1. Comparing the size of the middle class using the alienation component of polarization
by André-Marie Taptué
Université Laval
Conchita D’Ambrosio
Université du Luxembourg

2. The paper is very interesting, the topic is very relevant, the empirical findings offer a picture of the middle-classes in several countries in 2004.

3. AIM as presented by André-Marie:
The paper proposes a modification of the DER/ER index of polarization to compare the size of the middle class.
This index accounts only for the alienation component and not for the identification component.
Nonetheless this index is not a measures of inequality.
Statistical inference and test dominance between pairs of distributions are implemented.
The index is applied to 22 LIS countries.

4. AIM as I would have presented it:
The paper proposes a modification of the Wang and Tsui index of polarization by introducing a threshold under which distances from the median income are not considered.
Statistical inference and test dominance between pairs of distributions are implemented.
The index is applied to 22 LIS countries.

5. What I say is somewhere in the text of André-Marie but it is hidden in the lines written on DER/ER for a passion for DER/ER.
The abandoning of DER/ER would make the theoretical contribution neater, would simplify the analysis and would make it easier for the non-expert reader to appreciate its value.

6. Polarization
There are two distinct approaches to its measurement: DER/ER and FW. One cannot be obtained from the other.
DER/ER contributions are very nicely written, ER (1994) has a wonderful introduction and contains the original idea.
W (1994) (later FW) is more dry, with technical steps. The axiomatization and generalization of W is due to WT (2000).

7. Polarization: the Esteban and Ray Approach
Each individual feels alienated from others located at different points of the income scale:
if there is more than one individual with the same income level:

8. Polarization: the ER Approach
Each individual identifies with people having the same income, identification/alienation gives rise to effective alienation:
Polarization, in the whole society, is the sum of these sentiments of effective alienation:

9. The Esteban-Ray (absolute) measure The Duclos-Esteban-Ray measure
Polarization: the ER Approach
Each individual identifies with people having the same income, identification/alienation gives rise to effective alienation:
Polarization, in the whole society, is the sum of these sentiments of effective alienation:
The Esteban-Ray (absolute) measure
The Duclos-Esteban-Ray measure

10. Class of indices by Wang and Tsui (JPET, 2000)
Polarization: the Wolfson’s approach
Class of indices by Wang and Tsui (JPET, 2000)

11. Polarization: the Wolfson’s approach