1
00:00:00,040 --> 00:00:02,460
The following content is
provided under a Creative

2
00:00:02,460 --> 00:00:03,870
Commons license.

3
00:00:03,870 --> 00:00:06,910
Your support will help MIT
OpenCourseWare continue to

4
00:00:06,910 --> 00:00:10,560
offer high quality educational
resources for free.

5
00:00:10,560 --> 00:00:13,460
To make a donation or view
additional materials from

6
00:00:13,460 --> 00:00:17,390
hundreds of MIT courses, visit
MIT OpenCourseWare at

7
00:00:17,390 --> 00:00:18,640
ocw.mit.edu.

8
00:00:21,708 --> 00:00:24,990
PROFESSOR: Today, we're going
to be working on fall 2010,

9
00:00:24,990 --> 00:00:27,360
problem set number 2, and
we're going to be doing

10
00:00:27,360 --> 00:00:29,200
problem number 4.

11
00:00:29,200 --> 00:00:31,030
I'm going to start off by
reading the problem.

12
00:00:31,030 --> 00:00:32,970
There's a lot of information
here, so I've written up some

13
00:00:32,970 --> 00:00:36,220
of the key points on the board
for you guys already.

14
00:00:36,220 --> 00:00:40,240
It is exactly 24 hours before
Lauren's physics final.

15
00:00:40,240 --> 00:00:42,830
She has an economics final
directly after her physics

16
00:00:42,830 --> 00:00:45,810
final, and has no time
to study in between.

17
00:00:45,810 --> 00:00:48,390
Lauren wants to be a physicist,
so she places more

18
00:00:48,390 --> 00:00:50,630
weight on her physics
test score.

19
00:00:50,630 --> 00:00:55,560
Her utility function is
given right here.

20
00:00:55,560 --> 00:00:58,550
Where p is the score of her
physics final and e is the

21
00:00:58,550 --> 00:01:00,740
score of her economics final.

22
00:01:00,740 --> 00:01:02,665
Although, she cares more
about physics, she

23
00:01:02,665 --> 00:01:04,280
is better at economics.

24
00:01:04,280 --> 00:01:07,280
For each hour spent studying
economics, she will increase

25
00:01:07,280 --> 00:01:09,560
her score by three points.

26
00:01:09,560 --> 00:01:12,640
But her physics score will only
increase by two points

27
00:01:12,640 --> 00:01:15,830
for every hour spent
studying physics.

28
00:01:15,830 --> 00:01:18,170
Studying zero hours results
in a score of

29
00:01:18,170 --> 00:01:20,310
zero on both subjects.

30
00:01:20,310 --> 00:01:22,960
Although natural log of zero
is not defined, assume her

31
00:01:22,960 --> 00:01:27,070
utility for a score of zero
is negative infinity.

32
00:01:27,070 --> 00:01:28,730
Now, we're going to go ahead
and we're going to work

33
00:01:28,730 --> 00:01:33,300
through parts A, B, and C, A, B,
C, and D and then we'll do

34
00:01:33,300 --> 00:01:36,510
part E, which is a new
scenario afterwards.

35
00:01:36,510 --> 00:01:38,740
Part A, we're going to find the
constraints that Lauren

36
00:01:38,740 --> 00:01:41,790
faces in her test score
maximization problem.

37
00:01:41,790 --> 00:01:44,260
And part B, we're going to
find how many hours does

38
00:01:44,260 --> 00:01:47,340
Lauren optimally spend studying
physics, how many

39
00:01:47,340 --> 00:01:49,950
hours does she spent
studying economics.

40
00:01:49,950 --> 00:01:52,270
And hours are divisible, so
we don't need whole number

41
00:01:52,270 --> 00:01:55,120
solutions for the hours spent
studying physics and the hours

42
00:01:55,120 --> 00:01:58,550
spent studying economics.

43
00:01:58,550 --> 00:02:00,740
So for part A, all we're looking
for is we're looking

44
00:02:00,740 --> 00:02:01,810
for the constraints.

45
00:02:01,810 --> 00:02:04,020
And it sounds like the
constraint that Lauren is

46
00:02:04,020 --> 00:02:06,310
really facing in this scenario
is the amount of

47
00:02:06,310 --> 00:02:07,770
time that she has.

48
00:02:07,770 --> 00:02:10,509
It seems like Lauren's pretty
intense about studying, so in

49
00:02:10,509 --> 00:02:13,630
the 24 hours before the test,
she's not getting any sleep.

50
00:02:13,630 --> 00:02:16,370
She's going to spend all
of this time studying.

51
00:02:16,370 --> 00:02:19,980
So we're going to have two hours
variables to represent

52
00:02:19,980 --> 00:02:22,970
the hours she spends studying
physics and the hours she

53
00:02:22,970 --> 00:02:25,350
spends studying economics.

54
00:02:25,350 --> 00:02:33,360
And we know that the hours,
when we add them together,

55
00:02:33,360 --> 00:02:38,610
they're going to have to be
less than or equal to 24.

56
00:02:38,610 --> 00:02:41,160
Now, if this is our constraint,
we really know

57
00:02:41,160 --> 00:02:43,190
that if she's trying to maximize
her scores and if

58
00:02:43,190 --> 00:02:45,650
that's all she really cares
about, she's not going to

59
00:02:45,650 --> 00:02:48,120
spend less than 24
hours studying.

60
00:02:48,120 --> 00:02:55,320
So we can actually just say
that the sum of those two

61
00:02:55,320 --> 00:02:57,200
hours-- her hours spent studying
physics and her hours

62
00:02:57,200 --> 00:03:01,780
spent studying economics are
going to be equal to 24.

63
00:03:01,780 --> 00:03:05,000
So that's the first constraint
that Lauren is going to face.

64
00:03:05,000 --> 00:03:07,040
The second constraint that she's
going to face, and it's

65
00:03:07,040 --> 00:03:11,190
not necessarily an
intuitive one--

66
00:03:11,190 --> 00:03:14,940
or it is actually really
intuitive, but it's also a

67
00:03:14,940 --> 00:03:16,190
trivial one.

68
00:03:19,360 --> 00:03:22,810
It's just that she can't spend
less than zero hours studying

69
00:03:22,810 --> 00:03:24,880
physics or studying economics.

70
00:03:24,880 --> 00:03:28,210
So we're going to add those
constraints in as well.

71
00:03:28,210 --> 00:03:30,530
And the final constraints
are actually production

72
00:03:30,530 --> 00:03:32,100
constraints.

73
00:03:32,100 --> 00:03:33,640
If the hours spent studying--

74
00:03:33,640 --> 00:03:35,430
these aren't actually what
she's interested in.

75
00:03:35,430 --> 00:03:38,160
What she's interested in
are the p and the e.

76
00:03:38,160 --> 00:03:39,870
That's how she's going to
get her utilities--

77
00:03:39,870 --> 00:03:41,520
through the test scores.

78
00:03:41,520 --> 00:03:45,360
So what we need is we need to
find how she produces her

79
00:03:45,360 --> 00:03:46,720
physics score.

80
00:03:46,720 --> 00:03:50,530
If she gets two points for every
hour she spent studying,

81
00:03:50,530 --> 00:03:57,140
her physics score is going to
be p equals 2 times Hp--

82
00:03:57,140 --> 00:03:59,690
that's the production of
her physics score.

83
00:03:59,690 --> 00:04:03,640
The production of her economics
score is going to

84
00:04:03,640 --> 00:04:08,190
equal 3 times He, since we know
that she's a little bit

85
00:04:08,190 --> 00:04:10,820
better at economics
than physics.

86
00:04:10,820 --> 00:04:12,650
So these are the constraints
that we're going to face in

87
00:04:12,650 --> 00:04:14,370
the problem.

88
00:04:14,370 --> 00:04:19,680
Let's go ahead and move on to
part B. And for part B, what

89
00:04:19,680 --> 00:04:21,870
we're going to try to do is
we're going to try to take

90
00:04:21,870 --> 00:04:25,620
these constraints and we're
trying to maximize the amount

91
00:04:25,620 --> 00:04:30,180
of utility she's going
to get from studying.

92
00:04:30,180 --> 00:04:32,720
So what we're doing for part B
is we're going to maximize

93
00:04:32,720 --> 00:04:35,710
utility, and we're going to
plug in some of these

94
00:04:35,710 --> 00:04:36,840
constraints.

95
00:04:36,840 --> 00:04:39,530
Before we can do this, we really
need to get down to

96
00:04:39,530 --> 00:04:40,650
first-order conditions.

97
00:04:40,650 --> 00:04:44,620
We need to be able to take the
derivative with regards to

98
00:04:44,620 --> 00:04:45,670
only one variable.

99
00:04:45,670 --> 00:04:48,890
So we have to get-- instead of
p and e here, we're going to

100
00:04:48,890 --> 00:04:50,400
try to get only one variable.

101
00:04:50,400 --> 00:04:52,940
And the variable we're going to
get in there is going to be

102
00:04:52,940 --> 00:04:54,560
the hours spent study
economics.

103
00:04:54,560 --> 00:04:58,240
So we're going to get it
all in terms of He.

104
00:04:58,240 --> 00:05:04,810
So we need to find a way to
replace both e and p with He.

105
00:05:04,810 --> 00:05:08,740
In this case, replacing e
with He is really easy.

106
00:05:08,740 --> 00:05:11,740
We can just say that e is
going to equal 3He.

107
00:05:14,830 --> 00:05:19,150
Replacing p with the He or the
hours spent studying economics

108
00:05:19,150 --> 00:05:21,060
is a little bit more
difficult.

109
00:05:21,060 --> 00:05:23,120
We're going to go to
this equation.

110
00:05:23,120 --> 00:05:27,260
And we can say that
Hp is going to be

111
00:05:27,260 --> 00:05:33,470
equal to 24 minus He.

112
00:05:33,470 --> 00:05:36,930
And then for this Hp, we're
going to plug this into our

113
00:05:36,930 --> 00:05:39,310
production function for
the physics score.

114
00:05:39,310 --> 00:05:50,250
So we get that p is equal
to 2 times 24 minus He.

115
00:05:50,250 --> 00:05:55,650
So we're going to take these two
equations that we have in

116
00:05:55,650 --> 00:05:57,316
and we're going to substitute
for e and p

117
00:05:57,316 --> 00:05:58,566
in our utility function.

118
00:06:30,740 --> 00:06:32,430
And so we're going to maximize,
and the way we're

119
00:06:32,430 --> 00:06:35,720
going to maximize is by just
taking the first-order

120
00:06:35,720 --> 00:06:36,970
conditions or the FOC.

121
00:06:39,490 --> 00:06:41,970
And the FOC in this case
is just going to be the

122
00:06:41,970 --> 00:06:44,590
derivative with respect to He.

123
00:06:44,590 --> 00:06:47,040
When you take the derivative
with respect to He, what

124
00:06:47,040 --> 00:06:56,400
you're going to get is 0.6 times
negative 2 all over (48

125
00:06:56,400 --> 00:07:06,983
minus 2 He) plus 0.4 times
3 all over 3He.

126
00:07:10,220 --> 00:07:12,350
And the reason this first-order
condition is going

127
00:07:12,350 --> 00:07:14,800
to help us is because
we know to maximize.

128
00:07:14,800 --> 00:07:18,030
We just set the first-order
condition or the derivative

129
00:07:18,030 --> 00:07:22,810
with respect to He
equal to zero.

130
00:07:22,810 --> 00:07:25,514
And from here, we can
solve out for He.

131
00:07:29,780 --> 00:07:32,530
And when we solve for He, we
find that the hour she spends

132
00:07:32,530 --> 00:07:35,766
studying economics is going
to equal to 9.6.

133
00:07:38,390 --> 00:07:41,960
And we know that any time she
spends not studying the

134
00:07:41,960 --> 00:07:44,770
economics, she's studying
physics, since all she cares

135
00:07:44,770 --> 00:07:46,020
about is her test scores.

136
00:07:50,450 --> 00:07:53,070
She's going to spend
the remaining 14.4

137
00:07:53,070 --> 00:07:54,570
hours studying physics.

138
00:07:58,610 --> 00:08:02,960
This is our answer for part B.

139
00:08:02,960 --> 00:08:09,340
Now, part C just asks us, "What
economics and physics

140
00:08:09,340 --> 00:08:12,145
test scores will she achieve?"
so we're looking for the e

141
00:08:12,145 --> 00:08:13,840
star and the p star.

142
00:08:13,840 --> 00:08:16,950
We can just take the hour she
spent studying physics and the

143
00:08:16,950 --> 00:08:19,930
hours she spends studying
economics and we can plug

144
00:08:19,930 --> 00:08:23,550
these back into her production
functions for her physics

145
00:08:23,550 --> 00:08:27,300
score and an economics score.

146
00:08:27,300 --> 00:08:33,330
So, for part C, her economics
score is going to be equal to

147
00:08:33,330 --> 00:08:35,565
3 times 9.6.

148
00:08:38,320 --> 00:08:45,620
She's going to get a 28.8
on her economics test.

149
00:08:45,620 --> 00:08:54,040
And for her physics
test, she's also

150
00:08:54,040 --> 00:09:04,350
going to get a 28.8.

151
00:09:04,350 --> 00:09:09,310
Probably the not the best test
scores in the world.

152
00:09:09,310 --> 00:09:12,940
Let's go ahead and look at part
D. Part D asks us, "What

153
00:09:12,940 --> 00:09:16,400
utility level will she achieve
with these test scores that

154
00:09:16,400 --> 00:09:20,170
she has?" Now, to find the
utility levels, all we have to

155
00:09:20,170 --> 00:09:22,790
do is we have to go back to our
utility function that we

156
00:09:22,790 --> 00:09:26,460
wrote down in part A, and we're
just going to plug in

157
00:09:26,460 --> 00:09:29,850
the two test scores that she
received, and we can solve

158
00:09:29,850 --> 00:09:31,620
through for overall utility.

159
00:09:57,900 --> 00:09:59,940
And so when you grab a
calculator, and you solve

160
00:09:59,940 --> 00:10:02,350
through for this equation, what
you're going to find is

161
00:10:02,350 --> 00:10:12,850
you're going to find her overall
utility level is going

162
00:10:12,850 --> 00:10:15,210
to be equal to 3.36.

163
00:10:15,210 --> 00:10:17,090
And when we're measuring
utility, we don't have any

164
00:10:17,090 --> 00:10:18,190
units on this.

165
00:10:18,190 --> 00:10:21,010
The units, we usually think of
as utils, but we can just keep

166
00:10:21,010 --> 00:10:21,730
it like this.

167
00:10:21,730 --> 00:10:24,660
Just know that in relativistic
terms, her

168
00:10:24,660 --> 00:10:26,970
utility is that 3.36.

169
00:10:26,970 --> 00:10:32,290
And this is going to be useful
for part D. Because in part E,

170
00:10:32,290 --> 00:10:35,160
we have the option of sending
Lauren off to

171
00:10:35,160 --> 00:10:36,630
an economics tutor.

172
00:10:36,630 --> 00:10:39,090
And we're going to be comparing
her utility after

173
00:10:39,090 --> 00:10:44,040
going to the economics tutor
to this utility of 3.36.

174
00:10:44,040 --> 00:10:47,270
And by comparing her new utility
with the old utility,

175
00:10:47,270 --> 00:10:49,740
we can see if her utility's
higher after going the

176
00:10:49,740 --> 00:10:50,900
economics tutor.

177
00:10:50,900 --> 00:10:53,400
We can see if that's a good
decision for her.

178
00:10:53,400 --> 00:10:56,090
So now that we finished
calculating the utility for

179
00:10:56,090 --> 00:10:59,080
Lauren in the initial case where
she had 24 hours and no

180
00:10:59,080 --> 00:11:03,040
help, now we're going to move
on to part E. Part E says,

181
00:11:03,040 --> 00:11:05,660
"Suppose Lauren can get
an economics tutor.

182
00:11:05,660 --> 00:11:08,300
If she goes to the tutor, she
will increase her economics

183
00:11:08,300 --> 00:11:11,710
test score by five points for
every hour spent studying,

184
00:11:11,710 --> 00:11:13,290
instead of three points.

185
00:11:13,290 --> 00:11:15,380
But will lose four hours
of study time by

186
00:11:15,380 --> 00:11:16,810
going to the tutor.

187
00:11:16,810 --> 00:11:19,570
She cannot study while at the
tutor, and going to the tutor

188
00:11:19,570 --> 00:11:22,050
does not directly improve
her test score.

189
00:11:22,050 --> 00:11:25,090
Should Lauren go
to the tutor?"

190
00:11:25,090 --> 00:11:26,630
Now, for this problem,
we can go back

191
00:11:26,630 --> 00:11:28,730
to our initial scenario.

192
00:11:28,730 --> 00:11:31,560
Where we had our constraint
for the amount of

193
00:11:31,560 --> 00:11:33,010
time Lauren can spend.

194
00:11:33,010 --> 00:11:35,380
If Lauren goes to the tutor, it
sounds like she's going to

195
00:11:35,380 --> 00:11:38,540
spend about four hours in the
car driving to the tutor's

196
00:11:38,540 --> 00:11:40,470
house and back.

197
00:11:40,470 --> 00:11:44,150
So instead of having 24 total
hours of study time, her new

198
00:11:44,150 --> 00:11:48,630
constraint is that her total
study time Hp plus

199
00:11:48,630 --> 00:11:51,740
He is equal to 20.

200
00:11:51,740 --> 00:11:53,090
That's the downside.

201
00:11:53,090 --> 00:11:56,030
The upshot of it is that her
production function for

202
00:11:56,030 --> 00:11:59,970
economics test scores is no
longer three points for every

203
00:11:59,970 --> 00:12:06,210
hour she spent studying, now she
gets five point for every

204
00:12:06,210 --> 00:12:08,150
hour she spends studying.

205
00:12:08,150 --> 00:12:11,000
So we're going to go through the
exact same process for A,

206
00:12:11,000 --> 00:12:14,520
B, C, and D that we did before,
only now we're going

207
00:12:14,520 --> 00:12:17,090
to use our new constraints
to solve for the

208
00:12:17,090 --> 00:12:18,340
maximization problem.

209
00:12:24,160 --> 00:12:26,580
Before we start off for our
maximization problem and

210
00:12:26,580 --> 00:12:29,150
solving for the first-order
conditions, what we have to do

211
00:12:29,150 --> 00:12:31,690
is we have to get it all in
terms of one variable.

212
00:12:31,690 --> 00:12:34,000
And the variable we're going to
get it in terms of is the

213
00:12:34,000 --> 00:12:36,870
hours of economic
studying or He.

214
00:12:36,870 --> 00:12:45,340
So from the utility function, to
solve or to plug in for e,

215
00:12:45,340 --> 00:12:53,290
the He, we have that e is going
to equal five times He.

216
00:12:53,290 --> 00:13:00,510
For p, we know that Hp is
equal to 20 minus He.

217
00:13:00,510 --> 00:13:03,490
That just says that any time
she's not studying economics,

218
00:13:03,490 --> 00:13:05,770
she's going to be studying
physics.

219
00:13:05,770 --> 00:13:12,580
We can plug this into our
p equals 2 times Hp.

220
00:13:12,580 --> 00:13:21,200
So we know that p is going to
equal 2 times 20 minus He.

221
00:13:21,200 --> 00:13:25,500
Let's go ahead and plug this
and this into our utility

222
00:13:25,500 --> 00:13:27,830
maximization problem and solve
for the first-order

223
00:13:27,830 --> 00:13:29,080
conditions.

224
00:13:53,780 --> 00:13:55,230
So we're maximizing
utilities--

225
00:13:55,230 --> 00:13:56,510
so we're going to take
the derivative with

226
00:13:56,510 --> 00:13:58,020
respect to He again.

227
00:13:58,020 --> 00:13:59,610
When we do that,
we get our FOC.

228
00:14:25,030 --> 00:14:27,660
And again, for the maximization
problem, we can

229
00:14:27,660 --> 00:14:30,310
just set that equal to 0.

230
00:14:30,310 --> 00:14:33,000
Solving out for He, we find
that she's going to spend

231
00:14:33,000 --> 00:14:39,560
eight hours studying
economics.

232
00:14:39,560 --> 00:14:46,430
And we also find that any time
she's not spending studying

233
00:14:46,430 --> 00:14:50,320
economics, she's going to
be studying physics.

234
00:14:50,320 --> 00:14:56,080
So she's going to spend 12
hours studying physics.

235
00:14:56,080 --> 00:14:58,750
Now, we're not done with part
E because really all we have

236
00:14:58,750 --> 00:15:01,580
to do for this problem is we
have to solve for her new

237
00:15:01,580 --> 00:15:05,130
utility using these hours that
she spends studying.

238
00:15:05,130 --> 00:15:08,380
And we have to compare her new
utility to her old utility of

239
00:15:08,380 --> 00:15:11,980
3.36 to see if she's better
off or worse with the

240
00:15:11,980 --> 00:15:14,720
economics tutor.

241
00:15:14,720 --> 00:15:18,360
In her production function for
economics test score, we know

242
00:15:18,360 --> 00:15:22,980
that for each of those eight
hours you spend studying, her

243
00:15:22,980 --> 00:15:25,670
test score's going to improve
eight points, so her new

244
00:15:25,670 --> 00:15:29,850
economic score is 40.

245
00:15:29,850 --> 00:15:33,350
And we know that for each of
those 12 hours she spends

246
00:15:33,350 --> 00:15:38,910
studying physics, her score's
going to improve two points.

247
00:15:38,910 --> 00:15:44,510
So her physics score has dropped
a little bit to 24.

248
00:15:44,510 --> 00:15:51,280
When we plug in for e and p into
our utility function, we

249
00:15:51,280 --> 00:15:54,290
can find her utility level
with the economics tutor.

250
00:16:06,890 --> 00:16:08,960
When we solve through for these
natural logs, we're

251
00:16:08,960 --> 00:16:19,150
going to find that her new
utility is equal to 3.38.

252
00:16:19,150 --> 00:16:20,990
And since the problem asks
us to actually make

253
00:16:20,990 --> 00:16:22,220
a qualitative judgment--

254
00:16:22,220 --> 00:16:25,840
is she better off or worse off,
we need to compare her

255
00:16:25,840 --> 00:16:36,240
utility before of 3.36 to
her new utility of 3.38.

256
00:16:36,240 --> 00:16:38,460
This is the comparison that
we're interested in.

257
00:16:38,460 --> 00:16:40,910
And when we make this
comparison, we can see that in

258
00:16:40,910 --> 00:16:43,900
relative terms, she's a little
bit better off going to be

259
00:16:43,900 --> 00:16:47,415
economics tutor because her
utility has risen two

260
00:16:47,415 --> 00:16:50,010
hundredths of a point.

261
00:16:50,010 --> 00:16:53,840
So in this case, for part E,
we found that her utility

262
00:16:53,840 --> 00:16:56,640
increases even though she's
lost time because her

263
00:16:56,640 --> 00:16:59,680
production function for her
economics to score has gone

264
00:16:59,680 --> 00:17:03,050
up, and that she's better off
going to the economics tutor

265
00:17:03,050 --> 00:17:05,369
because of that increase
in utility.

266
00:17:05,369 --> 00:17:08,010
So the point of this problem was
to look at the different

267
00:17:08,010 --> 00:17:10,880
constraints that consumers face
when deciding where to

268
00:17:10,880 --> 00:17:14,079
spend their time or where to
spend their money, and to see

269
00:17:14,079 --> 00:17:16,530
how those different constraints
affect their

270
00:17:16,530 --> 00:17:17,780
utility levels.