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PROFESSOR: Hi, and welcome
back to the 14.01 problem

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00:00:24,990 --> 00:00:25,990
solving videos.

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00:00:25,990 --> 00:00:29,620
Today, we're going to work on
Fall 2010, problem set one,

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00:00:29,620 --> 00:00:31,250
problem number four.

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00:00:31,250 --> 00:00:33,070
And in this problem, we're
going to be working with

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elasticities.

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00:00:34,370 --> 00:00:36,580
But instead of starting with a
demand function, and starting

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with a supply function, and
calculating the elasticity

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00:00:39,960 --> 00:00:42,180
given those functions, we're
going to be given the

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00:00:42,180 --> 00:00:44,970
elasticity of demand and the
elasticity of supply.

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00:00:44,970 --> 00:00:46,720
And we're going to have to
back out what the demand

21
00:00:46,720 --> 00:00:48,950
functions and the
supply functions

22
00:00:48,950 --> 00:00:49,740
should have looked like.

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00:00:49,740 --> 00:00:51,840
So we're basically just working
in reverse from what

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00:00:51,840 --> 00:00:53,570
we did in lecture.

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00:00:53,570 --> 00:00:55,850
Let's go ahead and read the full
problem up through part

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00:00:55,850 --> 00:01:00,040
A. You have been asked to
analyze the market for steel.

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00:01:00,040 --> 00:01:02,490
From public sources, you are
able to find that last year's

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00:01:02,490 --> 00:01:05,120
price for steel was
$20 per ton.

29
00:01:05,120 --> 00:01:07,410
At this price, 100 million
tons were sold

30
00:01:07,410 --> 00:01:08,750
on the world market.

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00:01:08,750 --> 00:01:11,620
From trade association data,
you are able to obtain

32
00:01:11,620 --> 00:01:15,030
estimates for their own price
elasticities of demand and

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00:01:15,030 --> 00:01:19,610
supply on the world markets as
negative 0.25 for demand and

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00:01:19,610 --> 00:01:21,610
0.5 for supply.

35
00:01:21,610 --> 00:01:24,800
Assume the steel has linear
demand and supply curves

36
00:01:24,800 --> 00:01:25,960
throughout.

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00:01:25,960 --> 00:01:29,010
Part A asks us to solve for the
equations of demand and

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00:01:29,010 --> 00:01:31,500
supply in this market,
and to sketch the

39
00:01:31,500 --> 00:01:34,010
demand and supply curves.

40
00:01:34,010 --> 00:01:37,660
So looking at the formal
definition of elasticity of

41
00:01:37,660 --> 00:01:41,170
demand and elasticity of supply,
we basically are going

42
00:01:41,170 --> 00:01:43,500
to have three different
parts to it.

43
00:01:43,500 --> 00:01:46,360
We have the derivative of
either demand or supply

44
00:01:46,360 --> 00:01:50,180
function with respect to P, in
this case the own price of P,

45
00:01:50,180 --> 00:01:52,300
or the price of steel.

46
00:01:52,300 --> 00:01:54,830
And we also have the equilibrium
price, or any

47
00:01:54,830 --> 00:01:57,270
price at the point on the
curve, and a quantity.

48
00:01:57,270 --> 00:02:00,420
In this case, it's going to be
the equilibrium quantity.

49
00:02:00,420 --> 00:02:04,250
So basically, what we have
now is we are given--

50
00:02:04,250 --> 00:02:07,080
for the elasticity of demand,
we're given three variables.

51
00:02:07,080 --> 00:02:09,810
We're given the price,
the quantity, and the

52
00:02:09,810 --> 00:02:11,390
elasticity of demand.

53
00:02:11,390 --> 00:02:15,120
And that means the only thing
that we don't know is the

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00:02:15,120 --> 00:02:19,260
derivative of the demand curve
with respect to P.

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00:02:19,260 --> 00:02:22,380
So if we can isolate this
derivative, then we can

56
00:02:22,380 --> 00:02:25,030
integrate the number
that we're able to

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00:02:25,030 --> 00:02:26,280
solve through for.

58
00:02:26,280 --> 00:02:28,600
And then we can solve out for
what our demand curve is going

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00:02:28,600 --> 00:02:29,510
to look like.

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00:02:29,510 --> 00:02:32,230
So let's go ahead and walk
through that process together.

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00:02:32,230 --> 00:02:49,510
Substituting in for the
elasticity of demand P and Q,

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00:02:49,510 --> 00:02:51,050
we're gonna have
this equation.

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00:02:51,050 --> 00:02:53,830
And the one thing that I want
you to notice is since the

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00:02:53,830 --> 00:02:58,280
derivative of the demand curve
with respect to P is negative

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00:02:58,280 --> 00:03:01,670
0.25, in this case, we know
that it's linear.

66
00:03:01,670 --> 00:03:04,400
But just because it's linear at
the point where price is 20

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and quantity is 100, that
doesn't necessarily mean it's

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00:03:07,620 --> 00:03:08,880
gonna be linear throughout.

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00:03:08,880 --> 00:03:12,780
So it's useful to know that at
any point on this line, it's

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always going to have
the derivative

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00:03:15,240 --> 00:03:16,720
equal to negative 0.25.

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So that's useful.

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00:03:17,390 --> 00:03:21,170
We know we can integrate and
have the correct answer.

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00:03:21,170 --> 00:03:35,450
Solving for dQD dP, we're gonna
have negative 1.25.

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00:03:35,450 --> 00:03:37,845
And we're just going to
integrate this with respect to

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00:03:37,845 --> 00:03:41,340
P. And after we integrate,
we're going to

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00:03:41,340 --> 00:03:42,590
be left with a constant.

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00:03:42,590 --> 00:03:54,190

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00:03:54,190 --> 00:03:57,980
In this case, we're going
to call the constant a.

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00:03:57,980 --> 00:04:00,630
This is how much the demand
curve has actually shifted up

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00:04:00,630 --> 00:04:03,220
to begin with, shifted
up or down.

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00:04:03,220 --> 00:04:06,200
And to solve for a, all we have
to do is we can just plug

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00:04:06,200 --> 00:04:10,410
back in for the $20 and the 100
quantity, and we can solve

84
00:04:10,410 --> 00:04:12,970
through for what a is
going to be equal.

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00:04:12,970 --> 00:04:16,329
When you solve through plugging
in Q and P, you're

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00:04:16,329 --> 00:04:32,610
going to find that a
is equal to 125.

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00:04:32,610 --> 00:04:35,260
So we're gonna have that our
final demand function is gonna

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00:04:35,260 --> 00:04:40,500
be negative 1.25P plus 125.

89
00:04:40,500 --> 00:04:44,280
Now we can go through this exact
same process with the

90
00:04:44,280 --> 00:04:47,340
elasticity of supply now.

91
00:04:47,340 --> 00:04:51,000
And all we have to do now is
use the number 0.5 instead,

92
00:04:51,000 --> 00:04:52,130
and we can solve through.

93
00:04:52,130 --> 00:04:53,270
We can integrate.

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00:04:53,270 --> 00:04:55,630
And then we're going to solve
for the other constant to,

95
00:04:55,630 --> 00:04:58,400
again, get our supply curve.

96
00:04:58,400 --> 00:05:10,460
Substituting in the information
we have, we're

97
00:05:10,460 --> 00:05:11,770
going to be left with
this equation.

98
00:05:11,770 --> 00:05:15,300
And we're gonna go ahead and
isolate the derivative that we

99
00:05:15,300 --> 00:05:29,510
have. And when we integrate
again, we have to remember

100
00:05:29,510 --> 00:05:31,440
that we are going to have a
constant that we're gonna have

101
00:05:31,440 --> 00:05:32,690
to solve for.

102
00:05:32,690 --> 00:05:40,850

103
00:05:40,850 --> 00:05:43,270
And I'm gonna just call
this constant c.

104
00:05:43,270 --> 00:05:46,990
Again, plug in the price of 20
and the quantity equal to 100

105
00:05:46,990 --> 00:06:02,300
and you're gonna find that the
supply curve is gonna be equal

106
00:06:02,300 --> 00:06:05,880
to 2.5P plus 50.

107
00:06:05,880 --> 00:06:09,350
And we can do a quick sketch
of this on our axes here.

108
00:06:09,350 --> 00:06:11,640
We're just gonna go ahead and
draw our upward-sloping supply

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00:06:11,640 --> 00:06:19,050
curve, our downward-sloping
demand curve.

110
00:06:19,050 --> 00:06:24,505
And we're gonna mark the
equilibrium point and label

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00:06:24,505 --> 00:06:26,280
the equilibrium quantities
and the

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00:06:26,280 --> 00:06:27,620
equilibrium prices, as well.

113
00:06:27,620 --> 00:06:33,540

114
00:06:33,540 --> 00:06:35,720
And before we move on to the
second part of this problem,

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00:06:35,720 --> 00:06:36,510
we can pause here.

116
00:06:36,510 --> 00:06:38,730
And we can think about what did
the elasticities that we

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00:06:38,730 --> 00:06:40,590
started with actually mean.

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00:06:40,590 --> 00:06:43,340
Well, if we were to look at this
point of intersection at

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00:06:43,340 --> 00:06:47,250
the equilibrium of the demand
curve, we're looking at the

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00:06:47,250 --> 00:06:51,830
percentage change at this
point in quantity per

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00:06:51,830 --> 00:06:53,960
percentage change in price.

122
00:06:53,960 --> 00:06:59,850
So we're basically just saying,
for that tiny change,

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00:06:59,850 --> 00:07:02,720
an infinitesimally small change
at this point for the

124
00:07:02,720 --> 00:07:05,970
demand curve, how much
does quantity change,

125
00:07:05,970 --> 00:07:08,380
percentage-wise, relative
to price?

126
00:07:08,380 --> 00:07:10,950
And that's also what
we're looking at

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00:07:10,950 --> 00:07:12,000
with the supply curve.

128
00:07:12,000 --> 00:07:14,500
So when you're given a
elasticity, if you have an

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00:07:14,500 --> 00:07:16,830
elasticity of supply, it makes
sense that it's gonna be

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00:07:16,830 --> 00:07:21,950
positive, in this case 0.5,
because when price increases,

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00:07:21,950 --> 00:07:24,080
suppliers are willing
to supply more.

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00:07:24,080 --> 00:07:26,620
And it makes sense that the
demand elasticity that we're

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00:07:26,620 --> 00:07:31,050
given is negative, or negative
0.25, because when price

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00:07:31,050 --> 00:07:34,460
begins to increase, the
consumers are gonna want less

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00:07:34,460 --> 00:07:37,410
of the product.

136
00:07:37,410 --> 00:07:39,550
Now, the second part of this
problem is going to give us

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00:07:39,550 --> 00:07:41,820
new elasticities of
demand and supply.

138
00:07:41,820 --> 00:07:44,270
And I'm gonna just quickly
run through the actual

139
00:07:44,270 --> 00:07:46,180
calculation, because it's
gonna be the same as our

140
00:07:46,180 --> 00:07:47,960
calculation that we just did.

141
00:07:47,960 --> 00:07:52,250
And instead, we're gonna think
about possible causes for the

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00:07:52,250 --> 00:07:56,220
shifts that we see in the supply
and the demand curve.

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00:07:56,220 --> 00:07:59,010
Part B says, suppose that you
discover that the current

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00:07:59,010 --> 00:08:02,810
price of steel is $15 per ton
and the current level of

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00:08:02,810 --> 00:08:07,130
worldwide sales of steel
is 150 million tons.

146
00:08:07,130 --> 00:08:10,400
The most recent elasticity
estimates from the trade

147
00:08:10,400 --> 00:08:14,730
association this year are
negative 0.125 for demand and

148
00:08:14,730 --> 00:08:16,820
0.25 for supply.

149
00:08:16,820 --> 00:08:19,140
Describe the change in the
supply and the demand curves

150
00:08:19,140 --> 00:08:22,950
over the past year using your
diagram from part A. What sort

151
00:08:22,950 --> 00:08:26,410
of events might explain
the change?

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00:08:26,410 --> 00:08:28,710
Now, I've given us the
information for this part of

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00:08:28,710 --> 00:08:31,260
the problem on this board.

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00:08:31,260 --> 00:08:33,950
And you'll notice that our
inputs, or our variables, have

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00:08:33,950 --> 00:08:34,570
changed now.

156
00:08:34,570 --> 00:08:37,380
The price has dropped
from $20.

157
00:08:37,380 --> 00:08:39,600
Now it's gonna be down to $15.

158
00:08:39,600 --> 00:08:42,419
You're gonna notice that the
quantity has actually

159
00:08:42,419 --> 00:08:45,720
increased from 100 up to 150.

160
00:08:45,720 --> 00:08:49,080
And our elasticities of demand
and elasticities of supply

161
00:08:49,080 --> 00:08:51,250
have changed, because the price
and the quantity are

162
00:08:51,250 --> 00:08:55,210
different, and we're at a
different point on our graph.

163
00:08:55,210 --> 00:08:58,120
The process we're gonna do to
solve for our demand curves

164
00:08:58,120 --> 00:09:01,340
and our supply curves are going
to be exactly identical.

165
00:09:01,340 --> 00:09:03,100
And when you follow
the same process--

166
00:09:03,100 --> 00:09:05,020
I'll just do the first
step up here--

167
00:09:05,020 --> 00:09:08,020
you're gonna substitute in for
the information that's given

168
00:09:08,020 --> 00:09:09,270
in the problem.

169
00:09:09,270 --> 00:09:45,330

170
00:09:45,330 --> 00:09:47,230
And all you're going to do is
you're, again, gonna solve

171
00:09:47,230 --> 00:09:48,170
through for the derivative.

172
00:09:48,170 --> 00:09:49,050
You're gonna integrate.

173
00:09:49,050 --> 00:09:50,960
And you're gonna find
the constants.

174
00:09:50,960 --> 00:09:55,970
After you do that entire
process, you're gonna find

175
00:09:55,970 --> 00:10:00,420
that the demand curve is
given by this equation.

176
00:10:00,420 --> 00:10:05,410

177
00:10:05,410 --> 00:10:07,590
And you're gonna find that the
supply curve is given by this

178
00:10:07,590 --> 00:10:08,840
new equation.

179
00:10:08,840 --> 00:10:24,100

180
00:10:24,100 --> 00:10:27,960
Now, if we look at this new
demand curve and this new

181
00:10:27,960 --> 00:10:32,510
supply curve, we'll actually
notice that the slope, with

182
00:10:32,510 --> 00:10:35,910
respect to P, is going to be
identical in both of the cases

183
00:10:35,910 --> 00:10:39,160
that we solved for, both the
beginning case and the case in

184
00:10:39,160 --> 00:10:40,860
the end of the problem.

185
00:10:40,860 --> 00:10:43,370
The only thing that's shifted
between our quantities

186
00:10:43,370 --> 00:10:46,990
demanded and our quantities
supplied, or the curves, is

187
00:10:46,990 --> 00:10:48,240
there's been a shift.

188
00:10:48,240 --> 00:10:51,610
And the shift for the
demand curve--

189
00:10:51,610 --> 00:10:54,430
it went from an intercept
of 125 now to an

190
00:10:54,430 --> 00:10:57,270
intercept of 168.75.

191
00:10:57,270 --> 00:11:02,200
So our demand curve is
shifting up and out.

192
00:11:02,200 --> 00:11:08,330
So we can represent this shift
in demand like this.

193
00:11:08,330 --> 00:11:11,040
Notice that the slope is going
to be exactly identical.

194
00:11:11,040 --> 00:11:14,650
I'm going to write a small
db for part B.

195
00:11:14,650 --> 00:11:16,900
And then we can do the same sort
of interpretation for our

196
00:11:16,900 --> 00:11:18,150
supply curve.

197
00:11:18,150 --> 00:11:22,420
Looking at our supply curve, the
intercepts, now, is 112.5.

198
00:11:22,420 --> 00:11:26,070
But before, it was only at 50.

199
00:11:26,070 --> 00:11:29,160
And what this means, this means
that the supply curve is

200
00:11:29,160 --> 00:11:32,320
going to shift in and down.

201
00:11:32,320 --> 00:11:40,150

202
00:11:40,150 --> 00:11:43,090
And so my graph with the
equilibrium price that I've

203
00:11:43,090 --> 00:11:45,095
drawn-- it's a little bit off,
but what you should see--

204
00:11:45,095 --> 00:11:49,290

205
00:11:49,290 --> 00:11:54,960
you should see that the new
equilibrium price has fallen.

206
00:11:54,960 --> 00:11:58,790
In this case, it's
fallen to 15.

207
00:11:58,790 --> 00:12:04,090
And the equilibrium quantity has
increased from 100 to 150.

208
00:12:04,090 --> 00:12:07,090
So since we had both a shift
in supply and a shift in

209
00:12:07,090 --> 00:12:09,850
demand, necessarily we see that

210
00:12:09,850 --> 00:12:12,060
quantity is going to increase.

211
00:12:12,060 --> 00:12:18,620
But if the demand curve had
shifted way up here, we could

212
00:12:18,620 --> 00:12:20,300
see that price could
have increased.

213
00:12:20,300 --> 00:12:23,370
So the effect on the price in
this market is ambiguous.

214
00:12:23,370 --> 00:12:26,400
We can say that, necessarily,
the effect on quantity is

215
00:12:26,400 --> 00:12:29,170
going to be clearly towards
an increase.

216
00:12:29,170 --> 00:12:31,940
So to wrap up this problem,
we saw that changes in

217
00:12:31,940 --> 00:12:35,720
elasticities can also represent
changes in the

218
00:12:35,720 --> 00:12:38,070
underlying demand and
supply functions.

219
00:12:38,070 --> 00:12:41,230
Let's wrap up by just thinking
about what could have caused

220
00:12:41,230 --> 00:12:43,040
the demand shift that
we've seen.

221
00:12:43,040 --> 00:12:47,210
And what could have caused the
supply shift that we saw?

222
00:12:47,210 --> 00:12:49,480
Now, there are a couple
of ideas that we

223
00:12:49,480 --> 00:12:50,730
can have for demand.

224
00:12:50,730 --> 00:12:53,750

225
00:12:53,750 --> 00:12:55,950
The first idea that we could
have is we could just have had

226
00:12:55,950 --> 00:12:58,750
an increase in the income
of a consumer.

227
00:12:58,750 --> 00:13:01,710
If a consumer has more income,
then they might be willing to

228
00:13:01,710 --> 00:13:03,940
spend more on steel.

229
00:13:03,940 --> 00:13:07,370
A second idea that we have, we
could have that the price of a

230
00:13:07,370 --> 00:13:08,010
substitute--

231
00:13:08,010 --> 00:13:10,160
perhaps you're considering
building a bridge out of iron

232
00:13:10,160 --> 00:13:11,450
instead of steel--

233
00:13:11,450 --> 00:13:15,060
if the price of the substitute
has increased, then perhaps

234
00:13:15,060 --> 00:13:17,850
the consumers are going to be
willing to pay more to get the

235
00:13:17,850 --> 00:13:21,460
steel since the iron
is more expensive.

236
00:13:21,460 --> 00:13:24,250
A third possible idea is that
the number of goods that you

237
00:13:24,250 --> 00:13:27,690
need to make from steel
is increasing.

238
00:13:27,690 --> 00:13:32,050
So if you suddenly find new uses
for steel, then the price

239
00:13:32,050 --> 00:13:34,660
that you're willing to pay
at any given point

240
00:13:34,660 --> 00:13:37,640
is going to be higher.

241
00:13:37,640 --> 00:13:40,250
Basically, to affect the demand
curve, you have to

242
00:13:40,250 --> 00:13:43,170
think about why would people
be more willing to pay more

243
00:13:43,170 --> 00:13:44,280
for a fixed quantity.

244
00:13:44,280 --> 00:13:47,340
And I just listed off a
couple of ideas there.

245
00:13:47,340 --> 00:13:51,930
We can also think about reasons
about why the supply

246
00:13:51,930 --> 00:13:56,850
curve could be shifting in.

247
00:13:56,850 --> 00:13:58,620
In this case, why is it--

248
00:13:58,620 --> 00:14:02,220
why are sellers willing to offer
a cheaper price at any

249
00:14:02,220 --> 00:14:03,730
fixed quantity?

250
00:14:03,730 --> 00:14:05,720
And one idea that we could have
for this is just that

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00:14:05,720 --> 00:14:08,100
there are more firms
in this market.

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00:14:08,100 --> 00:14:11,440
If this market isn't perfectly
competitive to start off with,

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00:14:11,440 --> 00:14:13,710
then increasing the number of
firms is gonna increase

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00:14:13,710 --> 00:14:15,820
competition, and the producers
are gonna have

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00:14:15,820 --> 00:14:17,790
to drop their prices.

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00:14:17,790 --> 00:14:21,660
A second idea for why we've seen
the supply curve shift

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00:14:21,660 --> 00:14:25,870
out and down could be the
fact that input price

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00:14:25,870 --> 00:14:27,340
for steel has dropped.

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00:14:27,340 --> 00:14:29,290
Perhaps the way of manufacturing
or getting the

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00:14:29,290 --> 00:14:31,900
raw material is cheaper because
the machine they're

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00:14:31,900 --> 00:14:34,350
using to get the steel
is cheaper.

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00:14:34,350 --> 00:14:36,090
Basically, when you're thinking
about the shift

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00:14:36,090 --> 00:14:39,540
that's making it cheaper for
suppliers to produce the good,

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00:14:39,540 --> 00:14:41,990
all you need to think about is
what could make it so that

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00:14:41,990 --> 00:14:45,090
they're more willing to produce
at a lower price.

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00:14:45,090 --> 00:14:47,960
So again, with this problem, we
went through working with

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00:14:47,960 --> 00:14:49,350
elasticities and demands.

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00:14:49,350 --> 00:14:53,000
We've seen that we can go from a
demand curve or supply curve

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00:14:53,000 --> 00:14:56,650
to elasticities, or we can go
from elasticities to demands.

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00:14:56,650 --> 00:14:59,880
And then, once we've had the
supply and the demand curves,

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00:14:59,880 --> 00:15:01,510
we looked at how
do we interpret

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00:15:01,510 --> 00:15:02,750
the shifts and shocks?

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00:15:02,750 --> 00:15:04,760
And we looked at possible
explanations for those shift

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00:15:04,760 --> 00:15:05,910
and shocks.

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00:15:05,910 --> 00:15:07,160
I hope you found this
problem helpful.

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00:15:07,160 --> 00:15:14,858