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GREG HUTKO: Hi, welcome
back to the 14.01

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00:00:23,560 --> 00:00:25,110
problem-solving videos.

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00:00:25,110 --> 00:00:29,370
Today I'm going to be working
on Fall 2010 Problem Set 5,

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Problem Number 4.

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00:00:30,990 --> 00:00:34,160
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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00:00:41,810 --> 00:00:43,280
Problem Number 3.

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00:00:43,280 --> 00:00:46,440
And in that problem, we work
through the production

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function, and we go through and
we find conditional supply

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and the conditional
demand curves.

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00:00:51,700 --> 00:00:54,420
But this problem is going to
have us looking at aggregated

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00:00:54,420 --> 00:00:56,390
supply in a market.

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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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00:01:28,850 --> 00:01:30,930
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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00:01:34,730 --> 00:01:37,230
to Sebastian's company, the
company that we've been

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dealing with earlier
in the problem.

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00:01:39,340 --> 00:01:41,370
Part E asks us, what
is the aggregate

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00:01:41,370 --> 00:01:43,250
supply in this market?

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00:01:43,250 --> 00:01:46,220
And if you look back earlier
in this problem, the other

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00:01:46,220 --> 00:01:47,910
piece of information that we're
going to need is we're

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00:01:47,910 --> 00:01:51,130
going to need this cost function
that gives all of the

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00:01:51,130 --> 00:01:54,980
cost in terms of the rental rate
of capital or how much it

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00:01:54,980 --> 00:01:58,740
cost to use each machine per
hour, the wage rate or how

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00:01:58,740 --> 00:02:02,990
much labor cost per hour, and q,
the quantity, that's output

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00:02:02,990 --> 00:02:05,920
by a specific firm.

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00:02:05,920 --> 00:02:08,820
Now, to find the aggregated
supply, what we're first going

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00:02:08,820 --> 00:02:11,290
to find is we're going to first
find the supply curve

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00:02:11,290 --> 00:02:13,380
for one firm within
this market.

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00:02:13,380 --> 00:02:19,500
And then we're going to set the
demand or the supply curve

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00:02:19,500 --> 00:02:21,530
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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00:03:17,150 --> 00:03:20,880
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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00:03:36,330 --> 00:03:37,710
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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00:04:00,190 --> 00:04:01,490
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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00:04:28,350 --> 00:04:35,770
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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00:05:01,370 --> 00:05:04,980
And what we need to know is,
what's the critical price at

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00:05:04,980 --> 00:05:07,890
which this will represent
the supply curve?

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00:05:07,890 --> 00:05:14,290
So when the average cost curve
crosses the marginal cost

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00:05:14,290 --> 00:05:18,560
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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00:05:27,670 --> 00:05:31,840
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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00:05:34,230 --> 00:05:36,860
So we're going to go back to our
marginal cost. And what we

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00:05:36,860 --> 00:05:39,250
have to do is we have to find
the average cost as well to

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00:05:39,250 --> 00:05:41,520
set it equal.

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00:05:41,520 --> 00:05:43,980
To get the average cost, we're
just going to go back up to

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00:05:43,980 --> 00:05:46,970
our cost curve and we're going
to divide through the whole

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00:05:46,970 --> 00:05:48,660
thing by q.

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00:05:48,660 --> 00:05:54,600
So the average cost is going to
be 4 divided by q plus 4q.

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00:05:57,260 --> 00:06:00,210
Now we're going to set average
cost and marginal cost equal.

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00:06:03,300 --> 00:06:06,080
And when we set marginal cost
and average cost equal to find

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00:06:06,080 --> 00:06:08,830
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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00:06:17,910 --> 00:06:24,680
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

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going to be greater than
or equal to 8.

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00:07:18,310 --> 00:07:21,260
And as we're solving through the
problem, if we end up with

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a price that's less than 8, then
we're just going to go

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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

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supply curve we just found
equal to the demand curve

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00:07:40,410 --> 00:07:41,660
that's given in the problem.

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Solving through for p, we're
going to find that the price

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00:07:50,680 --> 00:07:55,860
in this market is going
to be equal to 10.

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00:07:55,860 --> 00:07:59,210
And plugging in the price back
into the demand curve, you can

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00:07:59,210 --> 00:08:03,640
find that the aggregate
quantity is going

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00:08:03,640 --> 00:08:04,960
to be equal to 10.

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00:08:04,960 --> 00:08:08,270
So the price for each unit is 10
and the aggregate quantity

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00:08:08,270 --> 00:08:09,660
is 10 as well.

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00:08:09,660 --> 00:08:11,940
Now to find the quantity
produced by each of the eight

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00:08:11,940 --> 00:08:15,650
firms, all the firms, since
they have identical cost

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00:08:15,650 --> 00:08:18,630
structures, are going to be
producing the same amount.

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00:08:18,630 --> 00:08:22,940
So we're just going to divide
this quantity by 8 to find the

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00:08:22,940 --> 00:08:36,820
quantity produced by the
individual firms. So each firm

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00:08:36,820 --> 00:08:40,600
in this case produces
5/4 of a unit.

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00:08:40,600 --> 00:08:42,799
The last thing that we have to
do is we have to calculate the

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00:08:42,799 --> 00:08:47,140
economic profits for each of
the single firms. So the

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00:08:47,140 --> 00:08:51,910
profit is going to be equal to
the revenue, which is just

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00:08:51,910 --> 00:08:56,350
price, times the quantity
for a single firm.

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A big mistake here would be to
use the aggregated quantity.

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00:08:59,580 --> 00:09:01,310
And then you're going
to subtract out the

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00:09:01,310 --> 00:09:03,070
cost for each firm.

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00:09:03,070 --> 00:09:05,030
And we're just going to use
the cost function that was

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00:09:05,030 --> 00:09:09,140
given after plugging
in w and r.

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00:09:41,920 --> 00:09:44,670
And this is going to represent
the economic profit for each

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00:09:44,670 --> 00:09:47,630
of the firms. And this leads
right into the next part of

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00:09:47,630 --> 00:09:48,210
the problem.

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00:09:48,210 --> 00:09:51,850
It asks us, can this be a long
run equilibrium where we have

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these prices, quantities,
and profits?

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00:09:55,490 --> 00:09:56,850
And why or why not?

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00:09:56,850 --> 00:09:58,760
And how will the supply
side of the market

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00:09:58,760 --> 00:10:01,000
adjust in the long run?

168
00:10:01,000 --> 00:10:02,920
Now when other firms are
considering entering the

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00:10:02,920 --> 00:10:06,070
market, the only thing that
they're going to consider is

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00:10:06,070 --> 00:10:09,020
they're going to consider, is a
firm that's existing in the

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00:10:09,020 --> 00:10:11,720
market currently
making profit?

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00:10:11,720 --> 00:10:13,860
If they're not currently making
profit, then the firm

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00:10:13,860 --> 00:10:16,140
that's considering entering
would have to have a better

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00:10:16,140 --> 00:10:19,750
technology, a better way of
producing at lower cost, to

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00:10:19,750 --> 00:10:21,360
enter the market and
be able to actually

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00:10:21,360 --> 00:10:23,140
produce with a profit.

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00:10:23,140 --> 00:10:27,960
But if there is profit being
made, in this case 2.25 for

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00:10:27,960 --> 00:10:31,350
each firm, then more and more
firms are going to enter until

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00:10:31,350 --> 00:10:35,000
profits are driven down to 0.

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00:10:35,000 --> 00:10:38,240
So is this a long
run equilibrium?

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00:10:38,240 --> 00:10:39,680
The answer is no.

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And why not?

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00:10:40,700 --> 00:10:43,410
More firms are going to enter on
the supply side until we're

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driven to equilibrium.

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00:10:45,980 --> 00:10:47,680
The last part that we're going
to do is we're going to do

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00:10:47,680 --> 00:10:52,180
part H. Part H asks us, what
is going to be the price in

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00:10:52,180 --> 00:10:53,280
the long run?

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00:10:53,280 --> 00:10:54,940
How many firms will
be present in this

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00:10:54,940 --> 00:10:56,450
market in the long run?

190
00:10:56,450 --> 00:10:59,300
And how much will each
firm produce?

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00:10:59,300 --> 00:11:01,700
Now in the long run, we know
that profits are going to be

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00:11:01,700 --> 00:11:05,170
driven down to 0, and that each
firm is going to have to

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00:11:05,170 --> 00:11:06,250
be leaner and meaner.

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00:11:06,250 --> 00:11:09,110
They're going to have to produce
at maximum efficiency

195
00:11:09,110 --> 00:11:11,160
to be competitive within
the market.

196
00:11:11,160 --> 00:11:15,360
So if we go back to the graph
that we started with, we can

197
00:11:15,360 --> 00:11:20,730
look at, at which point are
firms operating optimally?

198
00:11:20,730 --> 00:11:24,400
It's when marginal cost is equal
to average cost. It's at

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00:11:24,400 --> 00:11:27,120
this critical p star that
we calculated to

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00:11:27,120 --> 00:11:29,820
be equal to 8 earlier.

201
00:11:29,820 --> 00:11:33,220
So in the long run, all the
firms are going to have to be

202
00:11:33,220 --> 00:11:34,090
lean and mean.

203
00:11:34,090 --> 00:11:36,540
They're going to have to operate
at a very low average

204
00:11:36,540 --> 00:11:39,070
cost. And we're going to
calculate how many firms are

205
00:11:39,070 --> 00:11:41,730
going to be in the market
producing at this point where

206
00:11:41,730 --> 00:11:46,020
marginal cost intersects
average cost.

207
00:11:46,020 --> 00:11:50,600
So to start off H, we know that
p is going to be equal to

208
00:11:50,600 --> 00:11:56,650
the minimum of average cost,
which we calculated in one of

209
00:11:56,650 --> 00:12:00,180
the earlier parts of the problem
to be equal to 8.

210
00:12:00,180 --> 00:12:04,740
When we have this equal to 8,
we can plug-in to our demand

211
00:12:04,740 --> 00:12:09,330
curve that price.

212
00:12:09,330 --> 00:12:11,810
And we can find that the
quantity demanded is going to

213
00:12:11,810 --> 00:12:21,250
be equal to 12.

214
00:12:21,250 --> 00:12:24,700
Now what we can do now is we can
go back and we can look at

215
00:12:24,700 --> 00:12:29,370
the individual supply
curve for each firm.

216
00:12:29,370 --> 00:12:32,120
And each firm, when we had our
individual supply curves that

217
00:12:32,120 --> 00:12:35,470
we calculated in the first part
of the problem, had a

218
00:12:35,470 --> 00:12:42,370
supply function that was equal
to q equals p divided by 8.

219
00:12:42,370 --> 00:12:45,240
And in this case, if we know
that the price when the firms

220
00:12:45,240 --> 00:12:52,070
are operating optimally is equal
to 8, then we know that

221
00:12:52,070 --> 00:12:54,980
each firm is going
to be producing

222
00:12:54,980 --> 00:12:58,180
one unit of the good.

223
00:12:58,180 --> 00:13:00,630
So to find the total number of
firms, we just have to take

224
00:13:00,630 --> 00:13:03,880
this aggregated amount, the
total amount that's being

225
00:13:03,880 --> 00:13:07,000
produced, and divide through
by the amount each firm is

226
00:13:07,000 --> 00:13:09,050
producing to find out the number
of firms that have to

227
00:13:09,050 --> 00:13:10,300
be producing.

228
00:13:31,080 --> 00:13:33,930
So in the long run, we're going
to have 12 firms each

229
00:13:33,930 --> 00:13:37,960
producing one unit at a price
of 8, which is the optimal

230
00:13:37,960 --> 00:13:41,170
price where marginal cost is
equal to the minimum of the

231
00:13:41,170 --> 00:13:44,870
average cost. So just to
summarize what this problem

232
00:13:44,870 --> 00:13:49,180
had us look at, we looked at the
case where we had instead

233
00:13:49,180 --> 00:13:50,990
of just one firm, we
had multiple firms

234
00:13:50,990 --> 00:13:52,450
operating in a market.

235
00:13:52,450 --> 00:13:55,740
We saw that when we have
multiple firms operating that

236
00:13:55,740 --> 00:13:58,350
if there's any economic profit
that more firms are going to

237
00:13:58,350 --> 00:14:01,930
enter until the firms that exist
in the market are forced

238
00:14:01,930 --> 00:14:05,420
to operate optimally with
no economic profit.

239
00:14:05,420 --> 00:14:07,300
I hope that you found this
problem helpful.

240
00:14:07,300 --> 00:14:10,320
And go ahead and again, you can
do the earlier parts, and

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you can look to PSET 4
Problem Number 3 for

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help on those problems.