Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data Income vector, cdf Today Inequality and economics Welfare economics Inequality measures

Inequality and Economics Q/ Inequality of what? Here – income, consumption, or a single dimensional achievement Later – 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?

Issues Statistical 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 Theorems 1.A Walrasian equilibrium is Pareto efficient 2.A Pareto efficient allocation can be sustained as a Walrasian equilibrium (given transfers) Pretty weak criterion Can’t compare allocations along contract curve!

Digression Is trade good? All justifications require more than Pareto efficiency Kaldor-Hicks criterion Improvement if there are transfers that could leave everyone better off But they are never made And criterion is inconsistent

Must fundamentally go beyond Pareto Welfare functions Aggregate preferences to obtain social ranking Problem If 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({R i }) 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 interpretation Paucity of information Need some notion of interpersonal comparability

Utilitarian welfare functions Suppose individual welfare (or utility) can be expressed as a function of income A utilitarian judges income distributions via W = [u 1 (x 1 ) +…+ u n (x n )]/n where say each u i is strictly concave Q/ What does this mean? A/ Diminishing MU of income. Note Complete ordering Q/How does this relate to inequality? A/ In “utility space” not at all. Highest sum irrespective of who gets what Can favor richest if most efficient at converting income to utility Theorem Suppose all u i are identical and strictly concave. Then W is maximized at equality.

Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data Income vector, cdf Today Inequality and economics Welfare economics Inequality 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 goods Edgeworth box Pareto efficiency reduces consideration to contract curve Q/ Marginal conditions? Person 1 Person 2

Ex Two persons, two goods Edgeworth box Can move from goods space to utility space Person 1 Person 2

Ex Two persons, two goods Edgeworth box Person 1 Person 2 Utility of 2 Utility of 1

Ex Two persons, two goods Edgeworth box Person 1 Person 2 Utility of 2 Utility of 1

Ex Two persons, two goods Edgeworth box Person 1 Person 2 Utility of 2 Utility of 1

Ex Two persons, two goods Edgeworth box Person 1 Person 2 Utility of 2 Utility of 1

Ex Two persons, two goods Edgeworth box Called the utility possibilities curve Person 1 Person 2 Utility of 2 Utility of 1

Ex Two persons, two goods Edgeworth box Could also construct via “indirect” utility Utility as a function of income u i (x i ) 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 tradeoffs Def A social welfare function assigns an overall social welfare level to each vector of individual utilities (u 1,...,u n ). 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 UtilitarianW(u 1,...,u n ) = (u u n )/n Graph utility of person 1 u2u2 u1u1 utility of person 2

Examples of SWF RawlsianW(u 1,...,u n ) = min(u 1,...,u n ) Graph utility of person 1 u2u2 u1u1 utility of person 2

Examples of SWF GeneralW(u 1,...,u n ) increasing in each u i symmetric, convex social indiff. curves Graph Ex W = (u 1 u 2 ) 1/2 Note: Symmetric, quasiconcave utility of person 1 u2u2 u1u1 utility of person 2

Graph Social constraint Social objective Social choice u* u1u1 u2u2 u1*u1* u2*u2*

Graph Social constraint Social objective Rawlsian Social choice u* u1u1 u2u2 u1*u1* u2*u2*

Graph Social constraint Social objective utilitarian Social choice u* Note Equality in utility in all cases Note Same would be true if u i (x i ) = x i u1u1 u2u2 u1*u1* u2*u2*

Graph Social constraint Social objective utilitarian Social choice u* Q/ What if u 1 (x 1 ) = 2x 1 and u 2 (x i ) = x 2 ? Q/ Implications for MU of income? u1u1 u2u2 u1*u1* u2*u2*

Rawlsian Equity vs efficiency? Q/ In income space? u1u1 u2u2 u1*u1* u2*u2*

General case Equity vs efficiency? Q/ Income space? u1u1 u2u2 u1*u1* u2*u2*

Utilitarian Equity vs. efficiency? Q/ income space? u2u2 u1*u1* u 2 * = 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 utilities Normative choice, not objectively given Typically assume identical with diminishing MU consistent with arbitrary preferences over goods u2u2 u1*u1* u 2 * = 0

Session 3 Review Distributions Pen’s parade, quantile function, cdf Size, spread, poverty Data Income vector, cdf Today Inequality and economics Welfare economics Inequality measures

Inequality Measures Notation x is the income distribution x i is the income of the ith person n=n(x) is the population size. D is the set of all distributions of any population size Definition An 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 Definition We 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

Def We 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

Def We say that x is obtained from y by a proportional change (or scalar multiple) if x= α y, 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

Def We say that x is obtained from y by a (Pigou-Dalton) regressive transfer if for some i, j: i) y i < y j ii) y i – x i = x j – y j > 0 iii) x k = y k 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

Def Any measure satisfying the four basic properties (symmetry, replication invariance, scale invariance, and the transfer principle) is called a relative inequality measure.