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GREG HUTKO: Hi, welcome
back to the 14.01
problem-solving videos.
Today I'm going to be working
on Fall 2010 Problem Set 5,
Problem Number 4.
And I'm only going to be working
the last few sections,
E, F, G, and H in this video.
But if you need help with the
earlier sections, you should
go ahead and should look
at PSET Number 4,
Problem Number 3.
And in that problem, we work
through the production
function, and we go through and
we find conditional supply
and the conditional
demand curves.
But this problem is going to
have us looking at aggregated
supply in a market.
We have to consider what
production should be occurring
in the market given a set number
of firms in the market.
And then we're going to think
about the case of perfect
competition where firms have
to be operating at the
absolute best efficiency
possible.
And we're going to look
at how that affects
the production level.
Part E is introduced by saying,
consider now that r
equals 4 and w equals 1.
And that the market demand for
coffee is given by quantity
demanded equals 20 minus p.
There are eight other companies
operating in this
market and all companies have
the cost structures identical
to Sebastian's company, the
company that we've been
dealing with earlier
in the problem.
Part E asks us, what
is the aggregate
supply in this market?
And if you look back earlier
in this problem, the other
piece of information that we're
going to need is we're
going to need this cost function
that gives all of the
cost in terms of the rental rate
of capital or how much it
cost to use each machine per
hour, the wage rate or how
much labor cost per hour, and q,
the quantity, that's output
by a specific firm.
Now, to find the aggregated
supply, what we're first going
to find is we're going to first
find the supply curve
for one firm within
this market.
And then we're going to set the
demand or the supply curve
in terms of quantity
in terms of price.
We're going to multiply by
eight to aggregate it.
And then we'll have our
aggregated supply curve.
But before we do that, we also
have to think of a limiting
case as well.
When we're representing the
costs for a firm, we're going
to represent both the marginal
cost and the average cost. The
marginal cost is the cost of
one single additional unit,
while the average cost tells us
all the costs including the
fixed costs divided by the total
that we're producing,
what does that look like?
If the price in a market is
below the minimum of the
average cost of a firm in the
market, they're not going to
produce in the market.
So if the price is below this
critical p star, since the
firm, even if they're producing
right at the minimum
of average cost, they can never
recover their cost. So a
firm is only going to produce
if this p where the price
that's being charged is above
the p star, the minimum of
average cost. So we're going
to find the supply curve in
two cases, one where the price
is above this minimum of
average cost. And two, we're
going to find it when the
price is below that minimum.
Let's start off by finding the
marginal cost to get our
supply curve.
Taking the marginal cost, the
derivative with respect to q.
Or before we can take the
marginal cost, sorry, let's
plug-in for the variables
w and r.
We're going to find that our
cost curve is given by 4 plus
4q squared.
Now we can find the marginal
cost, which will be our supply
curve for a single firm.
So in most cases, we know that
this supply curve is going to
represent the supply for a
single firm where marginal
cost, the price, is
going to equal 8q.
So in most cases, this will be
our supply curve for one firm.
Putting it in terms of q,
we'll have q equals p/8.
Now since we have eight firms,
we multiply by 8.
And in this case, we're going
to have the aggregated
quantity, which we represent by
a capital Q. So that's the
quantity produced by all eight
firms in the market.
And that's going
to equal price.
So this is one part of
the supply curve.
And what we need to know is,
what's the critical price at
which this will represent
the supply curve?
So when the average cost curve
crosses the marginal cost
curve, that's the minimum of the
average cost. It's always
like that for all cost curves
for a producer.
So if we set marginal cost equal
to average cost, we find
this critical p star at which
the firm is going to produce
at any price above
that p star.
So we're going to go back to our
marginal cost. And what we
have to do is we have to find
the average cost as well to
set it equal.
To get the average cost, we're
just going to go back up to
our cost curve and we're going
to divide through the whole
thing by q.
So the average cost is going to
be 4 divided by q plus 4q.
Now we're going to set average
cost and marginal cost equal.
And when we set marginal cost
and average cost equal to find
the intersection point on our
graph, what we're going to
find is we're going to find that
critical p star is going
to be equal to 8.
So Qs is going to equal p for
any price that's greater than
or equal to 8.
But what happens in the
case where the price
is less than 8?
In that case, no single firm can
make a profit by being in
the market.
So for any price less than 8,
the production level is going
to be equal to 0.
Now we're going to move
on to the next case.
Now we're going to take the
demand curve that we're given
and we're going to calculate
the equilibrium price, the
aggregate quantity sold, and
the quantity sold by each
firm, and the economic
profit of each firm.
So let's start off in solving
this problem, we're going to
just assume that the price,
the equilibrium price, is
going to be greater than
or equal to 8.
And as we're solving through the
problem, if we end up with
a price that's less than 8, then
we're just going to go
back and we're going to say,
OK, there's going to be no
production, and we're done.
But let's work with the
assumption that we're working
with this supply curve
to begin with.
All we have to do in this case
is we're just going to set the
supply curve we just found
equal to the demand curve
that's given in the problem.
Solving through for p, we're
going to find that the price
in this market is going
to be equal to 10.
And plugging in the price back
into the demand curve, you can
find that the aggregate
quantity is going
to be equal to 10.
So the price for each unit is 10
and the aggregate quantity
is 10 as well.
Now to find the quantity
produced by each of the eight
firms, all the firms, since
they have identical cost
structures, are going to be
producing the same amount.
So we're just going to divide
this quantity by 8 to find the
quantity produced by the
individual firms. So each firm
in this case produces
5/4 of a unit.
The last thing that we have to
do is we have to calculate the
economic profits for each of
the single firms. So the
profit is going to be equal to
the revenue, which is just
price, times the quantity
for a single firm.
A big mistake here would be to
use the aggregated quantity.
And then you're going
to subtract out the
cost for each firm.
And we're just going to use
the cost function that was
given after plugging
in w and r.
And this is going to represent
the economic profit for each
of the firms. And this leads
right into the next part of
the problem.
It asks us, can this be a long
run equilibrium where we have
these prices, quantities,
and profits?
And why or why not?
And how will the supply
side of the market
adjust in the long run?
Now when other firms are
considering entering the
market, the only thing that
they're going to consider is
they're going to consider, is a
firm that's existing in the
market currently
making profit?
If they're not currently making
profit, then the firm
that's considering entering
would have to have a better
technology, a better way of
producing at lower cost, to
enter the market and
be able to actually
produce with a profit.
But if there is profit being
made, in this case 2.25 for
each firm, then more and more
firms are going to enter until
profits are driven down to 0.
So is this a long
run equilibrium?
The answer is no.
And why not?
More firms are going to enter on
the supply side until we're
driven to equilibrium.
The last part that we're going
to do is we're going to do
part H. Part H asks us, what
is going to be the price in
the long run?
How many firms will
be present in this
market in the long run?
And how much will each
firm produce?
Now in the long run, we know
that profits are going to be
driven down to 0, and that each
firm is going to have to
be leaner and meaner.
They're going to have to produce
at maximum efficiency
to be competitive within
the market.
So if we go back to the graph
that we started with, we can
look at, at which point are
firms operating optimally?
It's when marginal cost is equal
to average cost. It's at
this critical p star that
we calculated to
be equal to 8 earlier.
So in the long run, all the
firms are going to have to be
lean and mean.
They're going to have to operate
at a very low average
cost. And we're going to
calculate how many firms are
going to be in the market
producing at this point where
marginal cost intersects
average cost.
So to start off H, we know that
p is going to be equal to
the minimum of average cost,
which we calculated in one of
the earlier parts of the problem
to be equal to 8.
When we have this equal to 8,
we can plug-in to our demand
curve that price.
And we can find that the
quantity demanded is going to
be equal to 12.
Now what we can do now is we can
go back and we can look at
the individual supply
curve for each firm.
And each firm, when we had our
individual supply curves that
we calculated in the first part
of the problem, had a
supply function that was equal
to q equals p divided by 8.
And in this case, if we know
that the price when the firms
are operating optimally is equal
to 8, then we know that
each firm is going
to be producing
one unit of the good.
So to find the total number of
firms, we just have to take
this aggregated amount, the
total amount that's being
produced, and divide through
by the amount each firm is
producing to find out the number
of firms that have to
be producing.
So in the long run, we're going
to have 12 firms each
producing one unit at a price
of 8, which is the optimal
price where marginal cost is
equal to the minimum of the
average cost. So just to
summarize what this problem
had us look at, we looked at the
case where we had instead
of just one firm, we
had multiple firms
operating in a market.
We saw that when we have
multiple firms operating that
if there's any economic profit
that more firms are going to
enter until the firms that exist
in the market are forced
to operate optimally with
no economic profit.
I hope that you found this
problem helpful.
And go ahead and again, you can
do the earlier parts, and
you can look to PSET 4
Problem Number 3 for
help on those problems.
