1. Welfare: The Social-Welfare Function
Prerequisites
Almost essential
Welfare: Basics
Welfare: Efficiency
Welfare: The Social-Welfare Function
MICROECONOMICS
Principles and Analysis
Frank Cowell
December 2006

2. 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

3. Overview... What is special about a social-welfare function?
Welfare: SWF
The Approach
What is special about a social-welfare function?
SWF: basics
SWF: national income
SWF: income distribution

4. A sketch of the approach
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

5. Using a SWF • U ub W(ua, ub,... ) ua W defined on utility levels
Take the utility-possibility set
Social welfare contours
A social-welfare optimum?
W(ua, ub,... )
W defined on utility levels
Not on orderings
Imposes several restrictions…
…and raises several questions • U ua

6. 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?

7. Overview... Where does the social-welfare function come from?
Welfare: SWF
The Approach
Where does the social-welfare function come from?
SWF: basics
SWF: national income
SWF: income distribution

8. 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

9. 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

10. “Equal ignorance”: formalisation
payoffs I would get if I were assigned identity 1,2,3,... in the Great Lottery of Life
The individualistic welfare model:
W(u1, u2, u3, ...)
use theory of choice under uncertainty to find the shape of SWF W
vN-M form of the utility function:
åwÎW pwu(xw)
Equivalently:
åwÎW pwuw
pw: probability assigned to w
u: cardinal utility function, independent of w
uw: utility payoff in state w
Replace W by the set of identities {1,2,...nh}:
åh phuh
welfare is expected utility from a "lottery on identity“
A suitable assumption about “probabilities”? nh 1
W = — S uh
nh h=1
An additive form of the welfare function

11. Questions about “equal ignorance”
Construct a lottery on identity
The “equal ignorance” assumption... ph
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? | 1 | 2 | 3 | | nh
identity h
Or is the "truth" somewhere between...?

12. 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

13. Overview... Conditions for a welfare maximum Welfare: SWF The Approach
SWF: basics
SWF: national income
SWF: income distribution

14. 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

15. From an allocation to social welfare
From the attainable set...
(x1a, x2a)
(x1b, x2b)
...take an allocation
Evaluate utility for each agent
Plug into W to get social welfare
ua=Ua(x1a, x2a)
ub=Ub(x1b, x2b)
But what happens to welfare if we vary the allocation in A?
W(ua, ub)

16. Varying the allocation
The marginal utility derived by household h from good i
Differentiate w.r.t. xih :
duh = Uih(xh) dxih
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
The marginal impact on social welfare of household h’s utility
Changes in utility change social welfare .
Differentiate W with respect to uh: nh
dW = S Wh duh
h=1
Weights from the SWF
Weights from the utility function
So changes in allocation change welfare.
Substitute for duh in the above:
nh n
dW = S Wh SUih(xh) dxih
h= i=1

17. Use this to characterise a welfare optimum
Write down SWF, defined on individual utilities.
Introduce feasibility constraints on overall consumptions.
Set up the Lagrangean.
Solve in the usual way
Now for the maths

18. The SWF maximum problem
Utility depends on own consumption
Individualistic welfare function
First component of the problem:
W(U1(x1), U2(x2), U3(x3), ...)
The objective function
All goods are private
Feasibility constraint
Second component of the problem: nh
F(x) £ 0, xi = S xih
h=1
Usual Lagrange multiplier
The Social-welfare Lagrangean: nh
W(U1(x1), U2(x2), ...) - lF(S xh )
h=1
Note: 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

19. Solution to SWF maximum problem
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
From the first-order conditions :
Uih(xh) Uiℓ(xℓ)
——— = ———
Ujh(xh) Ujℓ(xℓ)
Also from the FOCs:
Wh Uih(xh) = Wℓ Uiℓ(xℓ)
This is new!
social marginal utility of toothpaste equated across all h.
Marginal utility of money
This is valid if all consumers optimise
Relate marginal utility to prices:
Uih(xh) = Vyhpi
Social marginal utility of income
At the optimum the welfare value of a $ of income is equated across all h. Call this common value M
Substituting into the above:
Wh Vyh = Wℓ Vyℓ

20. 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

21. 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}: nh
dW = S Wh duh
h=1
Changes in utility and change social welfare … nh
= S WhVyh dyh
h=1
Change in total incomes - i.e. change in “national income”
...related to income nh
dW = M S dyh
h=1
Differentiate the SWF w.r.t. pi : nh
dW = SWhVihdpi
h=1
Follows from Roy’s identity
Changes in utility and change social welfare … nh
= – SWhVyh xihdpi
h=1
Change in total expenditure nh
dW = – M S xihdpi
h=1
...related to prices . .

22. 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?

23. Overview... A lesson from risk and uncertainty Welfare: SWF
The Approach
A lesson from risk and uncertainty
SWF: basics
SWF: national income
SWF: income distribution

24. 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 functions V
Express utility in terms of prices p and incomes y
Assume prices p are given
“Equivalise” (i.e. rescale) incomes y
allow for differences in people’s needs
allow for differences in household size
Then you can write welfare as
W(ya, yb, yc, … )

25. Income-distribution space: nh=2
The income space: 2 persons
An income distribution
Bill's
income
line of perfect equality
Note the similarity with a diagram used in the analysis of uncertainty y
45°
Alf's
income O

26. Extension to nh=3 • Charlie's income line of perfect equality
Here we have 3 persons
An income distribution.
Charlie's income
line of perfect equality
Alf's income
Bill's income • y O

27. Welfare contours  Ey x y  yb ya x Ey
An arbitrary income distribution
Contours of W
Swap identities
Distributions with the same mean 
equivalent in
welfare terms
Equally-distributed-equivalent income
Anonymity implies symmetry of W.
Ey is mean income
Richer-to-poorer income transfers increase welfare. Ey
higher
welfare x  y
x is the income that, if received uniformly by all, would yield same level of social welfare as y. ya
Ey x is the income that, society would give up to eliminate inequality x Ey

28. 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”

29. Special form of the SWF W = — S z(yh) nh h=1 z(y) = —— y1 – i 1 – i
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:
z(y) = —— y1 – i
1 – i
where i is the index of inequality aversion
works just like r,the index of relative risk aversion

30. Concavity and inequality aversion
The social evaluation function W
Let values change: φ is a concave transformation.
lower inequality aversion
z(y)
More concave z(•) implies higher inequality aversion i
...and lower equally-distributed-equivalent income
and more sharply curved contours
z(y)
z = φ(z)
higher inequality aversion y
income

31. Social views: inequality aversionyb ya O yb ya O
Indifference to inequality
i = 0
i = ½
Mild inequality aversion
Strong inequality aversion
Priority to poorest
“Benthamite” case (i=0): nh
W= S yh
h=1 yb ya O yb ya O
i = 2
i = 
General case (0<i<): nh
W = S [yh]1-i/ [1-i]
h=1
“Rawlsian” case (i=):
W= min yh h

32. 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

33. Social values and welfare optimumyb
The income-possibility set Y
Welfare contours ( i = 0)
Welfare contours ( i = ½)
Welfare contours ( i = )
Y derived from set A or U
Nonconvexity, asymmetry come from heterogeneity of households Y
y* maximises total income irrespective of distribution
y*** 
y** 
y** trades off some income for greater equality y* 
y*** gives priority to equality; then maximises income subject to that ya

34. 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:
Information on social values
Determining the income frontier
This requires a modelling of what is possible in the underlying structure of the economy...
...which is what Micro-Economic principles is all about