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GREG HUTKO: Hi, welcome
back to the 14.01

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problem-solving videos.

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Today I'm going to be working
on Fall 2010 Problem Set 5,

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Problem Number 4.

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And I'm only going to be working
the last few sections,

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E, F, G, and H in this video.

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But if you need help with the
earlier sections, you should

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go ahead and should look
at PSET Number 4,

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Problem Number 3.

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And in that problem, we work
through the production

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function, and we go through and
we find conditional supply

00:00:49.530 --> 00:00:51.700
and the conditional
demand curves.

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But this problem is going to
have us looking at aggregated

00:00:54.420 --> 00:00:56.390
supply in a market.

00:00:56.390 --> 00:01:00.050
We have to consider what
production should be occurring

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in the market given a set number
of firms in the market.

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And then we're going to think
about the case of perfect

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competition where firms have
to be operating at the

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absolute best efficiency
possible.

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And we're going to look
at how that affects

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the production level.

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Part E is introduced by saying,
consider now that r

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equals 4 and w equals 1.

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And that the market demand for
coffee is given by quantity

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demanded equals 20 minus p.

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There are eight other companies
operating in this

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market and all companies have
the cost structures identical

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to Sebastian's company, the
company that we've been

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dealing with earlier
in the problem.

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Part E asks us, what
is the aggregate

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supply in this market?

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And if you look back earlier
in this problem, the other

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piece of information that we're
going to need is we're

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going to need this cost function
that gives all of the

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cost in terms of the rental rate
of capital or how much it

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cost to use each machine per
hour, the wage rate or how

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much labor cost per hour, and q,
the quantity, that's output

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by a specific firm.

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Now, to find the aggregated
supply, what we're first going

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to find is we're going to first
find the supply curve

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for one firm within
this market.

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And then we're going to set the
demand or the supply curve

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in terms of quantity
in terms of price.

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We're going to multiply by
eight to aggregate it.

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And then we'll have our
aggregated supply curve.

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But before we do that, we also
have to think of a limiting

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case as well.

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When we're representing the
costs for a firm, we're going

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to represent both the marginal
cost and the average cost. The

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marginal cost is the cost of
one single additional unit,

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while the average cost tells us
all the costs including the

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fixed costs divided by the total
that we're producing,

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what does that look like?

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If the price in a market is
below the minimum of the

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average cost of a firm in the
market, they're not going to

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produce in the market.

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So if the price is below this
critical p star, since the

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firm, even if they're producing
right at the minimum

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of average cost, they can never
recover their cost. So a

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firm is only going to produce
if this p where the price

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that's being charged is above
the p star, the minimum of

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average cost. So we're going
to find the supply curve in

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two cases, one where the price
is above this minimum of

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average cost. And two, we're
going to find it when the

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price is below that minimum.

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Let's start off by finding the
marginal cost to get our

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supply curve.

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Taking the marginal cost, the
derivative with respect to q.

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Or before we can take the
marginal cost, sorry, let's

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plug-in for the variables
w and r.

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We're going to find that our
cost curve is given by 4 plus

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4q squared.

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Now we can find the marginal
cost, which will be our supply

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curve for a single firm.

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So in most cases, we know that
this supply curve is going to

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represent the supply for a
single firm where marginal

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cost, the price, is
going to equal 8q.

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So in most cases, this will be
our supply curve for one firm.

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Putting it in terms of q,
we'll have q equals p/8.

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Now since we have eight firms,
we multiply by 8.

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And in this case, we're going
to have the aggregated

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quantity, which we represent by
a capital Q. So that's the

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quantity produced by all eight
firms in the market.

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And that's going
to equal price.

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So this is one part of
the supply curve.

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And what we need to know is,
what's the critical price at

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which this will represent
the supply curve?

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So when the average cost curve
crosses the marginal cost

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curve, that's the minimum of the
average cost. It's always

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like that for all cost curves
for a producer.

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So if we set marginal cost equal
to average cost, we find

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this critical p star at which
the firm is going to produce

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at any price above
that p star.

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So we're going to go back to our
marginal cost. And what we

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have to do is we have to find
the average cost as well to

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set it equal.

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To get the average cost, we're
just going to go back up to

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our cost curve and we're going
to divide through the whole

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thing by q.

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So the average cost is going to
be 4 divided by q plus 4q.

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Now we're going to set average
cost and marginal cost equal.

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And when we set marginal cost
and average cost equal to find

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the intersection point on our
graph, what we're going to

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find is we're going to find that
critical p star is going

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to be equal to 8.

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So Qs is going to equal p for
any price that's greater than

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or equal to 8.

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But what happens in the
case where the price

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is less than 8?

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In that case, no single firm can
make a profit by being in

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the market.

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So for any price less than 8,
the production level is going

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to be equal to 0.

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Now we're going to move
on to the next case.

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Now we're going to take the
demand curve that we're given

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and we're going to calculate
the equilibrium price, the

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aggregate quantity sold, and
the quantity sold by each

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firm, and the economic
profit of each firm.

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So let's start off in solving
this problem, we're going to

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just assume that the price,
the equilibrium price, is

00:07:14.260 --> 00:07:18.310
going to be greater than
or equal to 8.

00:07:18.310 --> 00:07:21.260
And as we're solving through the
problem, if we end up with

00:07:21.260 --> 00:07:23.980
a price that's less than 8, then
we're just going to go

00:07:23.980 --> 00:07:25.980
back and we're going to say,
OK, there's going to be no

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production, and we're done.

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But let's work with the
assumption that we're working

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with this supply curve
to begin with.

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All we have to do in this case
is we're just going to set the

00:07:36.290 --> 00:07:40.410
supply curve we just found
equal to the demand curve

00:07:40.410 --> 00:07:41.660
that's given in the problem.

00:07:48.040 --> 00:07:50.680
Solving through for p, we're
going to find that the price

00:07:50.680 --> 00:07:55.860
in this market is going
to be equal to 10.

00:07:55.860 --> 00:07:59.210
And plugging in the price back
into the demand curve, you can

00:07:59.210 --> 00:08:03.640
find that the aggregate
quantity is going

00:08:03.640 --> 00:08:04.960
to be equal to 10.

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So the price for each unit is 10
and the aggregate quantity

00:08:08.270 --> 00:08:09.660
is 10 as well.

00:08:09.660 --> 00:08:11.940
Now to find the quantity
produced by each of the eight

00:08:11.940 --> 00:08:15.650
firms, all the firms, since
they have identical cost

00:08:15.650 --> 00:08:18.630
structures, are going to be
producing the same amount.

00:08:18.630 --> 00:08:22.940
So we're just going to divide
this quantity by 8 to find the

00:08:22.940 --> 00:08:36.820
quantity produced by the
individual firms. So each firm

00:08:36.820 --> 00:08:40.600
in this case produces
5/4 of a unit.

00:08:40.600 --> 00:08:42.799
The last thing that we have to
do is we have to calculate the

00:08:42.799 --> 00:08:47.140
economic profits for each of
the single firms. So the

00:08:47.140 --> 00:08:51.910
profit is going to be equal to
the revenue, which is just

00:08:51.910 --> 00:08:56.350
price, times the quantity
for a single firm.

00:08:56.350 --> 00:08:59.580
A big mistake here would be to
use the aggregated quantity.

00:08:59.580 --> 00:09:01.310
And then you're going
to subtract out the

00:09:01.310 --> 00:09:03.070
cost for each firm.

00:09:03.070 --> 00:09:05.030
And we're just going to use
the cost function that was

00:09:05.030 --> 00:09:09.140
given after plugging
in w and r.

00:09:41.920 --> 00:09:44.670
And this is going to represent
the economic profit for each

00:09:44.670 --> 00:09:47.630
of the firms. And this leads
right into the next part of

00:09:47.630 --> 00:09:48.210
the problem.

00:09:48.210 --> 00:09:51.850
It asks us, can this be a long
run equilibrium where we have

00:09:51.850 --> 00:09:55.490
these prices, quantities,
and profits?

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And why or why not?

00:09:56.850 --> 00:09:58.760
And how will the supply
side of the market

00:09:58.760 --> 00:10:01.000
adjust in the long run?

00:10:01.000 --> 00:10:02.920
Now when other firms are
considering entering the

00:10:02.920 --> 00:10:06.070
market, the only thing that
they're going to consider is

00:10:06.070 --> 00:10:09.020
they're going to consider, is a
firm that's existing in the

00:10:09.020 --> 00:10:11.720
market currently
making profit?

00:10:11.720 --> 00:10:13.860
If they're not currently making
profit, then the firm

00:10:13.860 --> 00:10:16.140
that's considering entering
would have to have a better

00:10:16.140 --> 00:10:19.750
technology, a better way of
producing at lower cost, to

00:10:19.750 --> 00:10:21.360
enter the market and
be able to actually

00:10:21.360 --> 00:10:23.140
produce with a profit.

00:10:23.140 --> 00:10:27.960
But if there is profit being
made, in this case 2.25 for

00:10:27.960 --> 00:10:31.350
each firm, then more and more
firms are going to enter until

00:10:31.350 --> 00:10:35.000
profits are driven down to 0.

00:10:35.000 --> 00:10:38.240
So is this a long
run equilibrium?

00:10:38.240 --> 00:10:39.680
The answer is no.

00:10:39.680 --> 00:10:40.700
And why not?

00:10:40.700 --> 00:10:43.410
More firms are going to enter on
the supply side until we're

00:10:43.410 --> 00:10:45.980
driven to equilibrium.

00:10:45.980 --> 00:10:47.680
The last part that we're going
to do is we're going to do

00:10:47.680 --> 00:10:52.180
part H. Part H asks us, what
is going to be the price in

00:10:52.180 --> 00:10:53.280
the long run?

00:10:53.280 --> 00:10:54.940
How many firms will
be present in this

00:10:54.940 --> 00:10:56.450
market in the long run?

00:10:56.450 --> 00:10:59.300
And how much will each
firm produce?

00:10:59.300 --> 00:11:01.700
Now in the long run, we know
that profits are going to be

00:11:01.700 --> 00:11:05.170
driven down to 0, and that each
firm is going to have to

00:11:05.170 --> 00:11:06.250
be leaner and meaner.

00:11:06.250 --> 00:11:09.110
They're going to have to produce
at maximum efficiency

00:11:09.110 --> 00:11:11.160
to be competitive within
the market.

00:11:11.160 --> 00:11:15.360
So if we go back to the graph
that we started with, we can

00:11:15.360 --> 00:11:20.730
look at, at which point are
firms operating optimally?

00:11:20.730 --> 00:11:24.400
It's when marginal cost is equal
to average cost. It's at

00:11:24.400 --> 00:11:27.120
this critical p star that
we calculated to

00:11:27.120 --> 00:11:29.820
be equal to 8 earlier.

00:11:29.820 --> 00:11:33.220
So in the long run, all the
firms are going to have to be

00:11:33.220 --> 00:11:34.090
lean and mean.

00:11:34.090 --> 00:11:36.540
They're going to have to operate
at a very low average

00:11:36.540 --> 00:11:39.070
cost. And we're going to
calculate how many firms are

00:11:39.070 --> 00:11:41.730
going to be in the market
producing at this point where

00:11:41.730 --> 00:11:46.020
marginal cost intersects
average cost.

00:11:46.020 --> 00:11:50.600
So to start off H, we know that
p is going to be equal to

00:11:50.600 --> 00:11:56.650
the minimum of average cost,
which we calculated in one of

00:11:56.650 --> 00:12:00.180
the earlier parts of the problem
to be equal to 8.

00:12:00.180 --> 00:12:04.740
When we have this equal to 8,
we can plug-in to our demand

00:12:04.740 --> 00:12:09.330
curve that price.

00:12:09.330 --> 00:12:11.810
And we can find that the
quantity demanded is going to

00:12:11.810 --> 00:12:21.250
be equal to 12.

00:12:21.250 --> 00:12:24.700
Now what we can do now is we can
go back and we can look at

00:12:24.700 --> 00:12:29.370
the individual supply
curve for each firm.

00:12:29.370 --> 00:12:32.120
And each firm, when we had our
individual supply curves that

00:12:32.120 --> 00:12:35.470
we calculated in the first part
of the problem, had a

00:12:35.470 --> 00:12:42.370
supply function that was equal
to q equals p divided by 8.

00:12:42.370 --> 00:12:45.240
And in this case, if we know
that the price when the firms

00:12:45.240 --> 00:12:52.070
are operating optimally is equal
to 8, then we know that

00:12:52.070 --> 00:12:54.980
each firm is going
to be producing

00:12:54.980 --> 00:12:58.180
one unit of the good.

00:12:58.180 --> 00:13:00.630
So to find the total number of
firms, we just have to take

00:13:00.630 --> 00:13:03.880
this aggregated amount, the
total amount that's being

00:13:03.880 --> 00:13:07.000
produced, and divide through
by the amount each firm is

00:13:07.000 --> 00:13:09.050
producing to find out the number
of firms that have to

00:13:09.050 --> 00:13:10.300
be producing.

00:13:31.080 --> 00:13:33.930
So in the long run, we're going
to have 12 firms each

00:13:33.930 --> 00:13:37.960
producing one unit at a price
of 8, which is the optimal

00:13:37.960 --> 00:13:41.170
price where marginal cost is
equal to the minimum of the

00:13:41.170 --> 00:13:44.870
average cost. So just to
summarize what this problem

00:13:44.870 --> 00:13:49.180
had us look at, we looked at the
case where we had instead

00:13:49.180 --> 00:13:50.990
of just one firm, we
had multiple firms

00:13:50.990 --> 00:13:52.450
operating in a market.

00:13:52.450 --> 00:13:55.740
We saw that when we have
multiple firms operating that

00:13:55.740 --> 00:13:58.350
if there's any economic profit
that more firms are going to

00:13:58.350 --> 00:14:01.930
enter until the firms that exist
in the market are forced

00:14:01.930 --> 00:14:05.420
to operate optimally with
no economic profit.

00:14:05.420 --> 00:14:07.300
I hope that you found this
problem helpful.

00:14:07.300 --> 00:14:10.320
And go ahead and again, you can
do the earlier parts, and

00:14:10.320 --> 00:14:12.980
you can look to PSET 4
Problem Number 3 for

00:14:12.980 --> 00:14:14.730
help on those problems.