Now, what can we do, we have already talked
about substitution effect and income effect
in graphical sense. So, we have already talked
about you know ah , this is basically we are
going to get the Slutsky equation. An equation
that gives relationship between compensated
demand and Marshallian demand. So, we have
already done it qualitatively, not quantitatively
ok. So, we have already looked at it through
graphs, but now we are going to do it mathematically.
So, what we can do here, because we are talking
about vary P 1. How do we get the x 1, x 1
is quantity demanded, either in Marshallian
sense or in compensated sense and to get generate
the demand function what do we need? We need
to get quantity demanded as function of P
1.
hum
So, basically P 1 is changing and we are studying
its effect on quantity demanded. So, here;
of course, we will take P 2 u naught you know
everything.
Constant
Constant u naught constant in this case, but
u naught here will, here u will change the
utility achieved will change, but basically
we are varying P 1. And if we differentiate
both side with respect to P 1 what do we get?
Remember here P 1 u naught is given; that
is given to the system, its per, its you know
you P changing P 1 will not change u naught,
but changing P 1 will change the expenditure
to achieve the u naught.
hum.
Is it clear. So, changing P 1 will affect
the quantity demanded in direct way and here
in.
Indirect
Indirect way. So, how can we write , this
is going to be , this is the direct effect
and also what we will have here is.
Indirect effect.
Di basically this is I ok.
Hum
With respect to the third argument and this
is going to be.
dP 1.
With respect to.
P 1
P 1.
hum
Fine, and how about here , this is what we
will get of course, P 1 will not change P
2 and P 1 will also not change u naught.
hum
Fine and here rather than using I, what we
are doing basically here is we have P 1 comma
P 2 u naught, is it clear ok. Now if you pay
attention what is this. This is the optimal
level of expenditure or minimum level of expenditure
to reach, to have at least u naught level
of.
utility.
Utility. So, basically what it is ; of course,
this is not perfectly correct, but we havent
discussed that much of mathematics in this
class. So, I am taking a kind of, you know
ah that doesnt look completely right, but
what this is is basically P 1 x 1 plus P 2
x two .
hum
Or here it is . This is what the total expenditure
is, and this is not for any amount, this is
for the optimal amount.
hum
So, and how did we get this of course, from
the minimization problem that I discussed
earlier. So, if I differentiate e with respect
to P 1 what will happen .
um.
This is, let me tell you, again this is not
perfect way, but this is going to be equal
to x 1 H.
hum
Of course you can say that x two is also a
function of P 1.
P 1 hum
But at optimal level it does not matter ok,
that is what its little bit advance ok, but
not that advanced. You can get this using
the equation ok and that will be your one
of the homework problem ok. Fine, you can
get it from here. And this we already know,
this we already know of course, P 1 P 2 and
here we have u naught. This is we already
know is equal to x 1 M P 1 P 2 e of P 1 P
2 u naught fine. So, we can put it back there
what do we get ? dx 1 M del, this is partial
derivative of x 1 with respect to P 1, what
is x 1? Its Marshallian demand.
So, what we are trying to get here is, the
slope of Marshallian demand function with
respect to P 1.
P 1.
What is it equal to .
ah.
x 1 M, this is x 1 M and this is the optimal
amount, I am writing in shortcut . This is
the Slutsky equation , this is called Slutsky
equation .
.
We will spend some time talking about what
does it mean. Although we have already studied,
but again it will be a revision math. Is clear
to you, how we got this. So, I can rearrange
it , I can rearrange it what I can write this
is , this has to be equal to 
and what does this represent? This is substitution
effect . Fine? What is substitution effect,
what we are doing.
hum
We are changing P 1 here and we are studying
its effect on x 1 M. How do we get the substitution
effect, that we change the P 1, but we do
not change the utility label, we remain on
the same utility label. So, this represents
x 1 H represents, you know its compensated
demand in the sense that utility remains the
same. So, this is giving us substitution effect.
And this term is giving us income effect . Let
us spend little more time and let us look
at, let us pause this income effect side x
1 M is a non negative number ok, it is quantity
demanded, it cannot be negative, quantity
demanded cannot be negative.
Hum
And what is this .
This we have already studied if it is greater
than zero then it is .
Normal good.
Normal good and if it is less than zero .
Inferior good.
It is inferior good . Fine. So, if P 1 goes
up .
hum.
If P 1 goes up , this we know this is so,
we are certain about, because of convexity
of the preferences ok , this is shaped like
this. So, if it is rotation P 1 is increasing,
what is happening if P 1 increases, then it
will rotate like this.
hum
Budget curve will become steeper.
hum
Because what is the slope of budget curve
minus P 1 divided by P 2.
hum
So, if P 1 goes up it will become steeper
and if it becomes steeper, because this particular
shape, the consumption of good one even in
the compensate, even in the compensated sense
would come down. So, P 1 goes up, this is
negative, as P 1 is increasing x 1 H is decreasing.
Now, let us look at this part x 1 M dx 1 M
del x 1 M with partial derivative of x 1 M
with respect to i. What is happening here?
This is negative, this is positive or non
negative and what is this for normal good,
for normal good what is this, for normal good
this is.
Positive . So, overall this is negative. So,
for normal good substitution effect is negative
and income effect is.
negative
Negative. We have already learned this, this
is just a repetition of the same fact and
how about inferior good, this is for normal
good .
hum
P goes up , just we have already established
this is negative and minus x 1 M this is.
zero.
Positive.
Because this is negative , this is positive
and this is.
negative.
Negative. So, negative negative positive and
this has two possibility that either you get
minus or .
plus.
Plus if you get minus fine, but if you get
plus.
Giffen good.
Then you get Giffen good . So, one requirement
for Giffen good is.
inferior good
That the good has to be inferior, the second
requirement is that income effect has to be
large.
large
Sir can you Giffen good ah.
Giffen good is a kind of good. So, now, what
is happening, if it is plus then what is happening
P 1 is going up and Marshallian demand of
that good is also increasing . So, here demand
curve is a an upward sloping curve. Why it
is happening let us think about it substitution,
we have already seen, substitution effect
is always it price goes up ok, compensated
demand would definitely go down.
Now, we have to look at the income effect.
Income we as already we discussed that income
its not necessary that when a income goes
up you consume more of.
A particular good.
hum
One example could be that you know potato
example that I gave you earlier, that income
goes up what would happen? You will decrease
the amount of potato that you would consume,
because potato people, people consume potato,
because they do not have enough money, they
will probably substitute it by meat or some
similar product or milk ok.
So, income goes up consumption of a good may
come down.
yes sir.
Ok.
hum
Fine. So, here it can be, it can move in any
direction, either negative direction or positive
direction and for inferior good income effect
is, a price, price goes up what would happen
that the purchasing power.
comes down.
Comes down, purchasing power comes down, it
means real income is coming down, but that
is why you will consume more of that good,
because if your income is coming down you
can no longer afford milk, mutton or some
other products. So, you have, you will have
to consume.
potato
More of potato. So, that is what is happening
here. So, now, the scenario here is, that
for normal good that we have figured out that
substitution effect and income effect they
work in the same direction , but for inferior
good they work in the opposite direction.
So, there is a theoretical possibility that
for inferior good that not only, you know
for ah that that income effect is larger than
substitution effect. And in that case what
would happen, the price would go up and consumption
of that good will also go up.
hum
And that sort of goods are called Giffen goods.
hum
Its very difficult to get Giffen goods in
the real life, because the condition that
income effect has to be large.
Large.
And its difficult to have large income effect,
because you know you do not spend really significant
amount of income on one particular good .
hum
You distribute your income over a large number
of goods ok. So, that is why its difficult
to get Giffen good. Its clear, Slutsky equation
is clear and substitution effect an income
effect with help of substitute Slutsky equation
is also clear. So, we have done it using graph,
graphs and also mathematically, fine .
