00:00:01
um okay guys in this episode we're going
00:00:04
to talk about
00:00:05
core allocation all right so what do we
00:00:07
mean by coral location
00:00:09
so at our location is called core
00:00:12
allocation if the following three
00:00:15
properties hold one
00:00:17
it is feasible while normally pretty
00:00:19
optimality
00:00:20
already uh includes the feasibility i
00:00:23
mean if
00:00:24
if you're talking about a proto-optimal
00:00:26
allocation it should be feasible anyway
00:00:28
but i just wanted to uh sort of
00:00:31
underline it
00:00:32
i wanted to underline it anyway so a
00:00:35
core allocation
00:00:36
must be feasible so that's the first
00:00:38
property second it must be pretty
00:00:41
efficient and then finally third it
00:00:44
should give
00:00:45
both agents higher utility than their
00:00:48
initial endowments
00:00:50
all right so i'm going to sort of
00:00:52
formally write it down
00:00:53
uh but let me just say you know a few
00:00:56
things before going to more formal
00:00:58
treatment
00:00:59
well up until this point i mean when we
00:01:01
define
00:01:03
protoefficiency we did not really use
00:01:06
uh initial endowments we used initial
00:01:09
endowments
00:01:10
just to create the edgework box and
00:01:13
determine
00:01:14
the set of feasible allocations all
00:01:16
right so
00:01:17
therefore per the optimality i mean in
00:01:20
order to be able to find predator
00:01:21
optimal allocations
00:01:23
you have to know the initial endowments
00:01:25
the agents bring to the market because
00:01:27
otherwise you cannot
00:01:29
know the feasible allocations however
00:01:32
proto-optimality or predator efficiency
00:01:34
does not require uh to
00:01:37
so i mean the definition of pred
00:01:39
optimality
00:01:40
has nothing to do with the initial
00:01:42
endowments
00:01:44
again it indirectly is affected by
00:01:47
uh initial endowments because we care
00:01:50
about feasibility
00:01:51
but otherwise it's not directly related
00:01:54
however
00:01:54
coral location is directly related to
00:01:58
the initial endowments
00:01:59
all right so remember the our idea was
00:02:02
the following
00:02:03
uh you know two agents or more agents
00:02:06
get together in a market environment
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they bring some good one and good too
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all right
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there's no production and so if they
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want they can trade these goods
00:02:16
why do they trade well to make
00:02:18
themselves happier
00:02:19
well that was the idea so if they trade
00:02:23
where are they going to end up well they
00:02:25
may end up
00:02:26
in their uh with their initial
00:02:28
endowments meaning they actually don't
00:02:30
trade any goods uh if this is what they
00:02:33
prefer well fine
00:02:34
but they may actually end up somewhere
00:02:36
else by trading good one with good two
00:02:39
well for those you know what should be
00:02:41
the solution of a
00:02:43
potential trade we said well it should
00:02:45
be feasible the final outcome it should
00:02:47
be pretty efficient
00:02:48
meaning the agent should improve their
00:02:51
situations
00:02:52
up to the point where there's no further
00:02:55
improvements
00:02:56
all right so they shouldn't end up some
00:02:58
allocation where
00:02:59
actually improvement is possible all
00:03:01
right so if they're going to end up and
00:03:02
say well
00:03:03
i'm happy with this trade they say well
00:03:06
that was i mean that is
00:03:07
you know in some sense the best we can
00:03:09
achieve
00:03:10
uh at least jointly and then finally
00:03:13
obviously
00:03:14
the trade is uh is not compulsory it's
00:03:17
it's
00:03:18
optional if you don't want to trade you
00:03:20
can always
00:03:21
take your initial endowments go home and
00:03:24
and enjoy consuming them
00:03:26
so therefore if you're going to trade
00:03:28
and end up with this initial bundle
00:03:30
you should prefer this bundle all right
00:03:33
to your initial endowments so that's why
00:03:36
we use the idea of core
00:03:39
well so more formally in an environment
00:03:42
where
00:03:42
there are two agents with those
00:03:44
preferences with these initial
00:03:46
endowments
00:03:47
what do we mean by the allocation is a
00:03:50
core allocation as i said has to be
00:03:52
feasible
00:03:53
has to be pareto efficient and then
00:03:55
third the agent's
00:03:57
utility agent a and b's utility when
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they consume this allocation
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should be greater than or equal to
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doesn't have to be strictly
00:04:06
all right greater than or equal to to
00:04:09
their
00:04:09
unit utilities when they consume their
00:04:12
initial endowments all right so as
00:04:15
simple as this
00:04:16
let me give you one sort of simple
00:04:18
example so consider
00:04:19
two agents a and b again two goods x1
00:04:22
and x2
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the utility functions are as follows the
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first guys utility function is called
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douglas type
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and so it has a convex uh indifference
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curse
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and agent b is utility function however
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so these two goods are perfect
00:04:36
substitute for him
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so nevertheless we can use margin rate
00:04:41
of substitution a equals marginal rate
00:04:43
of substitution b
00:04:44
to find uh pretty optimal allocations
00:04:48
uh because uh i mean you'll see and
00:04:51
let's suppose the initial endowments are
00:04:53
as follows all right so the total unit
00:04:56
uh for good one and good two are three
00:04:58
units
00:04:59
well let's first find the predo optimal
00:05:02
allocations all right so
00:05:03
question is again is that are these guys
00:05:05
going to make any trade or are they
00:05:08
going to
00:05:08
consume what what they are already
00:05:10
holding
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if they are going to trade what is the
00:05:14
best possible solution
00:05:15
well so let's suppose the i mean
00:05:19
not suppose but let's try to find the uh
00:05:21
pretty optimal allocations first
00:05:24
well the pareto optimality means margin
00:05:26
rate of substitution of agent a which is
00:05:28
this term
00:05:29
equals to marginal rate of substitution
00:05:30
of agent b which is simply -1
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so they must be equal for predator
00:05:36
efficiency so
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basically that means whenever agent a's
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consumption of good one equals to uh
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agent a's
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consumption of good two well that point
00:05:46
is pretty optimal
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all right so if you drove the edgeworth
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bucks it's a three by three
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uh box right a square
00:05:56
so agent a is on this corner b is on
00:05:58
this corner
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so all those points where x1 is equal to
00:06:02
xa
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is basically this 45 degree line so this
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45 degree line
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starting from this point up until this
00:06:10
point
00:06:10
is the contract curve so this is the set
00:06:13
of all predator optimal allocations
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contract curve all right so remember
00:06:20
uh our final outcome should be feasible
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meaning
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it has to be in this box it has to be
00:06:26
pretty efficient so i reduced
00:06:29
my attention to only those points
00:06:32
that are on the contract curve but which
00:06:35
one of them exactly
00:06:36
well so basically initial allocation is
00:06:39
not pretty
00:06:40
efficient all right meaning uh they
00:06:42
should actually do trade because if they
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don't do
00:06:45
trade uh they are going to regret it
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because
00:06:49
uh improvement for both agents was
00:06:52
possible is possible so they should
00:06:54
trade all right
00:06:55
but trade and end up what well for this
00:06:59
we use the third concept core allocation
00:07:03
well how do we find it well i mean
00:07:05
simple you can do it graphically or you
00:07:08
can do it
00:07:09
algebraically so i'm going to do both
00:07:12
but let me just do it
00:07:13
mathematically or algebraically so
00:07:16
how do i do that well it's going to be
00:07:18
some consumption bundle
00:07:20
right x1a and x2a where the utility of
00:07:24
agent a
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has to be greater than or equal to
00:07:27
utility
00:07:28
of this agent if he consumes his initial
00:07:31
endowment which is 2
00:07:33
and 1. remember his utility function is
00:07:35
x1 times
00:07:36
x2 so therefore initially his utility
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was 2.
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so therefore i'm looking for x1a times
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x2a
00:07:43
greater than or equal to 2. don't forget
00:07:46
pretty optimal allocations must satisfy
00:07:48
this
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right so what does that mean that means
00:07:53
you know whenever you see x2 a you can
00:07:55
just write x1a
00:07:57
so x1a times x1a which is by the way
00:08:00
x1a squared has to be greater than or
00:08:03
equal to 2.
00:08:04
so therefore x1a should be greater than
00:08:06
or equal to
00:08:07
square root of 2. all right hmm
00:08:12
well uh what else well here's the
00:08:15
second thing uh
00:08:18
the agent b should also prefer this
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bundle right
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so what does that mean agent b is
00:08:23
utility of x2
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x1 b and and x2 b
00:08:28
should be greater than or or equal to
00:08:30
her initial utility which is
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uh one plus two right so three
00:08:35
all right well what is this well that
00:08:39
means
00:08:39
x one b because her utility function is
00:08:41
x one plus
00:08:42
x two so x one b plus x two b is greater
00:08:45
than or equal to
00:08:47
three all right what else do i know
00:08:50
well the other thing that i know is the
00:08:52
following uh so i mean here
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i would like to convert everything uh as
00:08:57
as
00:08:58
x1a and x2a all right um
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because i have more constraint or
00:09:04
condition uh
00:09:05
when we talk about agent a's consumption
00:09:08
bundles
00:09:09
so how can i do that well remember x1a
00:09:12
and and x1b same for good two
00:09:15
they are related because of feasibility
00:09:18
so
00:09:20
feasibility implies
00:09:22
if you remember uh agent a's consumption
00:09:25
on good one
00:09:26
and agent b's consumption on good one
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has to be equal to the total
00:09:30
initial endowment of good one which is
00:09:33
30.
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all right so therefore x 1 b is nothing
00:09:38
but 3 minus
00:09:39
x 1 a all right what else uh the same
00:09:42
thing
00:09:43
it must be true for good 2 right x
00:09:46
2 a plus x 2 b has to be equal to total
00:09:49
initial endowment which is 3
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so therefore x 2 b is nothing but 3
00:09:54
minus
00:09:54
x 2 a all right so that means what
00:09:57
then remember a core a location means
00:10:02
you know this sum is greater than or
00:10:04
equal to 3 but i can
00:10:05
write that sum uh by
00:10:08
basically uh plugging those uh
00:10:12
sort of uh things so that means
00:10:15
let me write it here so x1b instead of
00:10:18
it i'm going to write
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3 minus x1a plus
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instead of x2b i'm going to write 3
00:10:25
minus
00:10:26
x2a has to be greater than or equal to
00:10:28
30.
00:10:29
so let's do the following x1a and x2a
00:10:32
let's send
00:10:33
send them to the other side uh but you
00:10:35
know what before doing it
00:10:38
don't forget you know all these
00:10:40
efficient i'm sure all these uh
00:10:43
core allocations must be proto
00:10:44
proto-efficient and so therefore
00:10:46
whenever i see x2a
00:10:48
i can actually write x1a all right so
00:10:51
therefore x1a x1a
00:10:53
so it's 2x1a it's going to be positive
00:10:56
when i send them to the other side
00:10:57
and i have 2 3 here 1 3 here so it's
00:11:00
going to be 3. so what does that mean
00:11:01
that means
00:11:02
x 1a is less than or equal to 3 over 2.
00:11:07
all right um so therefore that means
00:11:10
x 1 a has to be greater than or equal to
00:11:13
squared of 2 which i believe is around
00:11:16
1.4 or something
00:11:18
and x1a has to be less than or equal to
00:11:21
3 over 2. so
00:11:23
let me bring them together
00:11:29
so x 1 a
00:11:32
has to be less than or equal to 3 over 2
00:11:35
and it has to be greater than or equal
00:11:37
to
00:11:38
square root of 2. all right um okay very
00:11:42
good
00:11:42
what else do i know
00:11:47
so is that it
00:11:50
i mean i couldn't pin down a specific
00:11:53
number for x1a
00:11:55
is that okay uh well yes it is okay
00:11:58
actually as you will see next uh when i
00:12:02
sort of find the
00:12:03
core allocations geometrically uh you're
00:12:06
gonna see that that's possible meaning
00:12:08
we may not have just one uh
00:12:11
a quora location we actually may have
00:12:14
you know
00:12:15
infinitely many core allocations so how
00:12:17
do we know that
00:12:18
well so remember this is the initial
00:12:20
endowment
00:12:21
uh if necessary i'm to come back here so
00:12:24
i'm not
00:12:24
quite done there so this is the initial
00:12:28
endowment
00:12:28
so what i want is for the core
00:12:32
allocation
00:12:33
i want pretty optimal allocations that
00:12:36
are going to give both agent and b
00:12:38
higher utility so what does that mean
00:12:40
that means i need to
00:12:42
find all the points on this contract
00:12:45
curve
00:12:45
which are going to give higher utility
00:12:48
to both agents
00:12:49
than this initial endowment so how can
00:12:52
figure out those
00:12:54
points well simple draw the indifference
00:12:57
curve
00:12:57
of agent a and b that are passing
00:13:00
through this initial endowment
00:13:02
because those indifference curves are
00:13:04
going to give
00:13:05
us what the initial utility of agent a
00:13:08
and b
00:13:09
are meaning if they consume their
00:13:12
initial endowments
00:13:13
that is i mean those indifference curves
00:13:16
indicate the level of utility they are
00:13:18
going to get
00:13:18
all right well we know that agent a's
00:13:22
utility
00:13:23
is is a douglas type so its indifference
00:13:26
curve is going to look something like
00:13:28
this
00:13:28
well what about agent b's utility
00:13:30
function well it's going to be
00:13:32
x1 plus x2 so it's a straight line so
00:13:35
let me use a blue color for this
00:13:37
and its indifference curve is going to
00:13:40
look something like this
00:13:42
okay so this is u of b at the blue one
00:13:45
and the black one is the u of a well one
00:13:49
question maybe is like why
00:13:50
they're not tangent to each other at you
00:13:53
know those points
00:13:55
well they can't be because i am drawing
00:13:58
indifference curves of agent a and b
00:14:02
that are passing through the initial
00:14:04
endowments
00:14:05
all right and so i know that initial
00:14:07
endowment is not
00:14:08
proto-efficient and so all i know is
00:14:11
that
00:14:12
when i draw these agents indifference
00:14:15
curve
00:14:15
passing at and on i mean
00:14:18
the initial endowment they're not going
00:14:20
to be tangent to each other
00:14:22
all right however if i draw the
00:14:25
indifference curve of
00:14:26
agent a on some point here
00:14:29
an indifference curve of agent b
00:14:32
well they are going to be tangent to
00:14:34
each other on this
00:14:36
contract curve but outside of it they're
00:14:38
not going to be tangent to each other
00:14:40
right that's the idea of mrs e equals
00:14:43
mrs b
00:14:44
gives us the proto-optimal locations so
00:14:47
anyway what does that mean that means uh
00:14:50
remember
00:14:51
the at least as good as set for agent b
00:14:54
is uh let me use uh black blue color
00:14:58
so all the allocations in this region
00:15:01
what is the at least as good as set for
00:15:04
agent b
00:15:05
a i'm sorry all the locations in this
00:15:08
uh region well remember
00:15:12
i am for coral locations i am looking
00:15:15
one
00:15:15
predo efficient i'm sorry feasible
00:15:18
allocations
00:15:19
fine two pretty efficient allocations so
00:15:23
i don't really care about those regions
00:15:25
i care about this 45 degree line
00:15:28
and the third thing is that both agents
00:15:32
should be getting higher utility that
00:15:34
means predo optimal allocations
00:15:37
that are in the intersection of the at
00:15:39
least as good as set of these two agents
00:15:42
are the core allocations all right so
00:15:44
what does that mean that means
00:15:46
this small portion of this contract
00:15:50
curve
00:15:51
indicates the set of all
00:15:55
set of all core allocations
00:15:59
in this problem all right
00:16:02
and in fact they are given by this
00:16:05
right so that what does that mean that
00:16:08
means in this region corresponds to
00:16:10
x1 which is uh greater than square root
00:16:13
of 2
00:16:14
less than or equal to 3 over 2. all
00:16:17
right
00:16:20
um
00:16:22
yes uh what else uh
00:16:25
well we also know that a by the way
00:16:28
square root of two is is is less than
00:16:30
uh three over two that that we know and
00:16:33
and
00:16:33
but obviously we should also figure out
00:16:36
what
00:16:36
x2 is right well remember x1 is equal to
00:16:40
x2 because those are the pretty optimal
00:16:42
allocations so therefore
00:16:44
you can just say x1a well let me say
00:16:47
this way
00:16:48
x1a x2a
00:16:51
comma x2a oh b i'm sorry x1b
00:16:56
x2b is
00:16:59
a core allocation
00:17:02
so this is the conclusion uh if and only
00:17:05
if
00:17:07
x one a equals x two a
00:17:10
which is less than or equal to three
00:17:12
over two greater than or equal to square
00:17:14
root of two
00:17:15
so all those points all right which are
00:17:18
represented
00:17:19
on this uh picture with those uh points
00:17:22
on this red
00:17:23
small uh uh you know
00:17:26
portion of this 45 degree line these are
00:17:30
all the core allocations
00:17:32
well the question is if these guys trade
00:17:35
which one exactly are they going to end
00:17:38
up
00:17:39
well we couldn't pin down a unique
00:17:42
one single allocation
00:17:45
what we said look if these guys really
00:17:49
come to this market with these initial
00:17:51
endowments
00:17:52
actually um they they shouldn't go home
00:17:56
without making any trade because they
00:17:58
are going to be
00:17:59
wasting an opportunity an improvement
00:18:02
uh if they trade they both can end up
00:18:06
at higher levels of utilities so they
00:18:08
should trade
00:18:10
that's the first lesson we learn well
00:18:12
the second lesson we learn is that
00:18:14
well if they do trade right they should
00:18:17
be
00:18:17
sort of end up somewhere in this box and
00:18:20
in fact they should end up somewhere on
00:18:22
this contract curve because
00:18:24
something outside of this contract curve
00:18:26
means
00:18:27
there's still an improvement all right
00:18:29
meaning
00:18:30
both agents can get even happier so we
00:18:33
don't want
00:18:33
that to be a final outcome we want final
00:18:36
outcome for that reason to be
00:18:38
proto-efficient so the third thing we
00:18:41
realized is that or the third lesson we
00:18:43
learned
00:18:44
is that they shouldn't end up some point
00:18:48
here or some point here because
00:18:52
those points are worse for agent uh
00:18:55
a then his initial endowment meaning if
00:18:58
they end up somewhere here
00:18:59
agent a is actually going to regret and
00:19:02
if
00:19:03
they end up on this part of the contract
00:19:05
curve
00:19:06
and this time agent b is actually going
00:19:08
to regret and
00:19:09
he's going to veto this trade he's going
00:19:11
to say hey you know what i mean i'm not
00:19:12
going to make this i'm not going to
00:19:14
accept this deal because i can go home
00:19:17
and consume my endowment
00:19:19
in which case i'm going to get higher
00:19:20
utility anyway you see what i mean so
00:19:22
they're not actually trade
00:19:24
if they trade they are i mean they are
00:19:26
supposed to end up
00:19:27
somewhere in this red region where
00:19:30
exactly
00:19:31
we don't say this we we need to impose
00:19:34
more structure
00:19:35
but all we can say they should end up
00:19:37
somewhere here i think that's
00:19:39
you know pretty narrow uh pre i mean
00:19:42
pretty narrow
00:19:43
a good prediction relatively given that
00:19:45
we initially started from this entire
00:19:48
uh you know uh uh box full of
00:19:51
allocations we ended up on a just very
00:19:54
small portion of a straight line
00:19:56
so if they end up they should end up
00:19:58
somewhere there
00:20:00
next we are going to talk about what if
00:20:03
so this is remember
00:20:04
a barter exchange there's no price the
00:20:07
question is
00:20:08
what if we introduce money and hence
00:20:11
uh price uh would this give us sort of a
00:20:14
sharper prediction
00:20:15
so it's coming up next
