Frank Cowell: Welfare -Social Welfare function
WELFARE: THE SOCIAL-
WELFARE FUNCTION
MICROECONOMICS
Principles and Analysis
Frank Cowell
Almost essential
Welfare: Basics
Welfare: Efficiency
Prerequisites
April 2018 1
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WELFARE: THE SOCIAL-
WELFARE FUNCTION
MICROECONOMICS
Principles and Analysis Frank Cowell
Almost essential Welfare: Basics Welfare: Efficiency
Prerequisites
April 2018^1
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
April 2018^2
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
April 2018^4
Using a SWF
υ a
υ b
𝕌𝕌
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 (υ a , υ b ,... )
April 2018^5
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Where does the social-welfare function come from?
April 2018^7
An individualistic SWF
The standard form expressed thus W (υ^1 , υ^2 , υ^3 , ...)
- 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
April 2018^8
“Equal ignorance”: formalisation
Individualistic welfare: W (υ^1 , υ^2 , υ^3 , ...)
use theory of choice under uncertainty to find shape of W vN-M form of utility function: ∑ω∈Ω πω u ( x ω) Equivalently: ∑ω∈Ω πω υω
πω: probability assigned to ω u : cardinal utility function, independent of ω υω: utility payoff in state ω
A suitable assumption about “probabilities”? n (^) h 1
W = — ∑ υ h
nh h =
welfare is expected utility from a "lottery on identity“
payoffs if assigned identity 1,2,3,... in the Lottery of Life
Replace Ω by set of identities {1,2,... nh }: ∑ h π h υ h
An additive form of the welfare function
April 2018^10
Questions about “equal ignorance”
π h
identity
| h n (^) 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: π h = 1 /nh But is this appropriate? Or should we assume that people know their identities with certainty? Or is the "truth" somewhere between...?
April 2018^11
Overview
The Approach
SWF: basics
SWF: national income
SWF: income distribution
Welfare: SWF
Conditions for a welfare maximum
April 2018^13
The SWF maximum problem
Take the individualistic welfare model W (υ^1 , υ^2 , υ^3 , ...)
Standard assumption
Assume everyone is selfish: υ h^ = U h ( x h ) , h = 1,2, ..., nh
my utility depends only on my bundle
Substitute in the above: W ( U^1 ( x^1 ), U^2 ( x^2 ), U^3 ( x^3 ), ...)
Gives SWF in terms of the allocation
a quick sketch
April 2018^14
Varying the allocation
Differentiate w.r.t. xih^ : dυ h^ = U (^) ih ( x h ) d xih marginal utility derived by h from good i
The effect on h if commodity i is changed Sum over i : n
dυ h^ = Σ Uih ( x h ) d xih
i =
The effect on h if all commodities are changed
Differentiate W with respect to υ h : n (^) h
d W = Σ Wh dυ h
h =
Changes in utility change social welfare.
Substitute for d n (^) h υ nh^ in the above:
d W = Σ Wh Σ Uih ( x h ) d xih
h =1 i =
So changes in allocation change welfare.
Weights from the SWF
Weights from utility function
marginal impact on social welfare of h ’s utility
April 2018^16
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
April 2018^17
Solution to SWF maximum problem
From FOCs: Uih (x h ) U (^) iℓ (x ℓ ) ——— = ——— U (^) jh (x h ) U (^) jℓ (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: Wh Uih ( x h ) = Wℓ Uiℓ ( x ℓ )
social marginal utility of toothpaste equated across all h
Relate marginal utility to prices: Uih ( x h ) = Vyh^ p (^) i
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
April 2018^19
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
April 2018^20