welfare: the social-welfare function

Frank Cowell: Welfare - Social Welfare function

WELFARE: THE SOCIAL-WELFARE FUNCTIONMICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential Welfare: BasicsWelfare: Efficiency

Prerequisites

July 2015 1


Social Welfare FunctionLimitations of the welfare analysis so far:Constitution approach

• Arrow theorem – is the approach overambitious?General welfare criteria

• efficiency – nice but indecisive• extensions – contradictory?

SWF is our third attemptSomething like a simple utility function…?

Requirements

July 2015 2


Overview

The Approach

SWF: basics

SWF: national income

SWF: income distribution

Welfare: SWF

What is special about a social-welfare function?

July 2015 3


The SWF approach Restriction of “relevant” aspects of social state to each person

(household) Knowledge of preferences of each person (household) Comparability of individual utilities

• utility levels• utility scales

An aggregation function W for utilities• contrast with constitution approach• there we were trying to aggregate orderings

A sketch of the approach

July 2015 4


Using a SWF

ua

ub

𝕌

Take the utility-possibility set

A social-welfare optimum? Social welfare contours

W defined on utility levelsNot on orderingsImposes several restrictions…..and raises several questions

W(ua, ub,... )

•

July 2015 5


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


Overview

The Approach

SWF: basics



Welfare: SWF

Where does the social-welfare function come from?

July 2015 7


An individualistic SWF The standard form expressed thus

W(u1, u2, u3, ...)• an ordinal function• defined on space of individual utility levels• not on profiles of orderings

But where does W come from...? We'll check out two approaches:

• The equal-ignorance assumption• The PLUM principle

July 2015 8


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 lotteries

• don't confuse w and q! Each individual has partial knowledge:

• knows the distribution of allocations in the population• knows the utility implications of the allocations• knows the alternatives in the Great Lottery of Life• does not know which lottery ticket he/she will receive

July 2015 9


“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


Questions about “equal ignorance”

ph

identity |

nhh |

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: ph = 1/nh

But is this appropriate?

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

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

July 2015 11


2: The PLUM principle Now for the second rather cynical approach Acronym stands for People Like Us Matter Whoever 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 power

There’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


Overview

The Approach

SWF: basics



Welfare: SWF

Conditions for a welfare maximum

July 2015 13


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


From an allocation to social welfare

From the attainable set...

AA

(x1a, x2

a)(x1

b, x2b) ...take an allocation

Evaluate utility for each agent

Plug into W to get social welfare

ua=Ua(x1a, x2

a)ub=Ub(x1

b, x2b)

W(ua, ub)

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

July 2015 15


Varying the allocation

Differentiate w.r.t. xih :

duh = Uih(xh) dxi

h

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) dxi

h 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) dxi

h

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


Use this to characterise a welfare optimum

Write down SWF, defined on individual utilities Introduce feasibility constraints on overall consumptions Set up the Lagrangian Solve in the usual way

Now for the maths

July 2015 17


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 xi

h 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 (...) Ui

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

h

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


Solution to SWF maximum problem From FOCs:

Uih(xh) Ui

ℓ(xℓ) ——— = ———

Ujh(xh) Uj

ℓ(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 Ui

h(xh) = Wℓ Uiℓ(xℓ)

social marginal utility of toothpaste equated across all h

Relate marginal utility to prices:Ui

h(xh) = Vyhpi

This is valid if all consumers optimise

Substituting into the above:Wh Vy

h = 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


To focus on main result... Look what happens in neighbourhood of optimum Assume that everyone is acting as a maximiser

• firms• households

Check what happens to the optimum if we alter incomes or prices a little

Similar to looking at comparative statics for a single agent

July 2015 20


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 incomechange in “national income”

Differentiate the SWF w.r.t. pi : nh dW = S WhVi

hdpi h=1

.

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

h xihdpi h=1

from Roy’s identity

nh dW = – M S xi

hdpi

h=1

...related to pricesChange in total expenditure

.

July 2015 21


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?

July 2015 22


Overview

The Approach

SWF: basics



Welfare: SWF

A lesson from risk and uncertainty

July 2015 23


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 incomes Do this by using indirect utility function V

• Express utility in terms of prices p and income y Assume prices p are given “Equivalise” (i.e. rescale) each income y

• allow for differences in people’s needs• allow for differences in household size

Then you can write welfare as W(ya, yb, yc, … )

July 2015 24


Income-distribution space: nh=2

Bill'sincom

e

Alf'sincome

O

The income space: 2 persons

An income distribution

· y

45°

line o

f perf

ect eq

uality

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

July 2015 25

Alf'sincome


Extension to nh=3

Here we have 3 persons

Charlie's

income

Alf's income

Bill's income

O

line of perf

ect

equality

• y

An income distribution.

July 2015 26


Welfare contours

x E y

ya

yb

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


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


Special form of the SWF It can make sense to write W in the additive form nh 1 W = — S z(yh) nh h=1

• where 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/nh , for all h

Constant relative-inequality aversion: 1 z(y) = —— y1 – i 1 – i

• where i is the index of inequality aversion• works just like r,the index of relative risk aversion

July 2015 29


Concavity and inequality aversion

W

z(y)

incomey

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)

July 2015 30


Social views: inequality aversion

i = ½

yb

yaO

i = 0

yb

yaO

i = 2

yb

yaO

i =

Indifference to inequality Mild inequality aversion

yb

yaO

Strong inequality aversion Priority to poorest

“Benthamite” case (i = 0): nh

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


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 curiosity Or as an essential component of welfare economics

• Uses 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 W

• In other words the shape of z• Or the value of i

Three examples

July 2015 32


Social values and welfare optimum

ya

yb 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


Summary The standard SWF is an ordering on utility levels

• Analogous to an individual's ordering over lotteries• Inequality- and risk-aversion are similar concepts

In ideal conditions SWF is proxied by national income But for realistic cases two things are crucial:

1. Information on social values2. Determining the income frontier

Item 1 might be considered as beyond the scope of simple microeconomics

Item 2 requires modelling of what is possible in the underlying structure of the economy...

...which is what microeconomics is all about

July 2015 34

welfare: the social-welfare function

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