Welfare Part4

8/10/2019 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

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,

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

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.

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

=

,,

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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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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 =

56

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 +

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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

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,

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

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Welfare Part4