Prerequisites

Almost essential Welfare: Basics Welfare: Efficiency

WELFARE: THE SOCIALWELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell

July 2015

Frank Cowell: Welfare - Social Welfare function

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Social Welfare Function  Limitations 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 attempt  Something like a simple utility function…? Requirements

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Frank Cowell: Welfare - Social Welfare function

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Overview Welfare: SWF

The Approach

What is special about a social-welfare function?

SWF: basics

SWF: national income SWF: income distribution July 2015

Frank Cowell: Welfare - Social Welfare function

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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

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Frank Cowell: Welfare - Social Welfare function

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Using a SWF υb  Take the utility-possibility set  Social welfare contours  A social-welfare optimum?

W(υa, υb,... )

W defined on utility levels Not on orderings

𝕌 July 2015

•

Imposes several restrictions… ..and raises several questions

υa

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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?

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Frank Cowell: Welfare - Social Welfare function

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Overview Welfare: SWF

The Approach

Where does the social-welfare function come from?

SWF: basics

SWF: national income SWF: income distribution July 2015

Frank Cowell: Welfare - Social Welfare function

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An individualistic SWF  The standard form expressed thus

W(υ1, υ2, υ3, ...)

• 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

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Frank Cowell: Welfare - Social Welfare function

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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 ω and θ!

 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

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Frank Cowell: Welfare - Social Welfare function

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“Equal ignorance”: formalisation  Individualistic welfare:

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

W(υ1, υ2, υ3, ...)

 vN-M form of utility function:

∑ω∈Ω πωu(xω) Equivalently:

∑ω∈Ω πωυω

 Replace Ω by set of identities {1,2,...nh}:

∑h πhυh

 A suitable assumption about “probabilities”? nh

W = — ∑ υh 1

nh

July 2015

use theory of choice under uncertainty to find shape of W πω: probability assigned to ω u : cardinal utility function, independent of ω υω: utility payoff in state ω welfare is expected utility from a "lottery on identity“ An additive form of the welfare function

h=1

Frank Cowell: Welfare - Social Welfare function

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Questions about “equal ignorance”  Construct a lottery on identity  The “equal ignorance” assumption...  Where people know their identity with certainty

πh

 Intermediate case

The “equal ignorance” assumption: πh = 1/nh But is this appropriate? |

| |

1 2 3

|

|

identity

h

nh

Or should we assume that people know their identities with certainty? Or is the "truth" somewhere between...?

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Frank Cowell: Welfare - Social Welfare function

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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

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Frank Cowell: Welfare - Social Welfare function

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Overview Welfare: SWF

The Approach

Conditions for a welfare maximum

SWF: basics

SWF: national income SWF: income distribution July 2015

Frank Cowell: Welfare - Social Welfare function

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The SWF maximum problem  Take the individualistic welfare model

Standard assumption

 Assume everyone is selfish:

my utility depends only on my bundle

 Substitute in the above:

Gives SWF in terms of the allocation

W(υ1, υ2, υ3, ...)

υh = Uh(xh) , h = 1,2, ..., nh

W(U1(x1), U2(x2), U3(x3), ...)

a quick sketch

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Frank Cowell: Welfare - Social Welfare function

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From an allocation to social welfare  From the attainable set...

(x1a, x2a) (x1b, x2b)

 ...take an allocation  Evaluate utility for each agent

A

A

 Plug into W to get social welfare

υa=Ua(x1a, x2a) υb=Ub(x1b, x2b)  But what happens to welfare if we vary the allocation in A? W(υa, υb)

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Frank Cowell: Welfare - Social Welfare function

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Varying the allocation  Differentiate w.r.t. xih : dυh = Uih(xh) dxih  Sum over i:

The effect on h if commodity i is changed

marginal utility derived by h from good i

n

dυh = Σ Uih(xh) dxih

The effect on h if all commodities are changed

i=1

 Differentiate W with respect to υh: nh

dW = Σ Wh dυh marginal impact on social welfare of h’s utility

h=1

 Substitute nfor dυnh in the above: dW = Σ Wh Σ Uih(xh) dxih h

h=1 Weights from the SWF July 2015

Changes in utility change social welfare .

i=1

So changes in allocation change welfare.

Weights from utility function Frank Cowell: Welfare - Social Welfare function

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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

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Frank Cowell: Welfare - Social Welfare function

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

The objective function

Utility depends on own consumption

Individualistic welfare

 Second component of the problem: n Φ(x) ≤ 0, xi = Σh=1 xih

Feasibility constraint

 The Social-welfare Lagrangian: n W(U1(x1), U2(x2),...) - λΦ (Σh=1 xh )

Constraint subsumes technological feasibility and materials balance

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

From differentiating Lagrangean with respect to xih

 And if xih = 0 at the optimum: Wh (...) Uih(xh) − λΦi(x) ≤ 0

Usual modification for a corner solution

h

All goods are private h

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Frank Cowell: Welfare - Social Welfare function

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Solution to SWF maximum problem 

From FOCs:

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

MRS equated across all h

Uih(xh) Uiℓ(xℓ) ——— = ——— Ujh(xh) Ujℓ(xℓ)

We’ve met this condition before - Pareto efficiency

 Also from the FOCs:

Wh Uih(xh) = Wℓ Uiℓ(xℓ)



Relate marginal utility to prices:

Uih(xh)

= Vy

hp

i

social marginal utility of toothpaste equated across all h

This is valid if all consumers optimise

Marginal utility of money



Substituting into the above:

Wh Vyh = Wℓ Vyℓ

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Social marginal utility of income

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

Frank Cowell: Welfare - Social Welfare function

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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

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Frank Cowell: Welfare - Social Welfare function

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Changes in income, social welfare 

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

Differentiate the SWF w.r.t. {yh}:

Changes in utility and change social welfare …



nh

nh

dW = Σ Wh dυh = Σ WhVyh dyh h=1

h=1 nh

dW = M Σ dyh h=1



...related to income change in “national income”

Differentiate the SWF w.r.t. pi : nh

nh

h=1

h=1

dW = Σ WhVihdpi= – ΣWhVyh xihdpi

from Roy’s identity

nh

dW = – M Σ xihdpi h=1 July 2015

Changes in utility and change social welfare …

Change in total expenditure

...related to prices .

.

Frank Cowell: Welfare - Social Welfare function

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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

Frank Cowell: Welfare - Social Welfare function

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Overview Welfare: SWF

The Approach

A lesson from risk and uncertainty

SWF: basics

SWF: national income SWF: income distribution July 2015

Frank Cowell: Welfare - Social Welfare function

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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, … )

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Income-distribution space: nh=2  The income space: 2 persons

Bill's income

An income distribution

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

•y 45° O

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Alf's Alf's income income Frank Cowell: Welfare - Social Welfare function

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Extension to nh=3 Charlie's income

 Here we have 3 persons An income distribution.

•y O

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Welfare contours  An arbitrary income distribution  Contours of W  Swap identities

yb

 Distributions with the same mean  Equally-distributed-equivalent income equivalent in welfare terms

 Anonymity implies symmetry of W

•

E

ξ

Ey

y is mean income

 Richer-to-poorer income transfers increase welfare

higher welfare

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

•y ya

E y −ξ is income that society would give up to eliminate inequality

ξ Ey July 2015

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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”

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Special form of the SWF  It can make sense to write W in the additive form nh

W=

1 — Σ nh h=1

ζ(yh)

• where the function ζ 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 ζ(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 1–ι ζ(y) = —— y 1–ι • where ι is the index of inequality aversion • works just like ρ,the index of relative risk aversion

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Concavity and inequality aversion W

The social evaluation function  Let values change: φ is a concave transformation.

ζ(y)

lower inequality aversion

ζ°(y) higher inequality aversion

ζ° = φ(ζ)

 More concave ζ(•) implies higher inequality aversion ι ...and lower equally-distributedequivalent income and more sharply curved contours

y income

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Social views: inequality aversion yb

 Indifference to inequality

yb

 Mild inequality aversion

ι=½

ι=0

 Strong inequality aversion  Priority to poorest

 “Benthamite” case (ι = 0): nh

ya

O

yb

ya

O

yb

ι=2

W= Σ yh h=1

ι=∞

 General case (0< ι< ∞): nh

W = Σ [yh]1-ι/ [1-i] h=1

O

ya

O

ya

 “Rawlsian” case (ι = ∞): W = min yh h

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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 ζ should be taken as

identical  “Optimal” social state depends crucially on shape of W • In other words the shape of ζ • Or the value of ι

July 2015

Frank Cowell: Welfare - Social Welfare function

Three examples

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Social values and welfare optimum yb

 The income-possibility set Y  Welfare contours ( ι = 0)  Welfare contours ( ι = ½)  Welfare contours ( ι = ∞)

Y derived from set A Nonconvexity, asymmetry come from heterogeneity of households

Y

 y* maximises total income irrespective of distribution

y***

•

•

 y** trades off some income for greater equality

y** y* • ya

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 y*** gives priority to equality; then maximises income subject to that

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Summary 

The standard SWF is an ordering on utility levels • •



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

 1. 2.

  

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

Information on social values 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

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