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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.