Session 2 Review Today Elements of the course (info cards) Global inequality High levels, no trend Global poverty (see update) High levels, downward trend Today Distributions Data Inequality and economics

Distributions Q/ How to depict distribution? A/ Pen’s Parade, cumulative distribution function, vector Jan Pen Dutch economist, Groningen Imagine a parade Persons walk by in an hour Least affluent first Heights proportional to income (wealth, consumption) Avg. height corresponding to avg. income Let the parade begin! Creates picture with words Graph

Pen’s Parade Source: Jan Pen, 1971

Pen’s Parade for Brazil: 1990 and 1995 Source: Ferreira and Litchfield, 1999, "Inequality, Poverty and Welfare, Brazil 1981-1995". London School of Economics Mimeograph

Pen’s Parade or Quantile Function Q(p) Income Cumulative population p

Cumulative Distribution Function F(s) Cumulative population Income s

Cumulative Distribution Function F(s) Cumulative population Income s Note Q is the inverse of F

Cumulative Distribution Function F(s) Cumulative population Income s Q/ What is the density function?

“Size” of the distribution Idea How far to the right? Median – The income at 0.5 m = Q(0.5) or the “middle” income

“Size” of the distribution 1 F(s) Cumulative population .5 Income s

“Size” of the distribution: Median 1 F(s) Cumulative population .5 m Income s

“Size” of the distribution Idea How far to the right? Median – The income at 0.5 m = Q(0.5) or the “middle” income Mean – The average income μ = sum of all incomes divided by number of people

“Size” of the distribution: mean 1 F(s) Cumulative population .5 μ Income s

“Size” of the distribution: mean 1 F(s) B Cumulative population .5 A μ Income s

“Size” of the distribution: mean 1 F(s) B Cumulative population .5 A μ Income s Note A = B

μ “Size” of the distribution: mean 1 F(s) Cumulative population .5 μ Income s mean = area to the left of F

“Size” of the distribution Q(p) Income Mean = area below Q Cumulative population p

“Size” of the distribution Idea How far to the right? Median – The income at 0.5 m = Q(0.5) or the “middle” income Mean – The average income μ = integral of Q(p) Note – There are many other possibilities Aggregation

“Spread” of the distribution Idea How far apart?

“Spread” of the distribution 1 F(s) Cumulative population .5 μ Income s

“Spread” of the distribution 1 F(s) Cumulative population .5 s1 μ s2 Income s

“Spread” of the distribution Idea How far apart? Depends on si’s relative to mean Aggregation There are many inequality measures

“Poverty” of the distribution Focus Incomes below a poverty line z

Cumulative Distribution Function F(s) Cumulative population z Income s

Cumulative Distribution Function F(s) Cumulative population H z Income s

“Poverty” of the distribution Focus Incomes below a poverty line z H - percentage of population below z How low are poor incomes? How unequal are poor incomes? Many ways to measure - aggregation

Session 2 Review Today Elements of the course (info cards) Global inequality High levels, no trend Global poverty (see update) High levels, downward trend Today Distributions Data Inequality and economics

Data Note Actual data are not smooth functions, but lists Identifier, income, age, and many other dimensions Focus here on first two Convenient representation: vector x = (x1, …, xn) xi = ith person’s income.. Ex x = (2, 8, 4, 1) Q/ Is there a cdf associated with x? A/ Yes, call it Fx(s)

Ex x = (2, 8, 4, 1) Fx(s) 1 Cumulative population .5 2 4 6 8 Income s

Ex x = (2, 8, 4, 1) Fx(s) 1 Cumulative population .5 2 4 6 8 Income s

Ex x = (2, 8, 4, 1) What is μ? Fx(s) 1 Cumulative population .5 2 4 6 8 Income s

Ex x = (2, 8, 4, 1) What is μ? Fx(s) 1 Cumulative population .5 2 4 6 8 μ =3.75 Income s

Suppose x = (1, 2, 4, 8) What is Fx? Fx(s) 1 Cumulative population .5 2 4 6 8 μ Income s

Suppose x = (1, 1, 2, 2, 4, 4, 8, 8) What is Fx? Fx(s) 1 Cumulative population .5 2 4 6 8 μ Income s

Ex x = (2, 8, 4, 1) Can view spread and poverty also Fx(s) 1 Cumulative population .5 2 4 6 8 μ =3.75 Income s

Ex x = (2, 8, 4, 1) Can view spread and poverty also Fx(s) 1 Cumulative population .5 2 4 6 8 μ =3.75 Income s

Ex x = (2, 8, 4, 1) Can view spread and poverty also Fx(s) 1 Cumulative population .5 2 4 6 8 z μ =3.75 Income s

Wait – Why should we care? Is size really important? Explain Is poverty? Explain Is spread? Explain Is inequality “bad”? What kind of inequality? How much? Controversial and contended Cf: Feldstein, "Reducing poverty, not inequality”, 1998 We begin with inequality

Session 2 Review Today Elements of the course (info cards) Global inequality High levels, no trend Global poverty (see update) High levels, downward trend Today Distributions Data Inequality and economics

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

Issues Can economics help? 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) Fundamental Welfare Theorems Note It’s a quasiordering Fundamental Welfare Theorems A Walrasian equilibrium is Pareto efficient 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 Problem 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({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 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 maximizes W = [u1(x1) +…+ un(xn)]/n where say each ui 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 ui are identical and strictly concave. Then W is maximized at equality.