1. 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

2. 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

3. Pen’s Parade
Source: Jan Pen, 1971

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

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

6. Cumulative Distribution Function
F(s)
Cumulative population
Income s

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

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

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

10. “Size” of the distribution1
F(s)
Cumulative population .5
Income s

11. “Size” of the distribution: Median1
F(s)
Cumulative population .5 m
Income s

12. “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

13. “Size” of the distribution: mean1
F(s)
Cumulative population .5 μ
Income s

14. “Size” of the distribution: mean1
F(s) B
Cumulative population .5 A μ
Income s

15. “Size” of the distribution: mean1
F(s) B
Cumulative population .5 A μ
Income s
Note A = B

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

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

18. “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

19. “Spread” of the distribution
Idea How far apart?

20. “Spread” of the distribution1
F(s)
Cumulative population .5 μ
Income s

21. “Spread” of the distribution1
F(s)
Cumulative population .5 s1 μ s2
Income s

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

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

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

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

26. “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

27. 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

28. 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)

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

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

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

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

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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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?

41. 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?

42. 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!

43. 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

44. 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

45. 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.