Welcome everyone. Welcome to the second
part of chapter four. in this chapter we
will we the different utility
functions and how to find optimal
choices so let's start with the
definition of the optimal choice the
optimal choice of good 1 and good 2 at
some set of prices and income is
called the consumers demanded bundle so
when we find the quantities at the
optimal choice so the highest
indifference curve which satisfied the
affordability constraint we will find
the consumers optimal choice which will
be called as the demand bundle so when
we change the prices or the income then
the consumers optimal choice will change
for sure so basically we say the demand
function is the function that shows the
relation between the optimal choice that
means the quantities demanded for
different values of prices and income
for instance if I maximize my utility x1
and x2 try to find the optimal
quantities of x1 and x2 subject to the
budget constraint I will find my x1 star
that depends on the prices and the
income and x2 star
same that depends on price and income
these are called as the demand function
so here the demand function try to
understand or try to find the answer to
such question how many units of x1 and
x2 to consume at a given prices and
income? so when the consumer choose the
best affordable bundle for sure. so for
this reason we can say that when we
write the demand functions that depend
on both prices and the income for each
different set of prices and income
there will be a different combination of
goods which we will call them as the
optimal choice of the consumer so
changing the prices and the income will
definitely affect the the quantity we
consume at our optimal choice for
instance if we find the demand function
for the good one
after we maximize the utility we find
the function as 5 P 2 M over P
1 if the prices are one and two dollars
an income is ten dollars so my x1 star
will be 5 multiplied by 2 multiplied by
10 divided by 1 so this will be 100
units but if the prices change so the
price one change to $10 two change to $1
and the income increase to $20 my x1
will be 50 units basically what we can
say that different prices and
income level will create the changes on
the quantity demanded and also the
different preferences such as the
perfect substitutes,
perfect complements etc that will give
us different demand functions so we have
two main points two important point to
understand the consumer demand one is
the prices and the income the second one
is the preferences or simply what is
the utility function so let's have a
look to some examples what happens if my
utility function is given by perfect
substitute so if x1 and x2 are perfect
substitutes and one unit of x1 is as
valuable as B over A unit of x2 that
means X 1 is equal to P over AX 2 then
we can write utility function as X 1
plus B over A X 2 or if I make the
monotonic transformation I will arrive
to my general
formula for the perfect substitutes it
will be A X1 plus B x2 so I'm going to
maximize such perfect complements
utility function subject to the budget
constraint and I need to check the
relation between the marginal rate of
substitution that is the slope of
indifference curve which is the ratio of
MU 1 over MU 2 so it is minus A over B
because when I take the derivative of my
utility according to x1 I have my MU 1
which is A and same idea mu 2 is B and
the relative price which is the slope of
the budget line will be minus p1 over p2
so here in this case I have three
potential solutions one is the absolute
values in absolute values my MRS is
larger than my price ratio or my MRS
can be smaller than my price ratio or it
can be equal because A and B are two
numbers that means my MRS the slope of
indifference curve is constant
and same as the price ratio it is
constant so I can check all
possibilities one is larger than the
other or smaller than the other so if my
MRS is smaller than the slope of budget
line so let's say that is the second
case here if my MRS is smaller than p1
over p2 my indifference curve which is
flatter than my budget line because the
slope of my indifference curve in
absolute value is smaller than the slope
of my budget line so here for instance I
have my budget line given by such line
and I'm trying to maximize my utility
function from passing one indifference
curve to the other one so here for
instance I'm on my indifference curve
one and there is an intersection with my
budget line
so here I have a corner solution M over
P 1 but what I know if I keep increasing
my indifference curve to indifference
curve 2 then I can pass to A higher
indifference curve higher utility when I
consume point at point B but also it is
possible to keep increasing my
indifference curve to IC3 and I
still have 1 intersection between the
indifference curve and the budget line
which is given by point A so point A B
and C they are all on the budget line
but A is the optimal bundle because the
A is giving me the highest utility level
so if my indifference curve is flatter
than the budget line I have a corner
solution where the consumer consumes the
whole income for for good 2 so in this
case I can write  0 consumption of x1
and spending the whole income for good
to an M over P 2 is my optimal bundle so
if you understand that idea it is not
that challenging for the opposite: MRS is
larger than the price ratio so
indifference curve slope is steeper than
the budget line then I will have only
one solution which is a corner solution
that the consumer spends the whole
income for good one and not buying any
good so my optimal choice will be M over
P 1 and 0 unit of x2 however if they are
equal if the slope of my indifference
curve is equal to the slope of budget
line has same slope of budget line so
any bundle on the budget line will be an
optimal choice it doesn't matter which
point we choose because they are all
giving me the highest possible utility
and they all satisfy the
affordability constraint so let's have a
look at a basic example of perfect
substitutes which is one-to-one basis x1
plus x2 like the red and blue pencil
example and here what I have my MRS is
minus 1 in this case if the price ratio
is smaller than 1 I have a budget line
which is flatter than my indifference
curve so i have the optimal choice where
the consumer spends whole income for
good one if the price ratio is larger
than one that means if price of good one
is larger than price of good 2 for
one to one substitution then i have a
budget line which is steeper than my
indifference curve and I have my optimal
bundle as a corner solution where the
consumer spends whole income for good
2 and not buying any good one if the
prices are equal it doesn't matter for
the consumer to choose any bundle
another example is the perfect
complement so the perfect complement
says x1 and x2 are perfect complements
so I cannot substitute I need both of
them and one unit of x1 is as valuable
as B over A unit of x2 which means X 1
is equal to B over A X 2 then I can
write my utility function as the minimum
of X 1 or B over AX 2 by making a
monotonic transformation I have my
general formula for perfect complements
minimum of AX 1 or BX 2 so in this case
I'm going to maximize my consumer
utility to find X 1 and X 2 levels
subject through the budget line so how I
can find that I know that the equality
between x1 and x2 according to the giveN
function it is x2 equal to A over B x1
it must be on the budget line so what we
know the optimal solution for the
perfect complements must be at the kink
point of the indifference curve so it is
that corner point of my L shape
indifference curve so that means at this
point I have the equality inside my
minimum parenthesis AX 1 equal to BX 2
and when I plug this into the budget
line I replace this X 2 here in my
budget line instead of writing X time
writing A over B X X 1 and when I solve
such equation I will have my optimal
quantity for X 1 which depends on A B
and P 1 and P 2 and because X 2 is A
over B x1 then I replace this X 1 with
X 1 star in order to find X 2 star which
is A M over B P 1 plus AP 2 that is my
optimal solution for perfect complements
so if I have one-to-one basis if I have
my minimum function equal to x1 x2 like
the left and right shoes what I have at
this kink point I have x1 equal to x2 so
let's solve it  by using some
numbers if the price of good one is $1
price of good 2 is 2 dollars and my
income is $30 so I have the equality
inside my min parentheses x1 equal to
x2 I plug this into my budget line so
instead of extra I'm going to write x1
and when I sold I have M over P 1 plus P
2 which is equal to x2 star as well so I
replace these prices here and the income
as well so I will get X 1 star equal to
X 2 star which is equal to 10 so my
optimal choice
we'll be ten and ten when the prices are
one two dollars and the income is $30
another example is the neutrals and bads
in this case what we know if the good is
a neutral the consumer will spend all
her money on the good that she likes and
will not purchase any of the neutral
good and that happens exactly the same
for the bad if a good is a bad the
consumer will not buy any unit of bad
but send the whole the whole income for
the other good so in this case if good
one is good is a good and good two is
something bad or neutral then the
consumer will spend all the income for
good one so it will be M over P 1 and
not purchasing any of bad or neutral
good and what happens if my goods are
discrete suppose that the good one is a
discrete good and available
only in integer units and good two is a
composite good so price is $1.00 and if
the consumer chooses one two three units
of good one the rest of the income will
go for the good 2 so let's have a
look if the consumer by one unit of x1
she will spend $1 let's say if the price
if p1 is let's say $1 then she will
spend $1 multiplied by p1 so it will be
just 1 $1 and the rest of the income M
minus p1 so it is M minus 1 dollars will
be spent on good 2 so let's cancel
that assumption let's say the price of
good 1 is just p1 and by buying two
units of x1 she will spend $2
multipled
by p $1 for good one and the rest of
the income M - 2 P 1 will be spent on
good - so we have a consumption bundle
here 1 and M minus P 1 or 2 and M - 2 P 1 etc so I keep finding the
consumption bundles and then I need to
compare the utility levels of consuming
one good one two units of good one and
units of good one and find the highest
utility possible in order to find the
optimal choice for the consumer so let's
say if this is my budget line my optimal
choice will be the unique optimal choice
is buying 0 unit of x1 and spending the
whole income for good 2 if that is my
budget line I have such point as the
optimal choice which means the the best
thing for the consumer is to buy 1 unit
of good one and spend the rest of the
money for good 2 what happens if my
preferences are not convex but they are
concave so for the concave preferences
for instance my utility can be X 1
square plus X 2 square so here my
indifference curves have such shape and
let's assume this is my budget line so
here I have a tangency condition at
point X and a corner point at point Z X
satisfy the tangency condition however I
can have my best or the highest utility
if I buy the consumption bundled Z so
let's solve it with some numbers my
utility function is given by x1 square
x2 square and the prices are $1 $2 and
income is $20
what is the best affordable consumption
bundle or what is the optimal choice for
the consumer so because I know that such
function is not
convex I cannot use the tangency
condition what happens if I use it if I
maximize my utility subject to the
budget line so p1 is $1 so it is x1 + 2
x2 equal to 20 that is my budget line by
using the tangency condition the
marginal utility 1 over  to the
marginal utility 2 equal to the price
ratio if I solve it I will find x1 equal
to 4 and X 2 equal to 8 so here let's
need to check whether it is a good point
for me or not if I checked optimal
choice of an interior solution 4 and 8
and a corner solution I compare these
optimal choices so by using the tangency
condition I find an interior solution
4 and 8 and I will check whether it is
definitely good for me to use that one
or a corner solution can be better
so the corner solution means i spend my
all income for good 2 i can buy 10 units
because I have 20 dollars and the price
is $2.00 and nothing from x1 and if i
consume all my income for good 1 then I
can afford 20 units of good 1 and 0
units of good 2 so let's check the
utilities 4 and 8 0 and 10 20 and 0 so
the utility from an interior solution
will give me 80 units the utility of
consuming just x2 giving me 100 units
and the utility just consuming x1 gives
me 400 units since the highest
utility is 400 then I can say that the
optimal solution is just consuming x1
and not consuming x2 at all and finally
let's have a look to the cobb-douglas
preferences so if x1 and x2 are neither
perfect complements nor perfect
substitutes
I can write my utility function as x1
power C x2 power D
then what I know my cobb-douglas is a
strictly convex preferences so the
tangency condition which means the MRS
equal to the price ratio is a sufficient
condition for me to find the optimal
choice so let's have a look to an
example if x1 x2 the utility is given by
x1 square x2 power 3 and the price are
two and three dollars and the income is
100 dollars I can solve that equation by
using the the MRS equal to the price
ratio so here let's use this general
formula first but what it says my
marginal utility is C x1 power C minus 1
and x2 power D and marginal utility 2 is
D X 1 power C X to power D minus 1 so my
MRS is basically C over D X 2 over X 1
right
so in this case what I can find my
optimal solution will be C X 2 over DX 1
equal to P 1 over P 2 so I have X 2
equal to DX 1 P 1 C P 2 and when I plug
this into the budget line instead of
using X2 if I use that equation after
solving I have X 1 star equal to C over
C plus d MP 1 and X 2 is D over C plus D
M P2 so if I have the prices through
three dollars and hundred dollars as the
income by replacing that is my C that is
my D these are the prices and the income
so here I replace C by 2 and C plus D by
five
income is 100 and price 1 is 2 I got 20 for
X 1 and I got 20 for X 2 which means that my
optimal solution is 20 and 20 or if I
make a monotonic transformation from X 1
power C X 2 power D I replace C over C
plus D by a I got my utility x1 x2 is x1
power A X to power 1 minus A or I can
use the logarithm which is A Ln X 1
plus 1 minus A Ln X 2 I have my tangency
condition when my MRS A over 1 minus A X
2 over X 1 equal to p1 over p2 I got the
relation between X 1 and X 2 and by
using that X 2 in my budget function
here I get x 1 equal to A M over P 1 and
X 2 equal to 1 minus A M over P 2 that
is my optimal choice so if I check the
graph here I have the tangent points at
this point I have my optimal choices of
X 1 and X 2 you can repeat the same
process for the logarithm so I have the
utility function given by the Ln
function and maximize this function
according to the budget line at the
tangent point I have A  over 1
minus a X 2 over X 1 equal to p1 over p2
and I just plug into my budget line then
I will find exactly the same optimal
choice as I found previously so if I
know the values for A and P 1 then I can
easily determine the quantities in real
numbers okay that's the end of part
2. Thanks for listening. We will meet in
part three shortly.
