Welfare: The Social-Welfare Function - PowerPoint Presentation

Slide1

Welfare: The Social-Welfare Function

MICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential Welfare: BasicsWelfare: Efficiency

Prerequisites

April 2018

1Slide2

Social Welfare Function

Limitations of the welfare analysis so far:Constitution approachArrow theorem – is the approach overambitious?General welfare criteriaefficiency – nice but indecisiveextensions – contradictory?

SWF is our third attemptsomething like a simple utility function…?RequirementsApril 20182Slide3

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

What is special about a social-welfare function?

April 2018

3Slide4

The SWF approach

Restriction of “relevant” aspects of social state to each person (household)Knowledge of preferences of each person (household)Comparability of individual utilitiesutility levelsutility scalesAn aggregation function

W for utilitiescontrast with constitution approachthere we were trying to aggregate orderingsA sketch of the approachApril 20184Slide5

Using a SWF

u

a

u

b

Take the utility-possibility set

A social-welfare optimum?

Social welfare contours

W

defined on utility

levels

Not on orderings

Imposes several restrictions…

..and raises several questions

W

(

u

a

,

u

b

,... )

•

April 2018

5Slide6

Issues in SWF analysis

What is the ethical basis of the SWF? What should be its characteristics? What is its relation to utility? What is its relation to income?April 2018

6Slide7

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

Where does the social-welfare function come from?

April 2018

7Slide8

An individualistic SWF

The standard form expressed thus W(u1

, u2, u3, ...)an ordinal functiondefined on space of individual utility levelsnot on profiles of orderingsBut where does W come from...?We'll check out two approaches:the equal-ignorance assumptionthe PLUM principleApril 20188Slide9

1: The equal ignorance approach

Suppose the SWF is based on individual preferences.Preferences are expressed behind a “veil of ignorance”It works like a choice amongst lotteriesdon't confuse w and

q!Each individual has partial knowledge:knows the distribution of allocations in the populationknows the utility implications of the allocationsknows the alternatives in the Great Lottery of Lifedoes not know which lottery ticket he/she will receive April 20189Slide10

“Equal ignorance”: formalisation

Individualistic welfare:

W(u1, u2, u3, ...)

use theory of choice under uncertainty to find shape of W

vN

-M form of utility function:

å

w

ÎW

p

w

u

(

x

w

)

Equivalently:

å

w

ÎW pw u

w

p

w: probability assigned to w

u : cardinal utility function, independent of w

uw: utility payoff in state wA suitable assumption about “probabilities”? nh

1

W = —

å

u

h

n

h

h

=1

welfare is expected utility from a "lottery on identity“

payoffs if assigned identity 1,2,3,... in the Lottery of Life

Replace

W

by set of identities {1,2,...

nh}: åh

ph u

h

An additive form of the welfare functionApril 201810Slide11

Questions about “equal ignorance”

p

h

identity

|

n

h

h

|

1

|

2

|

3

|

Construct a lottery on identity

The “equal ignorance” assumption...

Where people know identity with certainty

Intermediate case

The “equal ignorance” assumption:

p

h

=

1

/

n

h

But is this appropriate?

Or should we assume that people know their identities with certainty?

Or is the "truth" somewhere between...?

April 2018

11Slide12

2: The PLUM principle

Now for the second  rather cynical  approachAcronym stands for P

eople Like Us MatterWhoever is in power may impute:either their own viewsor what they think “society’s” views areor what they think “society’s” views ought to beprobably based on the views of those in powerThere’s a branch of modern microeconomics that is a reinvention of classical “Political Economy”concerned with the interaction of political decision-making and economic outcomesbut beyond our present courseApril 201812Slide13

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

Conditions for a welfare maximum

April 2018

13Slide14

The SWF maximum problem

Take the individualistic welfare model

W(u1, u2, u3, ...)

Standard assumption

Assume everyone is selfish:

u

h

=

U

h

(

x

h

) ,

h

= 1,2, ...,

n

h

my

utility depends only on

my

bundle

Substitute in the above:

W

(

U

1

(x1), U2(x2), U3(x3

), ...)

Gives SWF in terms of the allocation

a quick sketch

April 2018

14Slide15

From an allocation to social welfare

From the attainable set...

A

A

(

x

1

a

,

x

2

a

)

(

x

1

b

,

x

2

b

)

...take an allocation

Evaluate utility for each agent

Plug into

W

to get social welfare

u

a

=

U

a

(

x

1

a

,

x

2

a

)

u

b

=

U

b

(

x

1

b

,

x

2

b)W(ua, ub)

But what happens to welfare if we vary the allocation in A?April 201815Slide16

Varying the allocation

Differentiate w.r.t.

xih : duh = Uih(xh)

dxih

marginal utility derived by

h

from good

i

The effect on

h

if commodity

i

is changed

Sum over

i

:

n

d

u

h

=

S

U

i

h

(xh) dxih

i=1

The effect on

h

if all commodities are changed

Differentiate

W

with respect to

u

h

:

n

h

d

W

=

S Wh

duh h=1

Changes in utility change social welfare .

Substitute for d

uh in the above: nh

n dW = S Wh S Ui

h(xh) dxih h=1 i=1So changes in allocation change welfare.

Weights from the SWFWeights from utility functionmarginal impact on social welfare of h’s utilityApril 201816Slide17

Use this to characterise a welfare optimum

Write down SWF, defined on individual utilitiesIntroduce feasibility constraints on overall consumptionsSet up the LagrangianSolve in the usual way

Now for the mathsApril 201817Slide18

The SWF maximum problem

First component of the problem:

W(U1(x1), U2(x2), U

3(x3), ...)

Individualistic welfare

Utility depends on own consumption

The objective function

Second component of the problem:

n

h

F

(

x

)

£

0,

x

i

=

S

x

i

h

h=1

Feasibility constraint

The Social-welfare

Lagrangian

:

n

h

W

(

U

1

(

x

1

),

U

2(x2),...) - lF

(S xh

) h=1

Constraint subsumes technological feasibility and materials balance

FOCs for an interior maximum:

Wh (...)

Uih(xh

) − lFi(x) = 0From differentiating Lagrangean with respect to

xih And if xih = 0 at the optimum: Wh (...)

Uih(xh) − lFi(x) £ 0Usual modification for a corner solution

All goods are privateApril 201818Slide19

Solution to SWF maximum problem

From FOCs

: Uih(xh) Uiℓ(xℓ

) ——— = ——— U

j

h

(

x

h

)

U

j

ℓ

(x

ℓ

)

Any pair of goods,

i,j

Any pair of households

h, ℓ

MRS equated across all

h

We’ve met this condition before - Pareto efficiency

Also from the FOCs:

W

h

Uih(xh) = Wℓ Ui

ℓ(

x

ℓ

)

social marginal utility of toothpaste equated across all

h

Relate marginal utility to prices:

U

i

h

(

x

h

) =

V

y

hpi

This is valid if all consumers optimise

Substituting into the above:

Wh V

yh = Wℓ V

yℓ

At optimum the welfare value of $1 is equated across all h. Call this common value MMarginal utility of money

Social marginal utility of incomeApril 201819Slide20

To focus on main result...

Look what happens in neighbourhood of optimumAssume that everyone is acting as a maximiserfirmshouseholdsCheck what happens to the optimum if we alter incomes or prices a little

Similar to looking at comparative statics for a single agentApril 201820Slide21

Differentiate the SWF w.r.t. {

y

h}: nh dW = S Wh

duh

h

=1

Changes in income, social welfare

n

h

d

W

=

M

S

d

y

h

h

=1

n

h = S WhVyh dyh

h

=1

Social welfare can be expressed as:

W

(

U

1

(

x

1

),

U

2

(

x2),...) = W

(V1(p,y1

), V2(p,y2

),...)

SWF in terms of direct utility. Using indirect utility function

Changes in utility and change social welfare …

...related to income

change in “national income” Differentiate the SWF w.r.t. pi

: nh dW = S WhVihdpi

h=1 .

Changes in utility and change social welfare … nh = – SW

hVyh xihdpi h

=1 from Roy’s identity

n

h

d

W

= –

M

S

x

i

h

d

p

i

h

=1

...related to prices

Change in total expenditure

.

April 2018

21Slide22

An attractive result?

Summarising the results of the previous slide we have:THEOREM: in the neighbourhood of a welfare optimum welfare changes are measured by changes in national income / national expenditure

But what if we are not in an ideal world?April 201822Slide23

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

A lesson from risk and uncertainty

April 2018

23Slide24

Derive a SWF in terms of incomes

What happens if the distribution of income is not ideal? M is no longer equal for all h

Useful to express social welfare in terms of incomesDo this by using indirect utility function V express utility in terms of prices p and income yAssume prices p are given“Equivalise” (i.e. rescale) each income yallow for differences in people’s needsallow for differences in household sizeThen you can write welfare as W(ya, yb, yc, … )April 201824Slide25

Income-distribution space:

n

h

=2

Bill's

income

Alf's

income

O

The income space: 2 persons

An income distribution

y

45

°

line of perfect equality

Note the similarity with a diagram used in the analysis of uncertainty

April 2018

25

Alf's

incomeSlide26

Extension to

n

h = 3

Here we have 3 persons

Charlie's income

Alf's income

Bill's income

O

line of perfect equality

•

y

An income distribution.

April 2018

26Slide27

Welfare contours

x

E

y

y

a

y

b

x

E

y



y

An arbitrary income distribution

Contours of

W

Swap identities

Distributions with the same mean

Anonymity implies symmetry of

W

Equally-distributed-equivalent income

E

y

is mean income

Richer-to-poorer income transfers increase welfare



equivalent in

welfare terms

x

is income that, if received uniformly by all, would yield same level of social welfare as

y

higher

welfare

E

y



x

is income that society would give up to eliminate inequality

April 2018

27Slide28

A result on inequality aversion

Principle of Transfers : “a mean-preserving redistribution from richer to poorer should increase social welfare”THEOREM: Quasi-concavity of W implies that social welfare respects the “Transfer Principle”

April 201828Slide29

Special form of the SWF

It can make sense to write W in the additive form

nh 1 W = — S z(yh) nh h=1where the function z is the social evaluation function(the 1/nh term is unnecessary – arbitrary normalisation)Counterpart of u-function in choice under uncertainty

Can be expressed equivalently as an expectation: W = E z(yh)

where the expectation is over all identities

probability of identity

h

is the same, 1/

n

h

, for all

h

Constant relative-inequality aversion:

1

z(

y

)

= —— y

1 –

i

1 – iwhere

i is the index of inequality aversionworks just like r,the index of relative risk aversion

April 201829Slide30

Concavity and inequality aversion

W

z

(

y

)

income

y

z



(

y

)

The social evaluation function

Let values change:

φ

is a concave transformation.

More concave

z

(

•

)

implies higher inequality aversion

i

...and lower equally-distributed-equivalent income

and more sharply curved contours

lower inequality aversion

higher inequality aversion

z



=

φ

(

z

)

April 2018

30Slide31

Social views: inequality aversion

i =

½

y

b

y

a

O

i = 0

y

b

y

a

O

i = 2

y

b

y

a

O

i =



Indifference to inequality

Mild inequality aversion

y

b

y

a

O

Strong inequality aversion

Priority to poorest

“

Benthamite

” case

(

i

= 0)

:

n

h

W=

S

y

h

h

=1

General case (

0<

i

<



)

:

n

h

W =

S

[

y

h

]

1-

i

/

[1-i]

h

=1

“

Rawlsian

” case

(

i

=



)

:

W =

min

y

h

h

April 2018

31Slide32

Inequality, welfare, risk and uncertainty

There is a similarity of form between… personal judgments under uncertainty social judgments about income distributionsLikewise a logical link between risk and inequality

This could be seen as just a curiosityOr as an essential component of welfare economicsUses the “equal ignorance argument”In the latter case the functions u and z should be taken as identical“Optimal” social state depends crucially on shape of WIn other words the shape of zOr the value of i

Three examplesApril 2018

32Slide33

Social values and welfare optimum

y

a

y

b

The income-possibility set

Y

Welfare contours (

i

=

½)

Welfare contours (

i

=

0)

Welfare contours (

i

=



)

Y

derived from set

A

Nonconvexity

, asymmetry come from heterogeneity of households

y

*

maximises total income irrespective of distribution



y***

gives priority to equality; then maximises income subject to that



Y

y

*

y

***



y

**

y**

trades off some income for greater equality

April 2018

33Slide34

Summary

The standard SWF is an ordering on utility levels Analogous to an individual's ordering over lotteriesInequality- and risk-aversion are similar concepts

In ideal conditions SWF is proxied by national incomeBut for realistic cases two things are crucial:Information on social valuesDetermining the income frontierItem 1 might be considered as beyond the scope of simple microeconomicsItem 2 requires modelling of what is possible in the underlying structure of the economywhich is what microeconomics is all aboutApril 201834