馃攳
Lecture-59 Marginal Utility Vs. Marginal Rate of Substitution (MRS) - YouTube
Channel: unknown
[0]
[Music]
[10]
[Music]
[12]
now let's use the concept let's learn
[19]
about marginal utility although we
[23]
talked about marginal utility earlier
[25]
also but just what is marginal utility
[28]
so it is the amount of amount by which
[32]
total utility increases if we increase
[36]
one good in the bundle by one unit while
[41]
keeping all other goods in the bundle
[44]
fixed and then it is marginal utility
[48]
with respect to that particular good
[51]
which amount has been increased okay so
[55]
let's say what we mean here is we have
[58]
utility some utility let's say for a
[61]
bundle let's say this is of course if we
[63]
are using notation X comma by that it
[67]
donates ax but it represents a bundle
[70]
what's happening and if we we can use
[73]
the our earlier notation also X 1 comma
[75]
X 2 what we have here is X 1 comma X 2
[84]
okay so what we are saying that we keep
[89]
X 2 fixed while we increase X 1 by 1
[93]
unit so what's happening X 1 plus 1 this
[99]
will be the new utility and let's say if
[102]
it satisfies monotonicity then of course
[104]
in the new bundle we have same amount of
[108]
good 2 but more of good 1
[111]
so of course here utility will be higher
[114]
so the increase in utility is let us say
[118]
that change in utility is denoted by
[120]
delta u then it's going to be u x 1 plus
[125]
1 comma X 2 minus u of X 1 comma X 2
[132]
and denominator although we don't have
[134]
anything in the denominator or we have
[136]
one in the denominator I can write it
[139]
like X 1 plus 1 minus X 1 that is what
[143]
we have and this this is basically
[147]
defined as marginal utility with respect
[151]
to X fine but here what we are doing we
[156]
are changing X 1 by 1 unit what is the
[159]
marginal no wait we will be a talk here
[165]
basically is again if you have you are
[168]
talking about Delta u by Delta X let me
[170]
explain it to you how we reach there
[171]
basically what we are talking about here
[173]
is Delta u is in the numerator and in
[176]
the denominator we have 1 1 we can
[179]
express as X 1 plus 1 minus X 2 and
[185]
that's the work that's that's equivalent
[187]
to 1 and this is marginal utility with
[191]
respect to X 1 I will come back to the
[194]
calculus the definition that we we gave
[197]
earlier using calculus now let us look
[200]
at the marginal utility with respect to
[202]
X 2 what it means that we keep x1 fixed
[206]
and we increase X 2 by 1 unit and here
[212]
we have and of course denominator we can
[216]
leave it as it is does not matter this
[218]
is marginal utility with respect to X 2
[221]
but now what we are doing we are
[225]
changing X 1 by 1 unit what we can do
[229]
that rather than changing X 1 by 1 unit
[231]
let us look at the change in change in
[235]
utility if we change X 1 by Delta X 1
[238]
unit very small unit and then what we
[241]
have here is x1 plus Delta X 1 comma X 2
[245]
minus u x1 x2 fine
[250]
and what we have this Delta U is now
[253]
change because of Delta X 1 unit change
[257]
in amount of good 1 in the bundle fine
[261]
so but Delta X 1 can be anything so
[264]
rather than talking about
[266]
absolute change we should talk about
[268]
rate of change and how can we get the
[271]
rate of change if we divide it by X 1
[275]
then this is the rate of change and here
[279]
also we can divide it by we will have to
[281]
divide it by Delta X 1 and Delta X 1 can
[284]
be written as X 1 plus Delta X 1 minus X
[290]
1 and now if we can take limit okay
[297]
where Delta X 1 is going to 0 what will
[300]
if we get Delta u by Delta Delta X 1 is
[309]
equal to limit X 1 is going to 0 u X 1
[315]
plus Delta X 1 comma X 2 minus u of X 1
[320]
comma X 2 divided by X 1 plus Delta X 1
[325]
minus X 1 and this is nothing but the
[328]
partial derivative of U with respect to
[331]
X 1 so both definitions are fine this is
[335]
more precise here we are talking about
[337]
rate of change in U with respect to X 1
[341]
while keeping X 2 fixed in the bundle
[345]
this gives us marginal utility with
[347]
respect to X 1 here we are taking a
[350]
crude way because some time we don't
[353]
know calculus then we use this this
[357]
definite if we if we don't know calculus
[359]
then we can use this definition ok and
[363]
then we have 1 in the denominator
[364]
because Delta X 1 is 1 in that
[367]
particular case fine and this is
[370]
marginal utility but here is the problem
[372]
we have learned about marginal utility
[375]
let us look at the one particular
[377]
problem that marginal utility utility
[380]
leads to let me draw this problem and
[388]
then I will come back to the problem
[390]
that we intended to solve right in the
[393]
beginning of this topic ok so what we
[397]
will do we will come back to that topic
[399]
and we will solve
[399]
using some other technique but we should
[401]
also learn the problem with the marginal
[404]
utility concept now what is happening
[407]
let us draw the indifference map 4x plus
[412]
2y okay or if we want to convert it if
[417]
you want to denote it by X 1 + 2 X 1 the
[420]
problem would remain the same it's the
[422]
same problem doesn't matter huh
[424]
so you can change the variable because X
[427]
1 and X 2 they are just representing
[428]
their the name so it doesn't matter so
[430]
here we have X 2 and here we have X 1
[434]
when we draw it how would it look like
[436]
downward sloping with slope minus minus
[440]
1 by 2 something like this it would look
[445]
like okay fine
[447]
and let's say let's start if we can say
[452]
this is we have here K 1 K 2 K 3 K 4 now
[462]
instead of using this notation K 1 K 2 K
[465]
3 K 4 can I use this notation it is 2 K
[468]
1 2 K 2 2 K 3 and 2 K 4 or in other
[472]
words rather than using X 1 plus 2 X 2
[475]
can I use 2 X 1 plus 4 X 2 will it
[480]
represent the same preference nothing
[483]
would change
[484]
why because utility is ordinal in nature
[487]
it's about order fine so now what we are
[491]
saying that this these two utility
[495]
functions they represent the same
[497]
preference nothing different
[499]
now get the marginal utility for both of
[502]
these utility function marginal utility
[504]
with respect to X 1 what will you get
[508]
marginal utility in case of let's say in
[512]
shortcut mu 1 represent with respect to
[515]
first argument we have also written it
[518]
as mu X 1 ok and what we have is d X 1
[524]
plus 2 X 2 with respect to X 1 and we
[528]
get here 1 or what we are saying is in
[532]
other word
[534]
if we don't use the calculus definition
[536]
what we we can use our calculus
[539]
definition so if we increase X one by
[542]
one unit
[542]
how much will be increase in total
[544]
utility one same as this okay fine now
[551]
how about mu1 in the let us denote this
[556]
utility function as you and this utility
[560]
function as V so what will be M this is
[564]
U and this is V what will be the
[569]
marginal utility in this case it's to
[579]
again use non calculus definition if you
[581]
increase X one by one unit how much
[583]
increase will you get while keeping X 2
[585]
fix how much increase will you get in
[587]
the total utility 2 according to this
[591]
now what's happening in one case we are
[594]
getting 1 in another case we are getting
[596]
2 so it seems that marginal utility is
[600]
related to it somehow cardinally in
[603]
nature cardinal in nature it assumes
[611]
that the value attached to a particular
[614]
rank is can be double it can be haft
[621]
okay so remember earlier we discussed
[625]
that these these were quite important
[628]
when we studied utility function as
[632]
Cardinal in nature Cardinal utility
[634]
function but now we have figured out
[636]
that utility we don't need cardinality
[639]
of utility function ordinality will work
[643]
well but when we are talking about
[645]
ordinality we should not be moved by the
[648]
value of mu 1 or MV 1 because they are
[652]
cardinal in nature so be very very very
[656]
of using mu 1 and mb 1 marginal utility
[660]
in your practical problems because you
[662]
will reach to the wrong place if you
[666]
don't know
[668]
fine so what is the solution the
[671]
solution we will see immediately again
[673]
let us solve this problem using the
[676]
technique that we have learned in the
[677]
class okay
[679]
earlier we solved it using just
[681]
description and then table now let's
[684]
solve it using the techniques that we
[686]
have learned and what did we learn that
[689]
we learn that mrs should be equal to the
[695]
slope of the budget line or in other
[702]
word mrs is nothing but the slope of
[705]
slope of indifferent curve fine now
[714]
let's calculate mrs in both the cases
[716]
what does it equal to it's equal to the
[719]
exchange rate that you have in your mind
[722]
at that particular bundle of course in
[725]
some of the cases your exchange rate
[727]
that you are comfortable with in your
[729]
mind would change as you have different
[731]
bundles but in this particular case
[734]
what's happening exchange rate remains
[736]
the same it doesn't depend on how many
[739]
units of good one or good - you have
[743]
fine so how much is mrs if we use
[747]
calculus to calculate m RS is equal to
[753]
this is what it is equal to okay I have
[757]
used earlier X 4 X 1 and y 4 X 2 but
[760]
let's stick to that this fine and how
[763]
much is this - for the first utility
[766]
case for the first utility case from
[770]
here here what we can get mu 1 is 1 and
[776]
mu 2 is 2 fine and for the second case
[782]
for the second case that is 2 X 1 plus 4
[786]
X 2 2 + 4 so what we get in the first
[791]
case it's 1/2 and if we take another
[793]
utility function that is
[799]
what we have is minus two by four and
[803]
that is half so mrs mrs is independent
[808]
of the particular selection of the
[811]
utility function what we have learned
[814]
let me just emphasize this point once
[816]
again why we are getting something like
[818]
this so what we learned earlier that if
[822]
we have preference such that this then
[826]
it you will be able to represent of
[828]
course it should satisfy some axioms
[830]
that we have discussed that this and
[835]
then any monotonic transformation of
[838]
this utility function would also work
[843]
this is what we have of course this
[846]
symbol says if and only if if and only
[851]
if so both ways fine for all X 1 and X 2
[857]
in the conjunction set so in other word
[861]
in other word what is V X 1 V X 1 of
[866]
course right now let's choose some other
[869]
because it will lead to confusion we are
[871]
using X 1 and X 2 for not different
[874]
bundles for to denote the amount of a
[877]
particular good in the bundle so rather
[879]
than using X 1 and X 2 we can take it
[882]
here P and Q we have on Q let us take Q
[887]
and R sorry because P again is for price
[904]
is it clear this is for all Q & R in X
[915]
so now when we take V of Q of course V
[920]
is monotonic transformation of U so what
[924]
how can we write it this is nothing but
[926]
G of U of Q okay and where G Dash is
[932]
greater than 0 by our definition we have
[935]
discussed it in the class fine so now
[939]
let us calculate m RS using this
[943]
particular function and what is this
[945]
equal to minus D V with respect to X 1
[952]
because remember we are talking about
[955]
two good world Q has two goods X 1 and X
[958]
2 divided by partial derivative of V
[963]
with respect to X 2 fine and when we use
[968]
this what we will get - G - D u partial
[976]
derivative of U with respect to X 1 -
[980]
again here G - partial derivative of U
[985]
with respect to X 2 so this will get
[991]
cancelled and we are back to the mrs
[997]
which we calculate we calculated with
[1001]
the first utility function representing
[1003]
the preference of this particular person
[1007]
so mrs is independent of monotonic
[1010]
transformation of utility function so it
[1013]
doesn't depend on the particular
[1015]
valuation it's X - yes fine it's clear
[1020]
[Music]
[1039]
[Music]
Most Recent Videos:
You can go back to the homepage right here: Homepage