Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data...

Session 3

ReviewDistributions

Pen’s parade, quantile function, cdfSize, spread, poverty

DataIncome vector, cdf

TodayInequality and economicsWelfare economicsInequality measures

Inequality and Economics

Q/ Inequality of what?Here – income, consumption, or a single dimensional

achievementLater – Sen contends we should examined inequality in a

different space (capability space)

Q/ Which income?Among whom?Over what period of time?What about durable goods?Rich uncles?Bribes and black market income?

IssuesStatistical or normative measure?Cardinality or ordinality of income?Complete measure or “quasiordering”?

Can economics help?“New welfare economics” assumes different

persons’ utility cannot be added, subtracted or otherwise compared (L. Robbins)

Where does it leave us?What welfare criterion does not use interpersonal

comparisons?

Pareto efficiency (V. Pareto)Note It’s a quasiordering

Fundamental Welfare Theorems1. A Walrasian equilibrium is Pareto efficient2. A Pareto efficient allocation can be sustained as

a Walrasian equilibrium (given transfers)

Pretty weak criterionCan’t compare allocations along contract curve!

Digression Is trade good?All justifications require more than Pareto

efficiencyKaldor-Hicks criterionImprovement if there are transfers that could leave

everyone better offBut they are never madeAnd criterion is inconsistent

Must fundamentally go beyond ParetoWelfare functions

Aggregate preferences to obtain social ranking

ProblemIf require ordinal, non-comparable preference,

Arrow’s “Impossibility Theorem” applies.There is no SWF f aggregating individual

preference orderings into a social ordering R = f({Ri}) satisfying four basic conditions: U, P, I, D.

A second theorem of Sen loosens assumptions, and shows that the only possible aggregation procedure ranks all Pareto-incomparable states the same

One interpretationPaucity of informationNeed some notion of interpersonal comparability

Utilitarian welfare functionsSuppose individual welfare (or utility) can be

expressed as a function of incomeA utilitarian judges income distributions via

W = [u1(x1) +…+ un(xn)]/n

where say each ui is strictly concaveQ/ What does this mean?A/ Diminishing MU of income.Note Complete orderingQ/How does this relate to inequality?A/ In “utility space” not at all.

Highest sum irrespective of who gets whatCan favor richest if most efficient at converting income

to utility

Theorem Suppose all ui are identical and strictly concave. Then W is maximized at equality.

Session 3

ReviewDistributions




Welfare Economics

Q/ What is “best” allocation? (Normative)Of goods and services, of utility, of income?Note Three different spaces

We begin with “goods” space

Ex Two persons, two goodsEdgeworth box

Pareto efficiency reduces consideration to contract curveQ/ Marginal conditions?

Person 1

Person 2


Can move from goods space to utility space

Person 1

Person 2


Person 1

Person 2

Utility of 2

Utility of 1


Called the utility possibilities curve

Person 1

Person 2

Utility of 2

Utility of 1


Could also construct via “indirect” utilityUtility as a function of income ui(xi)

Person 1

Person 2

Utility of 2

Utility of 1

Note Utility possibilities curve is like a budget set Q/ How to choose?A/ If Pareto improvement, easy

If not, then there are tradeoffsDef A social welfare function assigns an overall social

welfare level to each vector of individual utilities (u1,...,un).

Note Weighs well being of one person against another; weighs efficiency vs equity

Def Satisfies Pareto principle if W is increasing in each utility level.

Def A social indifference curve is the set all utility vectors with with the same level of social welfare

Q/ Slope?

Examples of SWF

Utilitarian W(u1,...,un) = (u1 + ... + un)/n

Graph

utility of person 1u2

u1

utility of person 2

Examples of SWF

Rawlsian W(u1,...,un) = min(u1,...,un)

Graph


u1

utility of person 2

Examples of SWFGeneral W(u1,...,un)

increasing in each ui

symmetric, convex social indiff. curvesGraph

ExW = (u1u2)1/2 Note: Symmetric, quasiconcave


u1

utility of person 2

Graph Social constraintSocial objectiveSocial choice u*

u1

u2

u1*

u2*

Graph Social constraintSocial objective RawlsianSocial choice u*

u1

u2

u1*

u2*

Graph Social constraintSocial objective utilitarianSocial choice u*

Note Equality in utility in all casesNote Same would be true if ui(xi) = xi

u1

u2

u1*

u2*

Graph Social constraintSocial objective utilitarianSocial choice u*

Q/ What if u1(x1) = 2x1 and u2(xi) = x2?

Q/ Implications for MU of income?

u1

u2

u1*

u2*

Rawlsian Equity vs efficiency?Q/ In income space?

u1

u2

u1*

u2*

General case Equity vs efficiency?Q/ Income space?

u1

u2

u1*

u2*

Utilitarian Equity vs. efficiency?Q/ income space?

u2

u1*

u2* = 0

Weak Equity Axiom If person 1 has higher welfare than person 2 at all income levels, then the social choice should ensure that 2 has more income than 2.

Q/ Which satisfies?

Note Key issue is how to calibrate indirect utilitiesNormative choice, not objectively givenTypically assume identical with diminishing MU

consistent with arbitrary preferences over goods

u2

u1*

u2* = 0

Session 3

ReviewDistributions




Inequality Measures

Notation x is the income distributionxi is the income of the ith personn=n(x) is the population size.D is the set of all distributions of any population size

DefinitionAn inequality measure is a function I from D to R

which, for each distribution x in D indicates the level I(x) of inequality in the distribution.

Four Basic Properties

DefinitionWe say that x is obtained from y by a permutation

of incomes if x = Py, where P is a permutation matrix.

Ex

Symmetry (Anonymity) If x is obtained from y by a permutation of

incomes, then I(x)=I(y).

Idea All differences across people have been accounted

for in x

6

8

1

8

1

6

001

100

010

Pyx

DefWe say that x is obtained from y by a replication if

the incomes in x are simply the incomes in y repeated a finite number of times

Ex

Replication Invariance (Population Principle)If x is obtained from y by a replication, then

I(x)=I(y).

Idea Can compare across different sized populations

€

x = (y1, y1, y2, y2,......, yn , yn )

€

x = (6,6,6,1,1,1,8,8,8)

Def We say that x is obtained from y by a proportional

change (or scalar multiple) if x=αy, for some α > 0.

Ex

Scale Invariance (Zero-Degree Homogeneity)If x is obtained from y by a proportional change,

then I(x)=I(y).

Idea Relative inequality

€

y = (6,1,8)

€

x = (12,2,16)

DefWe say that x is obtained from y by a (Pigou-Dalton)

regressive transfer if for some i, j:i) yi < yj

ii) yi – xi = xj – yj > 0

iii) xk = yk for all k different to i,j

Ex

Transfer Principle If x is obtained from y by a regressive transfer, then

I(x) > I(y).

Idea Mean preserving spread increases measured

inequality

€

y = (2,6,7)

€

x = (1,6,8)

Def Any measure satisfying the four basic

properties (symmetry, replication invariance, scale invariance, and the transfer principle) is called a relative inequality measure.

Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data...

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