triangle was
the consumer surplus
plus producer surplus.
So now nobody's getting
that triangle.
But the producer surplus is much
bigger than it would have
been when it was just the space
below the price level to
the supply curve.
So the producers
are better off.
The consumers are going
to be worse off.
And society as a whole-- adding
together the producers'
and the consumers' surplus--
is going to be worse off.
Now, the way the producers
actually make their decision
on how much to produce is, when
they're moving this line
back and forth deciding how much
they want to constrict
the quantity that they're going
to supply, and when
they're supplying more, the
quantity they're supplying is
going to be increasing.
But as they supply more--
since the demand curve
is downward sloping--
the price is going
to be going down.
And now the way the producer
actually makes their
production decision is to say,
OK, I know I'm going to lose
some money if I'm producing
more, because the price is
going to be falling.
What I want to know is I want to
produce as much as I can so
that, at the margin, the cost
of producing that one
additional unit is the same as
the revenue that I'm going to
be taking in for that
additional unit.
The point where I'm producing,
and the additional cost of
that unit is more than the
additional money that I'm
taking in, I'm going to stop
assuming that there's no
competition.
So the monopolist firm is going
to set the marginal cost
equal to the marginal revenue.
So we have a total cost
function, so calculating the
marginal cost is pretty
straightforward.
I'm just going to take the
derivative with respect to Q,
and the marginal cost for
a monopolist firm
is going to be 2Q.
Now, it's tempting when we
look at this revenue
function-- revenue just being
the total quantity I'm
producing times the price
I'm receiving.
It's tempting to just take the
derivative here with respect
to Q, and say that marginal
revenue is going
to be equal to price.
But that's not what the
monopolist does.
Because in a competitive
situation, we were setting
marginal cost equal to P.
In the monopolist situation,
the monopolist is going to
look at this P right here, and
they're going to say I know
how the consumers are going to
respond based on my decision
to produce.
So I'm going to replace this
P with the demand curve, 24
minus Q. So I can plan how much
I'm producing based on
what I know the consumer's
response to my production
choice is going to be.
So instead of taking the
derivative of this function,
I'm going to plug-in 24 minus Q
and we're going to find the
marginal revenue using
this function.
When we do this, we find that
the marginal revenue is equal
to 24 minus 2Q.
And all we have to do now is
we have to set the marginal
revenue and the marginal cost
equal, and we can find the
quantity that's going
to be produced at
the monopolist outcome.
Solving for Q, you find that
the quantity is going to be
equal to 6.
And then we can solve for the
price just by going back to
the demand curve that's
given on our graph.
The price here in the
monopolist case
is going to be 18.
So now we can come back
to our graph.
We know that the monopolist
level of output is going to be
6, so we can label this 0.6.
We know the price that's
going to be charged--
which is not the intersection
with the supply curve, it's
going to be the intersection
with the demand curve--
this price is going to be 18.
And on our graph it's going to
also be useful to label two
more points.
That'll just make it easier for
us to calculate consumer
surplus, producer surplus,
and deadweight loss.
We're going to want to label
this point right here so the
equilibrium quantity is 8-- and
you'll see in a second why
I'm labeling that.
And you're also going to want
to label where, when the
quantity is 6, the intersection
with the supply curve.
So when the quantity is 6, you
know that the marginal cost
curve is given here.
So that means the intersection
right here is going to be 12.
And this is going to make our
calculations of the area of
PS, the area of CS, and
the area of DWL just
a little bit easier.
Now, to calculate consumer
surplus, I'm just going to
multiply the height of this
triangle right here by the
length of the triangle,
and I'm going to
take one half of that.
So consumer surplus in
this situation is
going to equal 18.
And now we're going to
do the same thing
for producer surplus.
We're going to add the area of
this rectangle to the area of
this triangle at the
bottom as well.
So the first term here that's
given is the area of the
rectangle, and the term here is
the area of the triangle.
Adding these together, we're
going to find that the
producer surplus is 72.
And now, to calculate the
deadweight loss you really
have two options.
One, you could find the total
producer and consumer surplus
at equilibrium--
so the area of this large
triangle right here.
And you could subtract out the
new consumer surplus and the
new producer surplus, and you'll
be left with only the
deadweight loss.
For our purposes, it's going to
be a little bit easier to
just take the height of the
triangle and the length of the
base, and to multiply through.
When we do that, we're going
to find that the deadweight
loss is going to
be equal to 6.
And so you can see that the
producer surplus is pretty
high in this situation.
And so the government's going
to come in in our next
problem, and they're going to
say we have an intervention
that might be able to correct
this problem that we see in
the market.
Part B says, "How does charging
the monopolist a
specific tax of $8 per unit
affect the monopoly optimum,
and the welfare of consumers,
the monopoly, and society,
where society's welfare or
surplus includes the tax
revenue?"
So, what's basically happening
in this new case is we're
going to start off with the same
sort of problem where the
monopolist gets to decide how
much they're going to output.
And we're interested in
the marginal cost and
the marginal revenue.
Now, the marginal revenue is
going to be represented by the
same equation, and we're going
to substitute in for price the
demand curve again.
And when we solve through,
substituting in for the demand
curve and taking the derivative,
we're going to
find that the marginal revenue
is again going to be equal to
24 minus 2Q.
So the marginal revenue
hasn't changed at all.
What is going to change is going
to be the total cost
curve for the monopolist. So
this was the cost curve that
we started off with, but now
for each unit Q that the
monopolist produces, it's
going to be taxed
at a rate of t.
So we can add in the
cost of the tax.
And in the next step, I'm going
to take the derivative
with respect to Q, and I'm going
to substitute in for t
the price of the tax, or 8.
So now our new marginal cost
is equal to 2Q plus 8.
And to solve for our new
equilibrium, we're just going
to set marginal cost and
marginal revenue equal.
And when we do that, we're
going to find that
Q is equal to 4.
And plugging into the demand
curve, you're going to find
that the price is equal to 20.
Now in this problem, you could
go through and you could go
ahead and you could calculate
the consumer surplus, the
producer surplus, the deadweight
loss and the tax
revenue, and you could figure
out quantitatively how much
they've changed.
But we're just going to draw a
new graph, and we're going to
look at the changes in consumer
surplus, producer
surplus, and deadweight loss,
and make a qualitative
assessment of how those
quantities have changed.
So on a new axes, I'm going to
draw the old supply curve and
the demand curve.
Now, what's essentially
happening is, since the
suppliers know for each unit
they're producing they're
going to have to pay a tax
of 8, the supply curve is
essentially shifting
up by 8 units.
And I'm representing the new
supply curve with an s prime.
So now, instead of the suppliers
making their
monopolist decision based on
this supply curve and the
demand curve, they're now
making it over here.
And so what's going to happen is
they're going to have some
new monopolist output.
And in this case, we know the
new monopolist output is 4.
We know that the new
monopolist price
is going to be 20.
And so we can see on this graph
that producer surplus is
going to be represented
by this four-sided
figure right here.
We know that the tax revenue
is going to be-- since the
distance from this point to this
point is 8, from 0 to 4--
this is going to represent the
tax, this box right here.
And we know that the consumer
surplus is going to be this
small triangle up top.
Meanwhile, the dead weight loss
is anything that's not
represented that would have been
in our original producer
surplus, plus consumer
surplus.
So the deadweight loss is this
large triangle over here.
So basically what's happened
is, compared to our initial
case, the government's come
in and for society--
since there's less
being produced--
we've basically shifted
the monopolist
quantity over to the left.
This means that, overall, the
deadweight loss, this triangle
over here, has increased
in size.
So if the deadweight loss is
increasing, we can say that
society is going to be worse
off in this situation.
In another case, it's also clear
to see that the consumer
surplus, since the consumers are
paying a higher price for
a lower quantity, we can also
say that the consumer surplus
is going to decrease.
So we can safely say that the
consumers are going to be
worse off as well.
And then the last interpretation
is knowing that
in the first case the producers
were allowed to make
their production decision just
given the demand curve and
their original supply curve.
Now their production decision
also has to take into account
the government taking away
some of their profits.
If the government is taking
some away some of their
profits, the producers
are necessarily going
to have less surplus.
So the producers are going
to be worse off as well.
So overall, the only person
who might possibly benefit
from this policy would
be the government.
But overall, the producers, the
consumers, and society are
going to be worse off.
Now, the last part of this
problem is part C. And instead
of implementing a tax on the per
unit production decision
for the producers, now the
government's going to consider
a different tax policy.
Part C says "How does imposing
a tax on profits--
profit after tax equals
1 minus t--
affect the monopoly optimum, and
the welfare of consumers,
the monopoly, and society?"
Now basically, what's happening
in this situation is
the government's going
to come in.
And they're going to say all
right, after you've made your
decision on how much to produce,
we're going to take a
set percentage of the
producer's surplus.
So if you get a producer's
surplus of this amount, then a
certain chunk of it is going
to go to Uncle Sam at a tax
percentage which we can
say is just x percent.
Now that percentage of tax, it
doesn't change the fact that
the producers want to have as
much surplus as possible.
Just because they're going to
lose, say, 10% of it because
of Uncle Sam, it's not actually
going to affect the
fact that they want their
producer surplus to be as big
as possible.
So what happens in this
situation is that this after
profit tax will not affect
the equilibrium at all.
And we're going to
be left with the
same consumer surplus.
Producer surplus is going to be
lower because of the tax,
but overall, society is going
to be left with the same
social welfare.
So really what this problem is
looking at through its three
parts, we look at the monopolist
situation, and we
look at how the government can
try to adjust with tax policy
what's happening
in the market.
And what we saw in our second
scenario is that when they
charge a per unit tax on the
producers, societal welfare is
going to go down.
But in the third case, when
they're just taking a set
percentage from the producer
surplus, the overall welfare
for the society is going
to stay the same.
So bundled in this problem we
had the monopolist situation,
setting marginal cost equal
to marginal revenue.
And we also looked at tax
implications on a per unit
basis, and on a profit basis
with a set tax after the
production decision is made.
I hope you found this
problem helpful.
