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PROFESSOR: Hi, and welcome
back to the 14.01 problem
solving videos.
Today, we're going to do Fall
2010, Problem Set 3,
Problem Number 5.
And we're going to go ahead
and we're going to work
through parts A, B, C, D, and
E, and then we're going to
finish up parts F and G.
Problem 5 says that Xiao spends
all her income on
statistical softwares
and clothes.
Her preferences can be
represented by the utility
function where her utility
equals 4 times the natural log
of S plus 6 times the natural
log of C, where S is software
and C is clothes.
Part A asks us to compute the
marginal rate of substitution
of software for clothes, asks us
if the MRS is increasing or
decreasing in S, and also asks
us how we interpret the MRS.
So before we start with this, we
should really think about,
conceptually, what the marginal
rate of substitution
of software for clothes looks
like on our graph.
So on this graph, I have
clothes on the y-axis--
the quantity of clothes--
and the quantity of software
on the x-axis here.
Looking at this graph, this line
that I've drawn is one
utility level.
So at this place, she might
have a utility equal to 1.
So she's indifferent on this
indifference curve between
being at point here or
at a point here.
And what the marginal rate of
substitution is really asking
us, it's asking us how much
clothing is she willing to
give up to get one more
unit of software?
So she's going to have to give
up a certain amount of clothes
to get one more unit
of software.
And the marginal rate of
substitution tells us exactly
how much clothing she's
willing to give up.
To calculate this algebraically,
all we're going
to do is we're going to take
the marginal utility of
software and divide it by the
marginal utility of clothes.
So we're going to take the
derivative with respect to
software and the derivative
with respect to
clothing and divide.
When we do this, we find that
the MRS is going to be equal
to 4 over S, which is our
marginal utility of software,
all over 6 divided by
C, which is our
marginal utility of clothes.
Solving through, we find that
our MRS is 4C over 6S.
Now, we have to think about,
conceptually, what happens
when software increases?
When we have S increase, since
it's in the denominator, we're
also going to have
the MRS decrease.
So what this means is as
software is increasing, or as
she has more software, she's
going to be willing to give up
fewer clothing, or less
clothing, to get another unit
of software.
So looking at our graph, when
she's at this point, she's
more willing to give up clothing
to get more software.
But when she has more software
down here, she's less willing
to give up the clothing.
Let's go ahead and move on to
Part B. Part B, find Xiao's
demand functions for software
and clothes--
so we're going to call
those QS and QC--
in terms of the price of
software PS, the price of
clothes PC, and Xiao's income.
Now, before we move on with
this, what we want to do is we
want to solve for one of the
variables C or S in terms of
the prices and the
other variable.
So to do this, we're going to
set the MRS equal to the price
of the software over the
price of the clothes.
From here, we can solve through
for C, and we find
that C is going to be equal to
3/2 times PS over PC times S.
Now, since we have
two variables--
we have a variable for clothes
and a variable for software--
we're going to have to introduce
another constraint
into this problem.
And the constraint that we're
going to introduce is going to
be the income function.
We know that Xiao has some sort
of income that's going to
be fixed, and she's going to
spend all of this on either
clothes or software.
Now, the amount of money she
spends on software is going to
be equal to the price of
software times how much
software she's going to buy.
The rest of her income is going
to be spent on clothes,
so the price of clothes times
the quantity of clothing.
Now, to solve for the demand
function for software, all
we're going to do is we're going
to plug in for C in the
income function here, and then
we're going to solve through
for S.
I know it's a little bit messy,
but this says PS times
3/2PS over PC, what we solved
for here, times S. Now, when
we solve through for S from this
equation, we're going to
find that the demand function
for software is going to be
equal to 2/2 times income over
the price of software.
Now, we can go through the same
process solving for the
demand function for clothing.
And all we'd have to do now is
we can take this S right here
that we just solved for, we can
plug this back into our
income function, and then we can
solve for C. When we solve
for C, we're going to find that
the demand function for
clothing is going to equal
3/5 times I over PC.
Part C asks us to draw the
Engel curve for software.
Now, all an Engel curve is, it's
a relationship between
the income and the quantity
that's demanded for a product.
And it shows us that as income
increases, it shows how the
quantity demanded is going to
change with changing income.
So to start off our Engel curve,
we're going to draw an
axes, we're going to put
software, or the quantity
that's demanded, on the x-axis,
and we're going to put
the income on the y-axis.
Now, the nice thing about the
software demand function that
we solved for is that it's
linear with respect to income.
Now, before we can graph this
equation, however, we have to
get it in terms of income.
So when we solve for this,
we're going to find that
income equals 5/2 PS times S.
So all our Engel curve is
going to look like is it's going
to be a straight line.
And the slope of that straight
line is going to be 5/2 PS.
And the way to interpret this
conceptually is to say that
with each one unit increase in
income, the amount that's
demanded is going to increase
by 2/5 divided by PS.
Let's go ahead and move on to
Part D. Part D says, suppose
that the price of software is PS
equals 2, and the price of
clothing is going to
equal PC equals 3.
And Xiao's income is
going to equal 10.
What bundle of software
and clothes
maximize Xiao's utility?
Now, we've already found the
conditional demand curves for
both software and clothes.
So we can start off this problem
by writing down those
conditional demand curves.
The conditional demand curve for
software was given by 2/5
I divided by PS.
And the conditional demand for
clothing was given by 3/5 I
divided by PC.
All we have to do now is we
have to plug in these
variables to solve for the
software and the clothing
that's going to be demanded.
When we plug those in, we're
going to find that she's going
to demand two units of both
software and clothes.
So this is in the scenario for
Part D. Part E gives us
another scenario that
we can solve for.
And all that's going to happen
now is that the price of
software is going to change.
And we're going to look at how
that affects the bundle that
maximizes her utility.
For Part E, it says, suppose
that the price of software
increases from PS equal to 2,
and now it's going to be PS is
going to equal 4.
What bundle of software and
clothes does Xiao demand now?
Again, we're just going
to solve through.
With our new PS equals 4, we're
going to solve for the
software and clothing
that Xiao demands.
We're going to find that S is
going to equal 1 now, and that
the amount of clothing
is going to equal 2.
So let's take a pause
right here.
And we're going to come
back in just a minute.
And we're going to look at the
more interesting case, which
is given the fact that she's
consuming less-- she has one
less unit of software
to consume--
how do we get her back to the
utility that she had before?
What amount of money or income
do we have to give her so that
she can be as happy as she was
in this initial scenario with
two units of both software
and clothes?
Welcome back.
So we're going to continue
onto Part F. Part F says,
given the price increase, how
much income does Xiao need to
remain as happy--
have the same utility--
as she was before the
price change?
What bundle of softwares and
clothes would Xiao consume if
she had the additional income
given the new prices?
So we want to find out, how can
we give her as much income
so she can be as happy as she
was to start off with?
To start this problem, the first
thing that we're going
to have to find out is we're
going to have to find out
exactly how happy Xiao
was to begin with.
So we need to know her
initial utility.
So let's start off with
that calculation.
To calculate her initial
utility, we're just going to
start off by saying that her
utility is equal to 4 natural
log of S plus 6 natural log of
C. And we can plug in 2 and 2
for S and C. In this case, when
we solve through, we find
that her initial utility
is 6.931.
So we're going to set her
utility equal to 6.931.
And what we want to find out
is we want to find out what
income we have to give her so
that she can get up to this
utility, given the new prices.
So we're going to take this
utility function, and we're
going to plug in the conditional
demand curves for
S and C so that income is now
a function, or one of the
inputs for her utility.
When we do that, we're going
to get this function.
And remember, we said that we
want, given this input and the
new prices--
so we're going to set this PS
is going to be equal to the
new price in the problem.
And we're going to also set the
PC equal to the price that
was in Part E as well.
And flipping back to the
problem, we know that the
price of clothing is going to
be equal to 3, and the price
of software for the second
part was equal to 4.
So we're going to plug in for PS
and PC, we're going to set
utility equal to 6.931, and
we're going to solve through
for I. When we do this, and when
we solve through for I,
what we're going to find is
we're going to have 6.931 is
going to be equal to 10 natural
log of I minus 4
natural log of 10 minus
6 natural log of 5.
Solving through, doing the
inverse natural log function
for I after isolating this
variable, we're going to find
that the new income that she
needs to be supplied is 13.19.
So the income that she needs to
be just as happy with these
prices has increased by 3.19.
Now, we can go back to our
conditional demand curves that
we had here.
We can plug in PS equals 4, PC
equals 3, and we can plug in
for income 13.19.
And we can solve S double
prime, which is the new
software that she's going to
demand, which will be 1.32,
and C double prime, the new
amount of clothes that she's
going to demand,
which is 2.64.
Now, the final part of the
problem, which we're going to
move on to now, which is Part
G, is actually the most
important part of the problem,
because what we're going to do
is we're going to tie
together the three
scenarios that we did.
We did this scenario where we
were giving her income so that
she would be just as happy.
We had our initial scenario
before the price increase.
And we had the scenario after
the price increase.
And we're going to look at this
conceptually on a graph,
and we're going to see, how do
we relate these three bundles
of consumption?
Part G says, going back to the
situation in Part E, where PS
equals 4 and I equals 10, we
need to decompose the total
change of softwares and
clothes demanded into
substitution and
income effects.
In a clearly-labelled diagram,
with softwares on the
horizontal axis, show the
income and substitution
effects of the increase in
the price of software.
Now, we're going to go back
to this graph that
we started off with.
And what we're going to do
here, is we're going to
illustrate the three bundles
that she selected.
I'll illustrate the first bundle
where she consumes 2
units of each.
And we already have our utility
curve, or indifference
curve, drawn up here.
Now we need to draw the budget
constraint that shows how much
she can spend on each product.
If she were to spend all her
money on clothes, she would be
up here at this corner
solution.
If she were to spend all her
money on software, she would
be down here.
When we connect a line through
here, this is her budget
constraint that shows all the
possible bundles of goods
where she could potentially
spend her money.
And this first bundle
is the point 2,2.
This is where she starts
off to begin with.
Now, when the price of software
increases, she's not
going to be able to buy as much
software with her money.
But she can still buy the
same amount of clothing.
So her new budget constraint in
this scenario is going to
look like this.
So in this scenario, which is
in our problem's Part E, her
utility has moved in
towards the origin.
And she isn't going
to be as happy.
And we can represent this on
a utility curve as well.
And we can see from this point
on the utility curve the way
that I've drawn it, that at
this point, she's still
consuming the same amount
of clothing.
But the amount of software
she's consumed
has been cut in half.
This is the total effect
of the price change.
It's the difference between
where she started and where
she's ending up without giving
her any money to change where
she actually is.
So the total effect
is just that she's
losing one unit of software.
Now, we can break down the
total effect in the
substitution effects
and income effects.
It's important that we really
understand conceptually how to
define substitution and
income effects.
And what we're going to think
about is we're going to think
about, on this graph, we're
going to represent the
substitution effect
by the movement--
if we were to just have the
price change, and we were to
give her income so she could
stay up at this utility level,
the substitution effect
is how her bundle
changes with that movement.
The income effect is going to
be-- since she's poorer
because the prices are
higher, it's going to
be the shift downward.
So I'm going to draw in the
scenario that we calculated in
Part F right here.
By drawing in this scenario with
the higher income level
and the price change, we can
represent this bundle as the
substitution effect.
So what this looks like is she's
going to have the same
budget constraint, only it's
going to be shifted back up.
This is going to be the bundle
in Part F. And we can label
the bundle 1.32, 2.64.
And this is where it's going
to get a little bit tricky.
The substitution effect is just
the movement from 2.2 to
the same utility curve but at
a different point with a
different bundle.
So it's when we've given her
income to keep her at the same
utility level, but we've
had the price change.
This is going to be the
substitution effect.
I'm going to label it 1.
Now, the income effect
is the next movement.
It's the movement that says,
well, we don't really give her
more income.
She's actually poorer.
It's the movement down from
1.32, 2.64, down to 1, 2.
And then the total effect
is just this
movement from 2.2 to 1.2.
So I can label the substitution
effect 1, the
income effect 2, and
the total effect 3.
So to calculate the substitution
effect, all it's
going to be is it's going to be
the difference between 2.2
and 1.32 and 2.64.
So in this case, our
substitution effect is going
to be equal to 0.68,
negative 0.64.
And you can see that we actually
had an increase in
the consumption of clothes for
the substitution effect.
And then the income effect,
using this equation and what
we calculated the substitution
effect and the total effect to
be, we find that the income
effect is equal to 0.32, 0.64.
So what this problem basically
had us do is it made us look
at the effect of a price change
on the consumption
decisions of a consumer.
So when a price increases,
two things
are basically happening.
The first thing that's happening
is the price of that
product is more, so the person,
in most cases, shifts
their buying away from that
product and towards the less
expensive product.
That's what this substitution
effect shows us.
The other effect that the person
feels is since the
price is higher, they can't
buy as much stuff with the
money that they have. So they
feel poorer, even though they
have the same amount of money,
because the prices are higher.
That's what this income
effect represents.
And the total effect is just the
summation of the fact that
the price is higher
for one good and
that they feel poorer.
And so we looked at the total
effect broken down into
substitution and income effect.
