Course: Economic Analysis of Inequality (with Capita Selecta)
Study guide number: 2300PSWEAI
Study: Sociology, SEW, elective course other directions (master)
Prof.: Koen Decancq
The exam is an open book exam, so you can take this summary to your exam. It contains all powerpoint slides (written out...
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Lecture 1: From principles to policy (27/9) ............................................................................................... 4
1.0. Practicalities...................................................................................................................................... 4
The Exam ............................................................................................................................................. 4
1.1. Setting the scene............................................................................................................................... 4
1.1.1 Hot topic ..................................................................................................................................... 4
1.1.2 Roadmap ..................................................................................................................................... 4
1.2. Normative reasons (because of our value judgments about inequality) ............................................. 8
Lecture 2: Charting Income Distributions (2/10) ..................................................................................... 10
2.1. Introduction .................................................................................................................................... 10
2.1.1 Concepts ................................................................................................................................... 10
2.2. 4 types of functions ......................................................................................................................... 11
2.2.1 Probability Density Function ...................................................................................................... 12
2.2.2 Cumulative Distribution Function (CDF) ..................................................................................... 16
2.2.3 Quantile Function ...................................................................................................................... 18
2.2.4 Lorenz Curve ............................................................................................................................. 19
Lecture 3: Partial Social Evaluations ....................................................................................................... 22
3.1. Social Welfare Functions ................................................................................................................. 22
3.1.1 Continuity (CONT) ..................................................................................................................... 22
3.1.2 Separability (SEP) ...................................................................................................................... 23
3.1.3 Population Replication (POP) ..................................................................................................... 24
3.1.4 Anonymity (ANOM) ................................................................................................................... 24
3.1.5 Monotonicity (MON) ................................................................................................................. 25
3.1.6 A class of Social Welfare Functions ............................................................................................ 25
3.2. Stochastic Dominance ..................................................................................................................... 26
3.2.1 First order stochastic dominance ............................................................................................... 26
3.2.2 Second order stochastic dominance .......................................................................................... 28
3.2.3 Third order stochastic dominance ............................................................................................. 32
3.2.4 And beyond… ............................................................................................................................ 34
Lecture 4: Complete Social Evaluations .................................................................................................. 35
4.0. Recall .............................................................................................................................................. 35
4.1. Iso-elastic social welfare functions .................................................................................................. 35
4.2. Equally Distributed Equivalent......................................................................................................... 41
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4.2.1 So far we obtained .................................................................................................................... 41
4.2.2 Equally Distributed Equivalent ................................................................................................... 41
4.3 Reading for this lecture .................................................................................................................... 43
Lecture 5: Partial inequality comparisons (18/10) .................................................................................. 44
5.1 The generalized Lorenz curve ........................................................................................................... 44
5.1.1 Generalized Lorenz Curve .......................................................................................................... 44
5.1.2 Generalized Lorenz dominance.................................................................................................. 45
5.2 The Lorenz curve .............................................................................................................................. 47
5.2.1 Lorenz dominance (equal means) .............................................................................................. 47
5.2.2 Lorenz dominance and invariance ............................................................................................. 50
Lecture 6: Complete inequality comparisons (23/10) ............................................................................. 56
6.0 Introduction ..................................................................................................................................... 56
6.1 The Atkinson Family ......................................................................................................................... 56
Remember ......................................................................................................................................... 56
Founding Family................................................................................................................................. 57
6.2 The Gini family ................................................................................................................................. 60
6.3 The Generalized Entropy family (NOT ON EXAM).............................................................................. 70
Subgroup decomposability ................................................................................................................. 71
In sum.................................................................................................................................................... 72
Lecture 7: Measuring Poverty (25/10) .................................................................................................... 73
7.0 Introduction ..................................................................................................................................... 73
7.0.1 Measuring poverty .................................................................................................................... 73
7.1 Poverty line ...................................................................................................................................... 73
7.1.1 Absolute poverty line ................................................................................................................ 73
7.1.2. Relative poverty line ................................................................................................................. 73
7.1.3. Weakly relative poverty line ..................................................................................................... 74
7.2 Poverty measures (complete poverty comparisons) ......................................................................... 75
7.2.1 Focus ......................................................................................................................................... 75
7.2.2 Monotonicity............................................................................................................................. 78
7.2.3 Transfer..................................................................................................................................... 79
7.3 Poverty dominance (partial poverty comparisons) ........................................................................... 82
7.3.1 Fixed measure, all poverty lines ................................................................................................. 82
7.3.2 All measures, fixed poverty line (NOT EXAMEN) ........................................................................ 83
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Lecture 8: Quantifying Redistribution (13/11) ........................................................................................ 84
8.0 Literature ......................................................................................................................................... 84
8.1 Income taxation ............................................................................................................................... 84
8.2 Impact of (income) taxation on inequality ........................................................................................ 90
Lecture 9: Optimal redistribution (15/11) ............................................................................................... 98
9.1 How much should we redistribute? .................................................................................................. 98
9.1.1 When Pigou-Dalton transfers are possible ................................................................................. 98
9.1.2 When Pigou-Dalton transfers are NOT possible ......................................................................... 99
9.2 The social planner wants to choose the level of redistribution so that a social welfare function 𝑊(. ) is
maximized given the constraints she faces. .......................................................................................... 102
X.Some conclusions/Interesting points................................................................................................. 103
X.0 Structure ........................................................................................................................................ 103
X.1. Lecture 1-3 .................................................................................................................................... 103
X.1.1 First order stochastic dominance ............................................................................................. 103
X.1.2 Second order stochastic dominance ........................................................................................ 104
X.1.3 Two person Graph ................................................................................................................... 105
X.2 Lecture 4-5 ..................................................................................................................................... 106
X.2.1 Lorenz dominance and invariance ........................................................................................... 106
X.2.2 Gini family ............................................................................................................................... 106
X.3 Lecture 6-7 ..................................................................................................................................... 107
X.3.1 Poverty deficit ......................................................................................................................... 107
X.4 Lecture 8 ........................................................................................................................................ 107
X.4.1 Impact of taxation on inequality .............................................................................................. 107
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Lecture 1: From principles to policy (27/9)
Core: different people may have different perspectives to look at welfare, wellbeing…
1.0. Practicalities
The Exam
Study material
o Articles on Blackboard (read it in a clever way)
o Slides on Blackboard
o Personal notes taken at lectures (most important)
Graphs you have to make it by yourself
o No reproduction, but understanding of material!
Open-book, written exam about material covered in core lectures
Take-home exam about material covered in additional lectures
o When you come to the exam of the written exam you give it back
1.1. Setting the scene
1.1.1 Hot topic
Not only more interest in the street, media, policymakers, but also in the academic world
o Ex. Pickety, Stiglitz
This is an economic course
1.1.2 Roadmap
1. What is inequality?
3 Concepts:
(Social) welfare: Evaluation of (income) distribution according to its size and inequality (see lecture
3-4): what is the better (most desirable) distribution, the core
o Grootte van de pizza + verdeling ervan
Inequality: Evaluation of (income) distribution according to its inequality (see lecture 5-6)
o Verdeling van de pizza (geen nadruk op de grootte)
Poverty: Evaluation of lower part of (income) distribution (see lecture 7)
Different concepts! (think of China nowadays, inequality raises and poverty decreases)
Creating conceptual clarity is one of the main aims of the course
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Survey exercise
More ethically intuitions/normative
How we can measure inequality knowing that there are different intuitions
We try to describe the normative side of economy
8
7
currency of "Hypothesia"
6
5
4 Programme X
Programme Y
3
2
1
0
Ann Bob Charlotte Danny
The pizza is bigger in Y, but more equal in X
6
5
currency of "Hypothesia"
4
3 Programme X
Programme Y
2
1
0
Ann Bob Charlotte Danny
X is equal
Y is not equal
If you take from Ann (X) to Danny (X) -> you reach the Y distribution (= transfer that is called Pigou-
Dalton: take from the poor give to the rich: inequality goes up)
Y is more unequal than X
, 6
6
currency of "Hypothesia"
5
4
3 programme X
Programme Y
2
1
0
Ann Bob Charlotte Danny
Now we compare pizza’s from another size
Reaching Y is doubling all people’s income from X
o Intuition 1: inequality is the same
o Intuition 2: inequality is increased
o Some think relative, some think absolute
2. Why should we care about inequality?
2.1 Instrumental reasons (because of the consequences of inequality on other outcomes)
Recent epidemiological insights (Wilkinson & Pickett, 2009)
o X-axe is always income inequality
o Y-axe is a bad thing (health problem, crime…)
o Positive relation, not at all a causal relation (maximum a correlation)
o http://www.youtube.com/watch?v=cZ7LzE3u7Bw (! Part of the study material)
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Economists are predominantly interested in the effect of inequality on economic output (GDP) or
its growth (not interested in crime, health…)
Founding father: Simon Kuznets
o Kuznets curve: horizontal: low to high GDP/output – gini first raises then decreases
o Unfortunately life is more complicated
Recent revival: Piketty (Capital), Milanovic (Global Inequality)
Establishing the (causal) effect of inequality on economic growth is a VERY DIFFICULT question
Because inequality is not an exogenous variable (given)
(it is not assigned randomly to countries)
Causal empirical statements are hard to make (see your econometrics course)
Also theoretically it is a hard question
Monopoly effect: the rich may gain a lot by protecting their acquired positions (rent seeking)
o The rich try to protect their rent, organize the society that the poor not becoming rich
o USA: education system: the rich don’t want the poor becoming rich
o USA: no progressive taxation system
Incentive effect: Inequality creates incentives for investment and entrepreneurs (a little bit of
inequality is good).
o Wealth generated by the rich trickles down to lower parts of distribution
Which of both effects, the incentive or the monopoly effect, is the biggest
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1.2. Normative reasons (because of our value judgments about inequality)
Inequality is unfair
When is a social situation fair? What is fair?
o We think inequality isn’t fair
o The central question in the work of John Rawls
Impartiality as crucial requirement for fairness: Rawls proposes to think about the society
behind a veil of ignorance
o Thought experience: jump out of a plane and fall in the shoes of a rich/poor person
healthy/sick person – the situation in the plane: sitting there, how we are going to organize
the society (fair society behind the veil of ignorance)
How would/should individuals behave behind such a veil of ignorance?
Answer of Rawls: use the difference principle = you look at the guy who is the worst off (ex. Plane-
shoe), he has to be best off
The difference principle: evaluate a social situation by looking at the worst-off individual
An example: in which society you wil land (plane-shoe exemple)
The worst thing that can happen in A (ind 1), in situation B (ind 3)
For Rawls it doesn’t matter only the person who is worst off
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Harsanyi
How would/should individuals behave behind such a veil of ignorance?
Answer of Rawls: use the difference principle
Answer of Harsanyi: use expected utility (as if it were a lottery)
Open philosophical (ethical) debate about what a fair income distribution is.
Yet, most people will agree about the ingredients of the problem:
o The size of economic output (the size of the cake) – mister average
o The inequality distribution of economic output (the distribution of the cake) – mister
Rawls
o How to trade off to the both??
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