Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data...
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Transcript of 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
Pen’s parade, quantile function, cdfSize, spread, poverty
DataIncome vector, cdf
TodayInequality and economicsWelfare economicsInequality measures
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
Ex Two persons, two goodsEdgeworth box
Can move from goods space to utility space
Person 1
Person 2
Ex Two persons, two goodsEdgeworth box
Person 1
Person 2
Utility of 2
Utility of 1
Ex Two persons, two goodsEdgeworth box
Person 1
Person 2
Utility of 2
Utility of 1
Ex Two persons, two goodsEdgeworth box
Person 1
Person 2
Utility of 2
Utility of 1
Ex Two persons, two goodsEdgeworth box
Person 1
Person 2
Utility of 2
Utility of 1
Ex Two persons, two goodsEdgeworth box
Called the utility possibilities curve
Person 1
Person 2
Utility of 2
Utility of 1
Ex Two persons, two goodsEdgeworth box
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
utility of person 1u2
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
utility of person 1u2
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
Pen’s parade, quantile function, cdfSize, spread, poverty
DataIncome vector, cdf
TodayInequality and economicsWelfare economicsInequality measures
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.