Welfare: The Social-Welfare Function
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Welfare: The Social-Welfare Function

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Welfare: The Social-Welfare Function




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Presentation on theme: "Welfare: The Social-Welfare Function"— Presentation transcript:

Slide1

Welfare: The Social-Welfare Function

MICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential Welfare: BasicsWelfare: Efficiency

Prerequisites

July 2015

1

Slide2

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…?

Requirements

July 2015

2

Slide3

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

What is special about a social-welfare function?

July 2015

3

Slide4

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 orderings

A sketch of the approach

July 2015

4

Slide5

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, ub,... )

July 2015

5

Slide6

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?

July 2015

6

Slide7

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

Where does the social-welfare function come from?

July 2015

7

Slide8

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 principle

July 2015

8

Slide9

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

July 2015

9

Slide10

“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: åwÎW pwu(xw) Equivalently: åwÎW pwuw

pw: probability assigned to wu : cardinal utility function, independent of wuw: utility payoff in state w

A suitable assumption about “probabilities”? nh 1 W = — å uh nh 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 phuh

An additive form of the welfare function

July 2015

10

Slide11

Questions about “equal ignorance”

ph

identity

|nh

h

|

1

|2

|3

|

Construct a lottery on identity

The “equal ignorance” assumption...

Where people know their identity with certainty

Intermediate case

The “equal ignorance” assumption:

p

h = 1/nhBut is this appropriate?

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

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

July 2015

11

Slide12

2: The PLUM principle

Now for the second  rather cynical approachAcronym stands for People Like Us MatterWhoever is in power may impute:...either their own views,... or what they think “society’s” views are,... or what they think “society’s” views ought to be, ...probably based on the views of those in powerThere’s a whole branch of modern microeconomics that is a reinvention of classical “Political Economy”Concerned with the interaction of political decision-making and economic outcomes.But beyond the scope of this course

July 2015

12

Slide13

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

Conditions for a welfare maximum

July 2015

13

Slide14

The SWF maximum problem

Take the individualistic welfare model W(u1, u2, u3, ...)

Standard assumption

Assume everyone is selfish: uh = Uh(xh) , h = 1,2, ..., nh

my utility depends only on my bundle

Substitute in the above: W(U1(x1), U2(x2), U3(x3), ...)

Gives SWF in terms of the allocation

a quick sketch

July 2015

14

Slide15

From an allocation to social welfare

From the attainable set...

A

A

(

x

1

a

,

x

2

a

)(x1b, x2b)

...take an allocation

Evaluate utility for each agent

Plug into

W

to get social welfare

u

a=Ua(x1a, x2a)ub=Ub(x1b, x2b)

W(ua, ub)

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

July 2015

15

Slide16

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 duh = S Uih(xh) dxih i=1

The effect on h if all commodities are changed

Differentiate W with respect to uh: nh dW = S Wh duh h=1

Changes in utility change social welfare .

Substitute for duh in the above: nh n dW = S Wh S Uih(xh) dxih h=1 i=1

So changes in allocation change welfare.

Weights from the SWF

Weights from utility function

marginal impact on social welfare of h’s utility

July 2015

16

Slide17

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 maths

July 2015

17

Slide18

The SWF maximum problem

First component of the problem: W(U1(x1), U2(x2), U3(x3), ...)

Individualistic welfare

Utility depends on own consumption

The objective function

Second component of the problem: nh F(x) £ 0, xi = S xih h=1

Feasibility constraint

The Social-welfare Lagrangian: nh W(U1(x1), U2(x2),...) - lF (S xh ) h=1

Constraint subsumes technological feasibility and materials balance

FOCs for an interior maximum: Wh (...) Uih(xh) − lFi(x) = 0

From differentiating Lagrangean with respect to xih

And if xih = 0 at the optimum: Wh (...) Uih(xh) − lFi(x) £ 0

Usual modification for a corner solution

All goods are private

July 2015

18

Slide19

Solution to SWF maximum problem

From FOCs: Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(xh) Ujℓ(xℓ)

Any pair of goods, i,jAny pair of households h, ℓ

MRS equated across all h We’ve met this condition before - Pareto efficiency

Also from the FOCs: Wh Uih(xh) = Wℓ Uiℓ(xℓ)

social marginal utility of toothpaste equated across all h

Relate marginal utility to prices: Uih(xh) = Vyhpi

This is valid if all consumers optimise

Substituting into the above: Wh Vyh = Wℓ Vyℓ

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

Marginal utility of money

Social marginal utility of income

July 2015

19

Slide20

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 littleSimilar to looking at comparative statics for a single agent

July 2015

20

Slide21

Differentiate the SWF w.r.t. {yh}: nh dW = S Wh duh h=1

Changes in income, social welfare

nh dW = M S dyh h=1

nh = S WhVyh dyh h=1

Social welfare can be expressed as: W(U1(x1), U2(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 = – SWhVyh xihdpi h=1

from Roy’s identity

nh dW = – M S xihdpi h=1

...related to prices

Change in total expenditure

.

July 2015

21

Slide22

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 expenditureBut what if we are not in an ideal world?

July 2015

22

Slide23

Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

A lesson from risk and uncertainty

July 2015

23

Slide24

Derive a SWF in terms of incomes

What happens if the distribution of income is not ideal? M is no longer equal for all hUseful 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, … )

July 2015

24

Slide25

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

July 2015

25

Alf's

income

Slide26

Extension to

nh=3

Here we have 3 persons

Charlie's income

Alf's income

Bill's income

O

line of perfect equality

y

An income distribution.

July 2015

26

Slide27

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

July 2015

27

Slide28

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”

July 2015

28

Slide29

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 uncertaintyCan be expressed equivalently as an expectation: W = E z(yh)where the expectation is over all identitiesprobability of identity h is the same, 1/nh , for all hConstant relative-inequality aversion: 1 z(y) = —— y1 – i 1 – iwhere i is the index of inequality aversionworks just like r,the index of relative risk aversion

July 2015

29

Slide30

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 incomeand more sharply curved contours

lower inequality aversion

higher inequality aversion

z = φ(z)

July 2015

30

Slide31

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 yh h=1

General case (0< i< ): nh W = S [yh]1-i/ [1-i] h=1

“Rawlsian” case (i = ): W = min yh h

July 2015

31

Slide32

Inequality, welfare, risk and uncertainty

There is a similarity of form between… personal judgments under uncertainty social judgments about income distributions.Likewise 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 examples

July 2015

32

Slide33

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 ANonconvexity, 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

July 2015

33

Slide34

Summary

The standard SWF is an ordering on utility levels Analogous to an individual's ordering over lotteriesInequality- and risk-aversion are similar conceptsIn 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 economy......which is what microeconomics is all about

July 2015

34