馃攳
Lecture-84 Elasticity of Substitution - YouTube
Channel: unknown
[0]
[Music]
[10]
[Music]
[14]
now
[15]
what we have is the elasticity of
[18]
substitution
[19]
what we have done let us look at mrts
[22]
what is mrts
[24]
mrt s is basically
[27]
slope of
[29]
iso quant and this is
[32]
mpl divided by mp
[36]
k
[37]
although we haven't done
[39]
but
[41]
look at this
[42]
isoquant ok when you want to produce
[47]
we will do this in more detail later on
[49]
the the concept that i am talking about
[51]
what we are talking about is here
[54]
that to produce q naught amount of
[56]
output all these combination of inputs
[59]
are
[61]
fine
[62]
all these combination of inputs are
[65]
efficient in the sense that we would not
[67]
be wasting any input to produce q naught
[70]
amount of output
[73]
but it does not mean that we can pick
[76]
any one of these
[77]
these are technically feasible we are
[80]
talking about technical constraint but
[82]
how how about the economic motive
[85]
you want to maximize your profit
[88]
or you want to minimize your cost to
[90]
produce the same amount of output we
[92]
haven't talked about cost minimization
[94]
or profit maximization that we will do
[96]
shortly but this is very simple that you
[99]
should be able to understand so a point
[101]
you will pick a point such that
[105]
the cost
[106]
of producing q naught amount of output
[108]
is minimized
[110]
so how will you pick at that point this
[114]
mrts will play a very important role if
[118]
you remember the concepts from consumer
[120]
theory what did we do
[122]
there
[124]
the counter part of mrts that is
[126]
marginal rate of substitution should be
[129]
equal to
[131]
the ratio of
[133]
market prices
[135]
here it is the same thing
[138]
okay
[139]
so what we are talking about is that it
[142]
has something to do with the
[144]
mrts should be equal to the price ratio
[148]
at the optimum level
[151]
fine
[153]
although i haven't discussed it in
[155]
detail okay but you do not need to know
[157]
this so mrts can proxy for the price
[160]
ratio at the optimal level at all the
[162]
optimal level
[164]
fine and what is elasticity
[167]
i am going to talk about it in little
[168]
differently also without using the
[170]
concept from profit maximize what is
[172]
elasticity uh how much does the value
[174]
deflect corresponding to something else
[177]
when we say
[178]
go back to if what we have learned price
[182]
elasticity of demand
[186]
what did we talk about the percentage
[188]
change in the quantity due to one
[190]
percent change in price price
[193]
okay that's what we have or it is rate
[196]
of proportional change in demand with
[198]
respect to proportional change in price
[201]
that's what we have talked about so what
[203]
we are talking about is elasticity of
[206]
substitution
[207]
we are not talking about we are not
[209]
saying here price elasticity of
[210]
substitution but what we are talking
[213]
about is
[214]
some sort of elasticity of substitution
[217]
so how the substitution
[219]
the proportional substitution changes
[222]
as
[223]
the proportional price price
[226]
proportional price change in the changes
[228]
in the market
[230]
and we do not have proportional price
[232]
what do we have to proxy for
[234]
proportional price we have mrts so what
[237]
we can say the proportional change in
[239]
substitution with respect to
[241]
proportional change in mrts and that is
[245]
elasticity of substitution
[248]
ok so let me write it here in this
[250]
particular
[251]
context it is
[258]
this is proportional change in the
[261]
inputs that you are using
[263]
this is instead of using this partial
[265]
derivative sign what you can do you can
[268]
write delta
[270]
ok proportional change in the ratio of k
[272]
l
[273]
ok
[275]
with respect to
[278]
proportional change in
[280]
m r t s instead of taking m r t s i am
[284]
taking the absolute value of m r t s
[286]
because m r t s is negative so does not
[289]
matter i am taking the proportional
[291]
value of m r t s
[293]
so in a sense it is very similar to the
[297]
what we had learned earlier
[299]
what we can if we can rewrite it what
[301]
will we get
[303]
you can rewrite it like this
[306]
k by l
[307]
divided by
[319]
fine that is one way to write it
[322]
another way to write it is
[324]
take log and you put here
[327]
and of course we are missing the minus
[329]
sign typically
[331]
ok
[332]
what we have here is basically
[336]
delta
[339]
l n
[341]
k by l
[343]
divided by
[344]
delta
[347]
l n
[348]
m r t s what is l n
[351]
natural log
[353]
ok so this can be written in the form of
[356]
natural log and that is what we get and
[359]
the denominator here in this part can be
[362]
written as
[363]
this
[364]
fine
[366]
ok
[367]
ok let me
[369]
let me not use the concept of profit
[372]
maximization because we haven't used it
[374]
yet
[375]
okay
[376]
without using the concept of profit
[378]
maximization i will again try to explain
[381]
what is the elasticity of substitution
[385]
and let us look at it here graphically
[389]
we have two graphs
[392]
fine and
[394]
let us take
[396]
one where we have cobb douglas function
[399]
this is isoquant in the case of
[402]
cobb douglas function
[404]
and let us take here
[407]
isoquant
[410]
in case of
[411]
perfect
[413]
complement
[414]
inputs
[416]
now let us say
[419]
this is the
[420]
mrts
[424]
at this particular point
[426]
which point will you choose again we are
[429]
again using concept from profit
[430]
maximizing but not
[432]
explicitly you you would not choose
[435]
choose a point here why because anyway
[437]
you are wasting this much of
[440]
labor
[441]
and labor is costly so you do not want
[443]
to use that point so you will always
[445]
produce here
[446]
at the corner
[448]
because you are not wasting any of the
[450]
inputs
[451]
so how can we define mrts over there it
[453]
can be any line
[455]
huh
[458]
good point i should not say it's mrts
[462]
what i should
[463]
let me change it little bit
[466]
let me change it little bit
[468]
what we have instead of
[471]
instead of steep corner sharp corner
[475]
what we have is a
[477]
curve
[480]
good point my mistake
[484]
fine
[486]
now we do not have a sharp corner
[488]
ok
[489]
its differentiable everywhere
[492]
fine again roughly
[494]
your
[498]
whenever you figure out
[501]
at where you want to produce you will
[503]
probably you will be producing probably
[505]
here
[506]
in this zone
[508]
fine
[509]
now lets say this is the mrts here
[513]
fine
[515]
if mrts changes
[518]
this is mrts old
[521]
and this is mrts
[524]
new
[526]
again
[528]
k by l will not change significantly
[532]
in this particular case
[534]
how about here
[539]
in this case let us say the change is
[541]
same i am trying to you know
[552]
old
[553]
mrts
[555]
nu
[557]
its not slight of fan its
[559]
this is here k by l would change
[562]
significantly in comparison to the
[565]
previous case
[567]
because from here you will move to this
[570]
point again one thing that i am silent
[572]
about why i am choosing this particular
[574]
mrts
[576]
why this mrts because what i am talking
[579]
basically talking about that earlier the
[582]
price ratio
[584]
was this because mrts is in your control
[587]
you change the point
[589]
the combination and your mrts will
[592]
change isn't it here you can say that i
[595]
will produce here i will produce here
[597]
and accordingly mrts will change but
[600]
which
[602]
where you will produce actually where
[604]
mrts is equal to the price ratio so
[607]
again i have gone back to the previous
[608]
concept that we have just talked about
[611]
okay
[612]
fine so in that case you are producing
[615]
here because this mrts represents the
[618]
price ratio in the market
[620]
so if mrts you need to change because of
[624]
price changes in the market
[626]
and this is the new mrts what will
[629]
happen the k by l will change
[631]
significantly
[633]
here k by l will not change
[635]
significantly because we are talking
[636]
about substitution remember these two
[639]
are
[640]
near perfect complement
[644]
ok so substitution is not taking place
[646]
that much but here substitution would
[649]
take place
[651]
fine is it clear
[653]
so basically if we go back and look at
[656]
the geometry what basically happening
[658]
when we are talking about mrts we are
[661]
trying to measure the slope of this
[664]
isoquant
[666]
but when we talk about elasticity of
[668]
substitution what we are trying to
[670]
measure the curvature
[673]
curvature of this isoquant it represents
[676]
the curvature of isoquant how
[680]
the different combination of how the
[682]
combination of input to produce the same
[685]
amount of output would change with
[688]
changing prices in the market
[691]
and that is what we talk about in the
[693]
elasticity of substitution is it clear
[697]
if you want we can come back to it again
[700]
when we talk about profit maximization
[703]
and there we will figure out that mrts
[706]
is nothing
[708]
where the production will take place it
[710]
is equal to p
[712]
what would it be equal to
[714]
p
[714]
l by p k
[716]
minus 1
[718]
fine
[719]
ok
[721]
so its doing this the proxy m r t s is a
[724]
kind of a proxy for p l by p k
[726]
and that we will see shortly so now let
[728]
us calculate
[733]
mrts we had calculated mrts for a
[736]
particular case of cobb douglas function
[739]
and do you remember what did we get
[743]
minus
[745]
minus
[747]
of course its always minus mpl by mpk
[750]
but what did we get in case of cobb
[752]
douglas function
[754]
check
[755]
a
[756]
k
[764]
fine
[765]
can we calculate now elasticity of
[767]
substitution how can we calculate the
[770]
elasticity of substitution re take
[773]
take here
[775]
first
[777]
the absolute value what do we get b a by
[781]
k l
[782]
and what we have learned that elasticity
[784]
of substitution is nothing but
[799]
this is what we have learned
[801]
fine so we have everything we can
[804]
calculate and let's see what what do we
[807]
get
[808]
from here if we take
[810]
log both side
[814]
we get log b by a plus
[818]
log k by l
[821]
and this is the property of
[823]
log
[824]
log
[825]
a multiplied by b is equal to log a plus
[829]
log b
[831]
fine
[832]
and that is what we have used
[835]
we differentiate
[837]
both side with respect to log of
[840]
absolute value of mrts what do we get
[843]
here we get 1
[846]
this if we differentiate with respect to
[849]
log of mrts what do we get 0
[852]
and then here what do we get
[855]
ln k by l
[859]
m rt mrts
[863]
so what we have got and this is our
[868]
elasticity of substitution so in the
[871]
case of cobb douglas it is equal to
[874]
1
[875]
and of course based on it we we can
[878]
figure out
[879]
a new production function where because
[882]
this is quite useful
[883]
that
[885]
a special class i am not going to
[886]
discuss it in detail a special class of
[889]
production function for which
[894]
elasticity of substitution is always
[896]
constant not just one always constant
[899]
some constant value
[901]
and of course then cobb douglas would be
[903]
a special case of that kind of
[905]
production function because for cobb
[907]
douglas also it is a constant value one
[911]
fine
[913]
so that
[915]
that particular class of
[918]
production function let me write it here
[920]
why don't you change it why don't you
[922]
check it calculate let me write it here
[927]
a
[928]
k
[930]
b
[931]
l
[935]
or let me write it this way
[943]
this is of course a production function
[945]
q is equal to
[947]
a k to the power rho
[950]
plus b
[951]
l to the power rho and whole
[953]
to the power one by rho and of course
[956]
there should be some restriction on rho
[958]
what should be that restriction
[960]
greater than zero
[962]
rho is greater than zero probably check
[964]
rho is greater than zero and what you
[966]
need to do now calculate the
[970]
elasticity of substitution
[972]
in this particular case
[975]
and then so that cobb douglas function
[978]
is nothing but a special case of this
[981]
production function that is the homework
[983]
you need to do
[984]
fine
[986]
okay
[987]
[Music]
[1006]
[Music]
[1019]
you
Most Recent Videos:
You can go back to the homepage right here: Homepage