welfare part4
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Part IIa: Paper 1Dr Snje Reiche
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Motivation
Casual use of Social Indifference curves
Where do these come from? How are the interests of different individuals weighed up in
creating such social indifference curves?
With trade, some winners and some losers ant a mec an sm or eva uat ng po cy
orma ve ssues
(rather than positive analysis of GE)
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Concepts of Welfare Economics Want social preference ordering over states
complete (defined over all states) transitive
yx f
zxzyf
f
Less controversial at individual level, but what does
this imply at society level?
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Distinguish:Bergson-Samuelson Social
Welfare Function
oc a we are unc on w c correspon s o
individual utility function)
Mechanism for achieving this social welfare
function (through combining individual
preferences)Arrow:
Social Choice Rule,
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orConstitution
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. esource a ocat ons s ou e u ge on y t roug t e r
effects on individuals in the economy
Goal of government is to achieve well-being of the individuals
who compose society
Individual rights crucial for well-being
o n oc e - , am m t - , o n-
Stuart Mill (1806-1873)
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Non-Paternalism
2. Individuals are the best judges of their ownwelfare
The effects of different resource allocations on an
individual depend only on that individuals preferenceorderings
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Alternative: Societ as an Or anismPlato (424-348 BC), Karl Marx (1818-1883)
Thomas Hobbes (1588-1679)
Rulers determine goals for the organism (e.g.
con uest lor of God trium h of ideolo
Individuals matter only insofar as they can be used
as tools to achieve those goals
. .,
importance beyond helping pump blood
Examples: Divine right of kings: Britain (formerly!)
Communism
8
Fascism
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Basic Value Judgements
Pareto PrincipleAllocation Y is socially preferred to allocation Z if Y is feasible
given resource constraints, all individuals are at least as well off
, .
mp c ssump ons
Individualism and Non-Paternalism Process Independence
Anonymity
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Define a utilit function to re resent an individuals
preferences over resource allocations
This is an ORDINAL utility function
1. Ordinally measurable
utilit numbers for an individual conve information about that
individuals welfare BUT only convey information on the order,
not intensity
2. Noncomparable (across individuals)
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cannot be compared
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ey: an or na ut ty unct on s un que up to a pos t ve,monotonic transformation
1 xyu a =
33
1 loglog yxu b +=
1c
all represent the same preferences
The units have no meaning: convey no information about then ens y o n v ua s pre erence
11
,
can compare utility to individual 2s level of utility
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Utilit Possibilit SetDifferent resource allocations between person 1 and 2 lead to different
utility levels for each (mapping from contract curve into utility space)
Pareto principle
Allows comparison between some
allocations (A and B)
2
Not between others (A and C)C
BUT: Pareto improvement possibleA
rom nter or, so necessary con t on
121u(although precise shape of UPS depend on the particular representation)
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Cannot rank Pareto efficient allocations
defines set of all efficient allocations
Need additional criteria (if equity matters)
One allocation is more desirable than another if the gainers couldhypothetically compensate the losers and still be better off
(potential Pareto improvement)
Uses willingness to pay (idea of CV): consider a price movement
fromp top,',,' mvCVmv =
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uxu + xyCVxyCV
No need for interpersonal value judgement!
But:
Only hypothetical reallocation. Otherwise: real pareto improvement. Implicit assumption that value of 1 is worth the same to all
individuals (so it does not matter who actually benefits).
Can lead to Paradox: Bothx andy can pass the compensation test.
New test: and ?Criterion is not transitive!
0),(),(21
>+ xyCVxyCV 0),(),(21
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roper es
Paretian 0
hu
W
Anonymous: identity of individual does not matter
Weak preference for equality (quasi-concave)
2u
161u
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- N
=h
hN
1
21 ...
2
u1
17This does not imply that giving 1 to individual 1has the same benefit as giving 1 to individual 2
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-Maximise the welfare of the worst off person in society
W = min u1,u2,..., uN[ ]u2
u1Utilit of individual 1 is a er ect com lement for the utilit of
18individual 2
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-Basis for showing how attitudes to inequality affect optimal allocations
u Utilitarian45aw s an
Egalitarian
19u1
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Summary
Evaluation of value to society of different allocations n v ua s c
Pareto
Pareto does not provide complete ranking
Compensation criterion: potential Pareto, implicit equal
wei ht
Ber son-Samuelson: com ares across individuals
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Outline: Arrow
Definition
Desirable Properties
Focus on information requirements for individual
ut ty unct ons
Books:
Varian: Ch. 33
21 Cowell: Ch. 9.2
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Where does Ber son-Samuelson Social Welfare function come
from?
how does society aggregate individual preferences into a social
Function of cardinal and individually
comparable utility values
B-S SWF: mechanism for making decisions about whichallocation is optimal given social preferences about inequality
Function of individual
preference orderings
generates these social preferences?
Social Welfare Functional or Constitution
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Exam le
3 Individuals in society, 3 alternativesA,B, C:
Preferences are given by CBA 11 ff
BAC 33
22
f
f
Utility representation:
u1( ) = 8 u1(B) = 6, u1(C) = 1
u A = 7 u B = 280 u B = 280u3(A) = 5 u3(B) = 5 u3(C) = 4
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W =W 8 75 = 4
Utilitarian B-S SWF W(B)=W(6,280,5) = 281
W(C)=W(1,280,4)= 283
Arrows SCR,
Example ff
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1. Universality (unrestricted domain) [U] Fmust e e ne over a poss e sets o n v ua pre erence
orderings
2. Pareto principle [P] if all individuals prefer allocation Y to Z, then social preference
ordering must rank Y above Z
3. Independence of irrelevant alternatives [IIA]
orderings of Y and Z and does not depend on other orderings of otherallocations
4. Non-dictatorship [ND] there is no individual whose preference ordering determines the
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The Social Choice Rule must map each set of preference
or er ngs nto a soc a or er ng over t e a ternat ves, or
example:
}{)()( BACVIFVF ff==
====
un
Pareto Princi al
The SCR that sets violates [P].}{)( BACIIIF ff=
Test: Does [P] say anything about F(IV)?
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,
must be the same!
,
F(III) must also rankA overB? What about F(IV)?
Non-Dictatorshi
The SCR that sets violates [ND].2
fF
Test: BABA 1
f
f
AB
BA 1
f
f
BA
AB 1
f
f
AB
AB 1
f
f
I: II: III: IV:
What possible SCRs satisfy all 4 requirements?
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Impossibility Theorem
In general, does there exist a social choice rule, F,
a sa s es ese es ra e requ remen s
rrow: more an
2 alternatives: No!
Needs to generate a social preference ordering from
ordinal utility / ordinal social welfare
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cba 11:1 ff
bac
acb 22
:3
:2
ff
ff
Condition U: satisfied for all admissible references
Condition P: would be satisfied! Conditions IIA and ND satisfied if social choice between an
pair of allocations is made by majority voting between that pair
cb f not transitive
30acf
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Solutions to Impossibility
Relax universality
works if single-peaked preferences
difficulties if multi-dimensional preferences
Relax independence of irrelevant alternatives
Im ose some cardinalit of utilit
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-Median voter always
utility
capitalists land owners
majority
workers
33tariff
tt t t
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cba 11:1 ff
?bac
acb 22
:3
:2
ff
ff
Individual 3
does not have
single peaked
pre erences
a cb34
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- - -
adding extra competitors affects the result (Ross Perot
in US, Le Pen in France)
Example :
CBCAB
BACCBA 2222
1111
ffff
ffff
CABABC 4444 ffff
BA
1. Individuals rankings betweenA andB are the same in
the two sets but the social ordering changes
35
.
affects the election betweenA andB.
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Imposing Cardinality
The only information for each individual allowedy rrow s an n v ua spre erence or er ng
Corresponding utility functions are ordinall measurable
non-comparable
ONC
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ons er t e - w t ut ty n ormat on
1 1 N
=u +2
m nuh u , u =N
uhh=1
Will the B-S SWF remain unchanged as individual utility is
transformed?
Allocation Y Allocation Z
u1 4 8
u2 6 4
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ordering):
=)
22
11
4
.
uu =
)
Allocation Allocation
u1 2 4
u2 24 16
W 7 1/2 7
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Social ordering is not invariant to (permissible)
transformations of utility
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Relaxing ONC
necessary to resolve Arrow?
Suppose individual utility:
cardinally measurable for each i
0>+= iiiii uu )
fully comparable across i (same transformation has to be applied to all
individuals))
Social choice rule exists and satisfies all 4 conditions: need extra
ii ,
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information on intensity and comparability
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Cardinally measurable for individual i: The difference
etween ut ty eve s or n v ua as mean ng
Take alternativesx,x,y andy.If the difference in utilit betweenx andx is 5 times bi er
than the difference in utility betweeny andy, this is true for
any cardinal utility representation of i s preferences.
)'()(
)'()(
)'()(
)'()(
)'()(
)'()(
yuyu
xuxu
yuyu
xuxu
yuyu
xuxu
ii
ii
iiiiii
iiiiii
ii
ii
=
+
+=
Comparable across individuals: Take alternativesx,x
If the difference in utility betweenx andx is 5 times bigger
for i than forj, this is true for any utility representation of i
and s references.
'
)'()(
'
)'()(
'
)'()( xuxuxuxuxuxu iiiiii =+
=
jjjjjj
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Arrow: how are individual preference orderings aggregated to
, , ,
information, so no intensity or comparability
Extra information enables aggregation to B-S SWF
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Conditions for Pareto Efficienc
First Fundamental Theorem
Efficienc vs E uit : The Second Fundamental Theorem
Im lications of the SWT
First Best vs Second Best
Lum sum taxes
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Books:
. . - . . , .
. . - . , . - .
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First Best and Second Best
First Best:
overall resource constraints and technology constraint
Second Best
an a oca on w c s are o e c en su ec o resource
constraints and additional constraints (eg. imperfect
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Nothing about prices in defining first-best allocation
All goods are private: rivalrous (if one person consumes a good, this uses up the good and so
excludable (one person can stop another from consuming the good)
Conditions for Pareto Efficiency: Production efficiency
Overall efficiency
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Pareto Efficiency in a Simple Economy
Two goods,x andy, two inputs KandL in fixed supply
wo consumers an w t uix,y , = ,
Two firms with technologyfj(K,L),j=x,y
subject to( )BAALKLKxx
yxu ,max
BBBB uyxu =),(
( ) BAyyyBAxxx
LLLKKK
yyLKfxxLKf
+=+=
+=+= ,,
Production Efficienc
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Production Efficienc
LKmax subject to: yyy yLKf =,LKLK yyxx ,,,
xyxy LLLKKK +=+= ;
xxyxxxLK
yxx
+ ,,max,
F.O.C.s
y
xxy
x
xxx
LL
=
,,
y
xxy
x
xxx
KK
=
,,
yyxx
LKfLKf ,,
( ) ( )yyy
xx
x
LKf
L
LKf
L
=
,,
48
yx KK
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OBy
rea o are o
improvement
Potential allocation of
consumption goods
BA RSRS =
OAx
u
B
BB
A
AA
x
yxu
x
yxu
=
),(),(
50BA y
yxu
y
yxu
,,
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Economics
Introduce prices: what are the welfareproperties of the market?
Assumption 1: Competitive Markets
Consumers and firms take prices as given
Consumers and firms o timise Markets clear
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Assumption 2: Complete Markets
property rights
individual utilities or firm production
No externalities
o pu c goo s
No asymmetries of information
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(Aside on externalities)
Consumption externality
BAAA xyxu ,,
Assumption is NOT that this sort of utility function is ruled
out
a er e assump on s a ere s a pr ce or eac goo
t ere s a pr ce mar et or eac goo , t en t eexternality becomes apecuniary externality
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First Fundamental Theorem
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First Fundamental Theorem
If assumption of complete and competitive marketso s, en a genera compe ve equ r um s
first-best Pareto efficient.
Intuition:
Optimising consumers each set :
y
xi
xy pMRS =
So, if all consumers face the same prices, then consumption efficiency
m ar y w op m s ng rms: eac se s
r
wMRTS
j
KL =
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y
x
xy p
p
MRT =
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Caveats
Agents might not be price-takers if markets are
monopo ze or monopson ze pr ces o not
measure the marginal value of all activities.
Markets ma not be com lete: how likel is it that
there is a market for clean air noise etc.? Future
markets markets for state contin entcommodities?
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Equity vs Efficiency
FT1: market leads to the Pareto frontier
Pareto optimality: which point on the Pareto frontiermax m ses soc a we are
Is there a cost between moving to a point on thefrontier which improves social welfare? Is there a
ra e-o e ween e c ency an equ y
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Assumption 3: Convexity
Individual indifference curves are convex
rules out increasing returns to scale
(Assumptions necessary for existence of competitive
e uilibrium
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-
With lump sum tax or transfer the receipt or payment of a sumof money is independent of individual choices
Transfer based on an individuals endowment (individual
Ke : im osition of a lum -sum tax leaves all individualsfacing commonprices but those common prices may depend on the lump sum taxes because
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Second Fundamental Theorem
If assumptions 1 to 3 hold, then any Pareto efficient
a ocat on can e ac eve y appropr ate
redistribution through lump-sum taxes by leaving
t e compet t ve mar et to n pr ces.
Prices will depend on endowments and the extent of
redistribution
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Pareto efficiency
FT2 in an Exchange Economy
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Competitive equilibrium,no interventionCompetitive equilibriumthat maximises SWF
B
y
A
y ee +
Initial
endowment
12
AxB
x
A
x ee + To achieve SWF*, tax is imposed on individualAs endowment, and this is
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rans erre o n v ua n orm o a rans er o goo x money
Tax and transfer are lump-sum since value of endowment is taken as given
by each individual
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-
y OB
o suppor e y
competitive equilibrium
p
areto c ent
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Implications of FT2
Decentralisation result: redistribute with limited
government n ormat ona requ rements
decentralised decisions will lead to a first-best Pareto
a ocat on sat s y t e marg na con t ons
Separate out equity from efficiency: any point on
the Pareto frontier can be achieved.
there is no conflict between redistribution and
achieving a first-best Pareto allocation
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Implications
There are two distinct reasons for governmentun er e assump ons o e
. o re s r u e: compe ve mar e may no ea o
the Pareto efficient point that maximises socialwelfare
2. To correct market failure (due to failure of one ofassumptions 1 to 3)
to return to first best (eg use of Pigouvian taxes to correctexternalities
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Value of FT2
FT2 provides rationale for why government
ntervent on may e es ra e
Assumptions (particularly on information) arever stron :
FT2 is not valuable to understand what form
interventions should take
second best analysis of policy intervention
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Focus on FB vs SB for efficienc i.e. kee assum tion 4 that
LST feasible)
Distortion to FB: there is one good where price does not equalmarginal cost (e.g. price of using a road is zero)
Competitive equilibrium will not be first best
RTRS
directly? If so, then move back to the first best.
try to satisfy as many Pareto conditions as possible (i.e. ensure efficiency
in all other markets ignoring distortion )
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or introduce offsetting distortion (second best policy)
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Car use (when congestion): socialcar
private
carcar MCMCp
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Deadwei ht loss
SMC
PMC p*
qq*
Reduction in
qq*
Introduced
deadweight loss from
marginal reduction in
deadweight loss from
marginal increase in
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q: - q: -
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Redistribution requires lump sum payments to vary across
. .endowments of ability)
These characteristics are private information to theindividual. Government onl observes behaviour.
information on characteristics through behaviour
Problem of incentive com atibilit
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Lump Sum Taxes and Incentive Compatibility
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2 goods: coconuts c and leisurez
individuals endowed withHunits of leisure
CRS technology: 1 unit of labour produces a units of coconuts
different types of labour: individuals of high ability, individuals of
low ability.lh aa >
Identical preferences over c andz
zvcu +
73
Assume v concave
Characterization of Pareto Efficiency
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Assume government maximises a Utilitarian SWF, i.e., itchooses consumption and leisure bundles for each person
subject to overall feasibility.
zzcc lhlh ,,,
( ) ( )llhhlh zHazHaccts +=+..
First-order conditions for consumption:
= h
hh
cc : =
l
ll
c
cuc :
Equate marginal utility of consumption across individuals
74
,
consumption.
First-order conditions for leisure
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( ) hh
h azvz =: ( ) ll
l azvz =:
( ) hhh azzv=
/
( ) lll azzv /
( ) ( )lh zvzv > lh zz haa
,in the social optimum.
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Competitive Equilibrium with LST
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prices arep=1 for coconuts and wh andwl for the two types
CRS firms make zero profits andwa = ll wa =
Consider a LST that depends on ability: ),( lhTT
Ti is the lump sum tax (which is a transfer received if
Budget constraint for individual with ability i is
( ) iiii TzHac =
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To achieve equal consumption, we must have
( ) ( ) lllhhh TzHaTzHa =
Since and we want we must havelh zz
, lh
High ability workers should work more and be taxed, low
ability workers should work less and receive a benefit.
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Im ossible to differentiate lum -sum taxes b unobservable
characteristics!
Taxes must be based on observed behaviour (or characteristics)
When characteristics are private information, redistribution must
involve taxes that are distortionary.
1. Some conditions for first best Pareto Efficiency will not hold
2. Theory of the second best suggests Pareto Efficiency in
remaining markets may not be first best neither
Policy prescriptions cannot be based on Second Welfare
Theorem: cannot separate out efficiency from equity
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Second Welfare Theorem: any Pareto efficient point is achievable
allows us to separate out efficiency from equity
2nd Best: if distortion in one market, it may be optimal to introducean extra distortion elsewhere to offset the first one (rather than
Unable to use lump sum taxes because of private information abouten owments o a ty . ou not try to ma nta n separat on
between equity and efficiency
80
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