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GREG HUTKO: 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 PSET 8,

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Problem Number 2.

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And we've seen the case with
the monopolist where we've

00:00:33.610 --> 00:00:36.290
only had one producer in a
market, and we've seen how

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that affects consumer surplus,
produce surplus, and

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deadweight loss.

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Now we're going to look
at the case where we

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have two firms competing.

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And specifically, for the first
part, we're going to

00:00:46.010 --> 00:00:48.360
have the Cournot equilibrium,
where they're going to be

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competing not by setting the
price of their product.

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Instead they're going to compete
by setting how much

00:00:53.170 --> 00:00:55.520
they're going to produce.

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Let's go ahead and read part
A of this problem.

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Consider a market in which
two firms produce

00:01:00.610 --> 00:01:02.320
a homogeneous product.

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Market demand is given by
quantity equals 200 minus p.

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The cost functions for Firm A
and Firm B, the total cost for

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A equals 5qA and the total cost
for b is going to equal

00:01:15.860 --> 00:01:19.680
1/2 qB squared, respectively.

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Find the Cournot equilibrium
quantity

00:01:21.490 --> 00:01:22.940
supplied by each firm.

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We're going to graph our
results using reaction

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functions and we're going to
find the market price, and

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then calculate the profits
for each firm.

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Now in the duopoly model, both
firms in the Cournot

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equilibrium are going
to set their

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quantities at the same time.

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But what they're going to do is
they're going to know what

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each other's revenues
and costs look like.

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So they can know, I know that
if I make this move and set

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this quantity, I know that my
competition is going to have

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this reaction.

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Since they can plan for each
other's reaction, they can

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decide how much quantity they're
going to produce at

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the same time and they're going
to reach an equilibrium.

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What this looks like on our
reaction curves, up here we're

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going to have qA.

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And down here is
going to be qB.

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We're going to graph the
reaction function for Firm A

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first. And we're now going to
graph the reaction function

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for Firm B.

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Now all this tells us is if
we're at this point, if we're

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Firm A and we decide to produce
this much, then we

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know that Firm B's best reaction
is going to produce

00:02:47.040 --> 00:02:50.410
this quantity here, the
intersection of how much I'm

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producing with their
reaction curve.

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Similarly, if Firm B is going
to decide to produce this

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much, then my best reaction
going over to Firm A's

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reaction function straight
across, is going to be to

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produce this much.

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Now since they're both choosing
at the same time,

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neither of the firms
can decide to

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declare a larger amount.

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And they have to kind of use
their intuition to determine

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what they're going to produce.

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So since they're choosing at
the same time, they have to

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plan for each other's
reactions.

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They're going to produce at
this point where the two

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reaction curves meet.

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And that's what we're going
to be calculating.

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We're going to be calculating
the quantity that Firm A

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produces and the quantity that
Firm B produces when the

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reaction curves are set
equal to each other.

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Now to get the reaction curves,
we're going to start

00:03:44.230 --> 00:03:48.490
off with our revenue function
for both Firm A and Firm B.

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And revenue is just going to
be the price times the

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quantity that either Firm A
or Firm B is producing.

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Now instead of saying that we
are just going to take the

00:03:59.170 --> 00:04:04.410
marginal revenue according to
this P times Q, we're going to

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plug-in for P from the demand
curve 200 minus a

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disaggregated quantity where
we disaggregate into the

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amount Firm A is producing
and the

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amount Firm B is producing.

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We're going to plug this in
to the revenue function.

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And so just like the monopolist
did where they're

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maximizing by setting marginal
revenue equal to marginal

00:04:24.350 --> 00:04:27.800
cost, Firm A and Firm B we're
going to do the same thing.

00:04:27.800 --> 00:04:29.840
We're going to set marginal
revenue equal to marginal

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cost. And then we're going to
solve through for qA in terms

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of the quantity that the other
firm is producing.

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That's why it's considered a
reaction curve because it's in

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terms of what the other
firm is producing.

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So solving through for Firm A's
marginal revenue, we're

00:04:50.010 --> 00:04:53.690
going to find that marginal
revenue for Firm A is equal to

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200 minus 2qA minus qB.

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And it makes sense that the more
that they're producing,

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Firm A and Firm B, the lower the
revenue that they're going

00:05:08.190 --> 00:05:10.320
to be taking in.

00:05:10.320 --> 00:05:12.410
Now we're going to also
calculate the marginal cost

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for Firm A by taking the
derivative with respect to the

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total cost function.

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We're going to find that the
marginal cost is going to be

00:05:19.980 --> 00:05:21.560
equal to 5.

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Now we're just going to set
the marginal cost and the

00:05:25.730 --> 00:05:28.400
marginal revenue for
Firm A equal.

00:05:28.400 --> 00:05:32.850
And we're going to solve
through for qA.

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When we do that, we're going
to find that qA is equal to

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97.5 minus 0.5 qB.

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And we're going to repeat this
exact same process for Firm B.

00:05:50.410 --> 00:05:52.410
But when we do it for Firm
B, instead of taking the

00:05:52.410 --> 00:05:54.570
derivative with respect to qA,
we're going to take the

00:05:54.570 --> 00:05:56.940
derivative with respect to qB.

00:05:59.540 --> 00:06:04.810
So the marginal revenue for Firm
B is going to be equal to

00:06:04.810 --> 00:06:12.520
200 minus qA minus 2 qB.

00:06:12.520 --> 00:06:19.695
And the marginal cost is just
going to be equal to qB.

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Again, we're going to set
marginal cost and marginal

00:06:26.540 --> 00:06:27.910
revenue equal to each other.

00:06:27.910 --> 00:06:31.630
And now we're going to
solve through for qB.

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And when we do this, we're
going to have Firm B's

00:06:35.590 --> 00:06:36.840
reaction curve.

00:06:52.240 --> 00:06:54.110
And now what we're going to
do since we have these two

00:06:54.110 --> 00:06:57.220
reaction curves, we have a
reaction for Firm A and a

00:06:57.220 --> 00:07:00.330
reaction for Firm B, all we're
going to do is we're going to

00:07:00.330 --> 00:07:06.140
plug-in for this qB, qB's
reaction curve.

00:07:06.140 --> 00:07:11.570
And when we do that, when we
plug-in 66.67 minus 0.33qA, we

00:07:11.570 --> 00:07:14.840
can solve through for just qA.

00:07:14.840 --> 00:07:18.180
Doing this we're going to find
that Firm A is going to

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produce approximately
77 units.

00:07:23.450 --> 00:07:27.080
And then taking this 77 and
plugging it in to Firm B's

00:07:27.080 --> 00:07:29.920
reaction function, we
can solve for Firm

00:07:29.920 --> 00:07:31.170
B's production amount.

00:07:34.200 --> 00:07:36.270
And we're going to find that
Firm B is going to produce

00:07:36.270 --> 00:07:41.380
approximately 41 units.

00:07:41.380 --> 00:07:43.530
Now to find the equilibrium
price, we're just going to

00:07:43.530 --> 00:07:47.470
come back up here to our
disaggregated demand function,

00:07:47.470 --> 00:07:50.460
and we're going to plug-in
for qA and qB.

00:07:50.460 --> 00:08:01.360
And we can solve through for the
price being equal to 82.

00:08:01.360 --> 00:08:06.050
Now what we've just done for
this problem is we solved on

00:08:06.050 --> 00:08:09.810
our graph for the intersection
of the two reaction functions.

00:08:09.810 --> 00:08:14.920
We found that qB is going to
be equal to 41 and we found

00:08:14.920 --> 00:08:18.790
that qA is going to
be equal to 77.

00:08:18.790 --> 00:08:23.500
So we just calculated the
intersection point for the

00:08:23.500 --> 00:08:25.500
Cournot equilibrium.

00:08:25.500 --> 00:08:27.940
Now the last part of this
problem asks us to calculate

00:08:27.940 --> 00:08:35.330
the profits for both the firms.
The profits for Firm A

00:08:35.330 --> 00:08:40.299
are just going to be price times
the quantity that A is

00:08:40.299 --> 00:08:49.130
producing minus the cost
as a function of qA.

00:08:49.130 --> 00:08:51.760
So we're just going to take the
total revenues minus the

00:08:51.760 --> 00:08:53.610
total costs.

00:08:53.610 --> 00:09:02.990
For Firm A, we're going to find
that the total profits

00:09:02.990 --> 00:09:06.350
are going to be about $5,929.

00:09:06.350 --> 00:09:10.530
And doing the same process for
Firm B, we can find that the

00:09:10.530 --> 00:09:13.960
profits for Firm B are
going to be equal to

00:09:13.960 --> 00:09:23.470
approximately $2,521.

00:09:23.470 --> 00:09:26.670
Now part B of this problem is
going to ask us instead of

00:09:26.670 --> 00:09:29.670
having this Cournot equilibrium
where neither firm

00:09:29.670 --> 00:09:32.670
can go ahead and produce a
higher quantity or move first

00:09:32.670 --> 00:09:34.960
in the market, we're going to
look at something different

00:09:34.960 --> 00:09:36.610
than the Cournot equilibrium.

00:09:36.610 --> 00:09:38.460
We're going to look at the case
where one of the firms

00:09:38.460 --> 00:09:40.570
gets to decide how much they're
going to produce

00:09:40.570 --> 00:09:42.080
before the other firm.

00:09:42.080 --> 00:09:45.010
And if you get to decide first,
you get to produce a

00:09:45.010 --> 00:09:48.100
higher quantity and get
more of the profits.

00:09:48.100 --> 00:09:51.450
Part B says, now suppose that
Firm A chooses how much to

00:09:51.450 --> 00:09:54.640
produce before firm B does.

00:09:54.640 --> 00:09:56.580
In this case, Firm A
is a Stackelberg

00:09:56.580 --> 00:09:58.470
leader and B a follower.

00:09:58.470 --> 00:10:01.220
We're going to calculate the
quantities, the market price,

00:10:01.220 --> 00:10:04.090
and the profit for each firm.

00:10:04.090 --> 00:10:07.920
Now coming over to this side of
the board, we see that I'm

00:10:07.920 --> 00:10:10.960
going to keep Firm B's reaction
function the same.

00:10:10.960 --> 00:10:13.930
So Firm B is going to be
reacting in the same way to

00:10:13.930 --> 00:10:15.730
Firm A's decision.

00:10:15.730 --> 00:10:18.770
Only now, the only difference is
when we calculate marginal

00:10:18.770 --> 00:10:24.940
revenue equal the marginal cost
for Firm A, instead of

00:10:24.940 --> 00:10:27.390
just saying the qB is going
to be random, we

00:10:27.390 --> 00:10:29.760
can't account for it.

00:10:29.760 --> 00:10:32.320
We're going to plug-in, we're
going to take into account

00:10:32.320 --> 00:10:36.350
Firm B's reaction when we're
maximizing or taking the

00:10:36.350 --> 00:10:38.720
derivative with respect to qA.

00:10:38.720 --> 00:10:41.440
So instead of having qB in here,
I'm going to plug-in

00:10:41.440 --> 00:10:43.350
this reaction function.

00:10:43.350 --> 00:10:45.020
So in this case, the equation
that I'm going to be

00:10:45.020 --> 00:11:06.730
maximizing is going to be
this one right here.

00:11:06.730 --> 00:11:11.480
I'm going to take the derivative
with respect to qA

00:11:11.480 --> 00:11:28.550
to find the marginal
revenue for A.

00:11:28.550 --> 00:11:30.890
And again, I'm just going to set
this equal to the marginal

00:11:30.890 --> 00:11:34.070
cost, which we found earlier
is equal to 5.

00:11:38.090 --> 00:11:40.460
And when we set these equal, we
can solve through for the

00:11:40.460 --> 00:11:42.960
quantity that Firm A is
going to produce.

00:11:42.960 --> 00:11:44.920
And we're going to find just
like we predicted that the

00:11:44.920 --> 00:11:46.310
leader is going to
produce more.

00:11:53.390 --> 00:11:55.160
In this case, Firm A
has increased their

00:11:55.160 --> 00:11:58.270
production to 96.25.

00:11:58.270 --> 00:12:02.610
And then plugging in this
quantity in to Firm B's

00:12:02.610 --> 00:12:09.410
reaction function, we can find
that Firm B in this case, is

00:12:09.410 --> 00:12:17.330
going to produce 34.6.

00:12:17.330 --> 00:12:20.840
Now in this case, we can again
calculate the price by taking

00:12:20.840 --> 00:12:27.410
the demand function that we
have. We can take the demand

00:12:27.410 --> 00:12:28.680
function that we're given
in the problem.

00:12:36.170 --> 00:12:40.240
Plugging in for qA and qB, we
find that the new price in the

00:12:40.240 --> 00:12:51.460
Stackelberg problem is
going to be 69.15.

00:12:51.460 --> 00:12:54.510
And again, we can calculate the
profits going through the

00:12:54.510 --> 00:12:58.270
same process of doing total
revenue minus total cost. And

00:12:58.270 --> 00:13:02.110
we're going to have that the
profit for Firm A is going to

00:13:02.110 --> 00:13:09.450
be equal to about 6,174.

00:13:09.450 --> 00:13:12.840
And the profit for Firm
B is going to be

00:13:12.840 --> 00:13:20.810
equal to about 1,794.

00:13:20.810 --> 00:13:23.480
And so what we can do here is
we can compare the profits

00:13:23.480 --> 00:13:26.660
that we had in the Stackelberg
case to the profits that we

00:13:26.660 --> 00:13:28.440
had at the start
of our problem.

00:13:58.600 --> 00:14:02.490
So before we can see that Firm
A was not as profitable when

00:14:02.490 --> 00:14:04.750
they had to choose their
quantity at the same time as

00:14:04.750 --> 00:14:08.360
Firm B. We can see that their
profits have increased.

00:14:08.360 --> 00:14:11.880
But we can see that Firm B,
their profits have actually

00:14:11.880 --> 00:14:13.630
decreased because they're
a follower in

00:14:13.630 --> 00:14:15.690
the Stackelberg model.

00:14:15.690 --> 00:14:17.560
Now the last thing, and the
thought I want to leave you

00:14:17.560 --> 00:14:20.680
with is, how do we actually
interpret this when we look at

00:14:20.680 --> 00:14:23.010
the reaction functions
on our graph?

00:14:23.010 --> 00:14:25.420
We're no longer at the point
where we're setting the two

00:14:25.420 --> 00:14:27.590
reaction functions equal.

00:14:27.590 --> 00:14:34.320
What's happening now is we're
way up here and qA is choosing

00:14:34.320 --> 00:14:37.840
their production way up here.

00:14:37.840 --> 00:14:44.770
And qB is forced to react by
choosing their production

00:14:44.770 --> 00:14:46.510
right here.

00:14:46.510 --> 00:14:49.040
And what happens is since they
both increased their

00:14:49.040 --> 00:14:53.760
production or since qA has
increased their production and

00:14:53.760 --> 00:14:56.110
qB has decreased their
production, but since

00:14:56.110 --> 00:15:01.610
production has increased
overall, the price has dropped

00:15:01.610 --> 00:15:03.720
compared to when they were at
the Cournot equilibrium.

00:15:03.720 --> 00:15:07.250
So total profits have actually
dropped as well.

00:15:07.250 --> 00:15:09.370
So really what the first two
parts of these problems were

00:15:09.370 --> 00:15:11.240
having us look at, they were
looking at two different

00:15:11.240 --> 00:15:13.520
situations of duopoly
where we have two

00:15:13.520 --> 00:15:15.190
competitors in the market.

00:15:15.190 --> 00:15:16.600
The first one they were
choosing their

00:15:16.600 --> 00:15:18.220
outputs at the same time.

00:15:18.220 --> 00:15:20.750
And in the second problem, one
of the firms had the advantage

00:15:20.750 --> 00:15:23.990
of getting to choose a higher
quantity and making a credible

00:15:23.990 --> 00:15:27.050
threat that they were
going to make that

00:15:27.050 --> 00:15:29.050
quantity to begin with.

00:15:29.050 --> 00:15:31.200
For the last parts of these
problems, you're going to go

00:15:31.200 --> 00:15:34.190
ahead and you can look at what
the implications are when we

00:15:34.190 --> 00:15:36.920
think about what the total
quantity is produced in a

00:15:36.920 --> 00:15:40.420
competitive market aggregating
the supplies of these two

00:15:40.420 --> 00:15:43.940
firms. And then you can compare
the output in the

00:15:43.940 --> 00:15:45.390
three different scenarios.

00:15:45.390 --> 00:15:47.150
But for now, I'm going
to leave you here.

00:15:47.150 --> 00:15:49.040
Go ahead and finish the rest of
the problem, and I hope you

00:15:49.040 --> 00:15:50.290
found this part helpful.