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PROFESSOR: Today, we're going
to be working on fall 2010,
problem set number 2, and
we're going to be doing
problem number 4.
I'm going to start off by
reading the problem.
There's a lot of information
here, so I've written up some
of the key points on the board
for you guys already.
It is exactly 24 hours before
Lauren's physics final.
She has an economics final
directly after her physics
final, and has no time
to study in between.
Lauren wants to be a physicist,
so she places more
weight on her physics
test score.
Her utility function is
given right here.
Where p is the score of her
physics final and e is the
score of her economics final.
Although, she cares more
about physics, she
is better at economics.
For each hour spent studying
economics, she will increase
her score by three points.
But her physics score will only
increase by two points
for every hour spent
studying physics.
Studying zero hours results
in a score of
zero on both subjects.
Although natural log of zero
is not defined, assume her
utility for a score of zero
is negative infinity.
Now, we're going to go ahead
and we're going to work
through parts A, B, and C, A, B,
C, and D and then we'll do
part E, which is a new
scenario afterwards.
Part A, we're going to find the
constraints that Lauren
faces in her test score
maximization problem.
And part B, we're going to
find how many hours does
Lauren optimally spend studying
physics, how many
hours does she spent
studying economics.
And hours are divisible, so
we don't need whole number
solutions for the hours spent
studying physics and the hours
spent studying economics.
So for part A, all we're looking
for is we're looking
for the constraints.
And it sounds like the
constraint that Lauren is
really facing in this scenario
is the amount of
time that she has.
It seems like Lauren's pretty
intense about studying, so in
the 24 hours before the test,
she's not getting any sleep.
She's going to spend all
of this time studying.
So we're going to have two hours
variables to represent
the hours she spends studying
physics and the hours she
spends studying economics.
And we know that the hours,
when we add them together,
they're going to have to be
less than or equal to 24.
Now, if this is our constraint,
we really know
that if she's trying to maximize
her scores and if
that's all she really cares
about, she's not going to
spend less than 24
hours studying.
So we can actually just say
that the sum of those two
hours-- her hours spent studying
physics and her hours
spent studying economics are
going to be equal to 24.
So that's the first constraint
that Lauren is going to face.
The second constraint that she's
going to face, and it's
not necessarily an
intuitive one--
or it is actually really
intuitive, but it's also a
trivial one.
It's just that she can't spend
less than zero hours studying
physics or studying economics.
So we're going to add those
constraints in as well.
And the final constraints
are actually production
constraints.
If the hours spent studying--
these aren't actually what
she's interested in.
What she's interested in
are the p and the e.
That's how she's going to
get her utilities--
through the test scores.
So what we need is we need to
find how she produces her
physics score.
If she gets two points for every
hour she spent studying,
her physics score is going to
be p equals 2 times Hp--
that's the production of
her physics score.
The production of her economics
score is going to
equal 3 times He, since we know
that she's a little bit
better at economics
than physics.
So these are the constraints
that we're going to face in
the problem.
Let's go ahead and move on to
part B. And for part B, what
we're going to try to do is
we're going to try to take
these constraints and we're
trying to maximize the amount
of utility she's going
to get from studying.
So what we're doing for part B
is we're going to maximize
utility, and we're going to
plug in some of these
constraints.
Before we can do this, we really
need to get down to
first-order conditions.
We need to be able to take the
derivative with regards to
only one variable.
So we have to get-- instead of
p and e here, we're going to
try to get only one variable.
And the variable we're going to
get in there is going to be
the hours spent study
economics.
So we're going to get it
all in terms of He.
So we need to find a way to
replace both e and p with He.
In this case, replacing e
with He is really easy.
We can just say that e is
going to equal 3He.
Replacing p with the He or the
hours spent studying economics
is a little bit more
difficult.
We're going to go to
this equation.
And we can say that
Hp is going to be
equal to 24 minus He.
And then for this Hp, we're
going to plug this into our
production function for
the physics score.
So we get that p is equal
to 2 times 24 minus He.
So we're going to take these two
equations that we have in
and we're going to substitute
for e and p
in our utility function.
And so we're going to maximize,
and the way we're
going to maximize is by just
taking the first-order
conditions or the FOC.
And the FOC in this case
is just going to be the
derivative with respect to He.
When you take the derivative
with respect to He, what
you're going to get is 0.6 times
negative 2 all over (48
minus 2 He) plus 0.4 times
3 all over 3He.
And the reason this first-order
condition is going
to help us is because
we know to maximize.
We just set the first-order
condition or the derivative
with respect to He
equal to zero.
And from here, we can
solve out for He.
And when we solve for He, we
find that the hour she spends
studying economics is going
to equal to 9.6.
And we know that any time she
spends not studying the
economics, she's studying
physics, since all she cares
about is her test scores.
She's going to spend
the remaining 14.4
hours studying physics.
This is our answer for part B.
Now, part C just asks us, "What
economics and physics
test scores will she achieve?"
so we're looking for the e
star and the p star.
We can just take the hour she
spent studying physics and the
hours she spends studying
economics and we can plug
these back into her production
functions for her physics
score and an economics score.
So, for part C, her economics
score is going to be equal to
3 times 9.6.
She's going to get a 28.8
on her economics test.
And for her physics
test, she's also
going to get a 28.8.
Probably the not the best test
scores in the world.
Let's go ahead and look at part
D. Part D asks us, "What
utility level will she achieve
with these test scores that
she has?" Now, to find the
utility levels, all we have to
do is we have to go back to our
utility function that we
wrote down in part A, and we're
just going to plug in
the two test scores that she
received, and we can solve
through for overall utility.
And so when you grab a
calculator, and you solve
through for this equation, what
you're going to find is
you're going to find her overall
utility level is going
to be equal to 3.36.
And when we're measuring
utility, we don't have any
units on this.
The units, we usually think of
as utils, but we can just keep
it like this.
Just know that in relativistic
terms, her
utility is that 3.36.
And this is going to be useful
for part D. Because in part E,
we have the option of sending
Lauren off to
an economics tutor.
And we're going to be comparing
her utility after
going to the economics tutor
to this utility of 3.36.
And by comparing her new utility
with the old utility,
we can see if her utility's
higher after going the
economics tutor.
We can see if that's a good
decision for her.
So now that we finished
calculating the utility for
Lauren in the initial case where
she had 24 hours and no
help, now we're going to move
on to part E. Part E says,
"Suppose Lauren can get
an economics tutor.
If she goes to the tutor, she
will increase her economics
test score by five points for
every hour spent studying,
instead of three points.
But will lose four hours
of study time by
going to the tutor.
She cannot study while at the
tutor, and going to the tutor
does not directly improve
her test score.
Should Lauren go
to the tutor?"
Now, for this problem,
we can go back
to our initial scenario.
Where we had our constraint
for the amount of
time Lauren can spend.
If Lauren goes to the tutor, it
sounds like she's going to
spend about four hours in the
car driving to the tutor's
house and back.
So instead of having 24 total
hours of study time, her new
constraint is that her total
study time Hp plus
He is equal to 20.
That's the downside.
The upshot of it is that her
production function for
economics test scores is no
longer three points for every
hour she spent studying, now she
gets five point for every
hour she spends studying.
So we're going to go through the
exact same process for A,
B, C, and D that we did before,
only now we're going
to use our new constraints
to solve for the
maximization problem.
Before we start off for our
maximization problem and
solving for the first-order
conditions, what we have to do
is we have to get it all in
terms of one variable.
And the variable we're going to
get it in terms of is the
hours of economic
studying or He.
So from the utility function, to
solve or to plug in for e,
the He, we have that e is going
to equal five times He.
For p, we know that Hp is
equal to 20 minus He.
That just says that any time
she's not studying economics,
she's going to be studying
physics.
We can plug this into our
p equals 2 times Hp.
So we know that p is going to
equal 2 times 20 minus He.
Let's go ahead and plug this
and this into our utility
maximization problem and solve
for the first-order
conditions.
So we're maximizing
utilities--
so we're going to take
the derivative with
respect to He again.
When we do that,
we get our FOC.
And again, for the maximization
problem, we can
just set that equal to 0.
Solving out for He, we find
that she's going to spend
eight hours studying
economics.
And we also find that any time
she's not spending studying
economics, she's going to
be studying physics.
So she's going to spend 12
hours studying physics.
Now, we're not done with part
E because really all we have
to do for this problem is we
have to solve for her new
utility using these hours that
she spends studying.
And we have to compare her new
utility to her old utility of
3.36 to see if she's better
off or worse with the
economics tutor.
In her production function for
economics test score, we know
that for each of those eight
hours you spend studying, her
test score's going to improve
eight points, so her new
economic score is 40.
And we know that for each of
those 12 hours she spends
studying physics, her score's
going to improve two points.
So her physics score has dropped
a little bit to 24.
When we plug in for e and p into
our utility function, we
can find her utility level
with the economics tutor.
When we solve through for these
natural logs, we're
going to find that her new
utility is equal to 3.38.
And since the problem asks
us to actually make
a qualitative judgment--
is she better off or worse off,
we need to compare her
utility before of 3.36 to
her new utility of 3.38.
This is the comparison that
we're interested in.
And when we make this
comparison, we can see that in
relative terms, she's a little
bit better off going to be
economics tutor because her
utility has risen two
hundredths of a point.
So in this case, for part E,
we found that her utility
increases even though she's
lost time because her
production function for her
economics to score has gone
up, and that she's better off
going to the economics tutor
because of that increase
in utility.
So the point of this problem was
to look at the different
constraints that consumers face
when deciding where to
spend their time or where to
spend their money, and to see
how those different constraints
affect their
utility levels.
