**PROGRAM FOR DAGUM’S GINI INDEX
DECOMPOSITION **

** **

Camilo Dagum^{1}, Stéphane
Mussard^{2}, Françoise Seyte^{2}, Michel Terraza^{2}

with SOCREES’ participation

^{1}UNIVERSITY of OTTAWA

^{2} LAMETA, UNIVERSITY of MONTPELLIER I

**BIOGRAPHY**

** **

**Camilo Dagum**, Emeritus Professor of the University
of Ottawa and Doctor Honoris Causa of the University of Montpellier I,
introduced in 1997 the decomposition of the Gini index by
population subgroups. This method allows you to compute overall income
inequality and to break it down into within-group and between-group income
inequality.

**Michel Terraza**** **is a science Professor of economics at
University of Montpellier I. He applied this decomposed measure when studying
the wages inequalities in the Languedoc-Roussillon region (see the bibliography). He did it in collaboration
with **Françoise Seyte** (Associate Professor) and **Stéphane Mussard**** **(Assistant Professor).

**The SOCREES ^{©} **corporation, company of statistics and economic studies, made the Gini decomposition software (first version).

**THE SOFTWARES**

** **

** **

**The softwares **you can download are Excel
macro. To try it, you only need income or wage panel data. You can get some
here below. The programs include the decomposition of the Gini index and either
the measures derived from the general entropy inequality measures such as **Theil,
Hirschman-Herfindahl and Bourguignon** that are additively decomposable or
the weakly decomposable measures such as **the
coefficient of variation squared and the α-Gini**. Indeed, the latter can
be decomposed according to the same method as the Gini index. All the
information connected with the programs running is easy and available on sheet
1 of the Excel files "The softwares" and in the paragraph "THE
USE" here below.

**The data**, at your disposal (strictly
positive reals), consist of three columns. The first one concerns the codes of
the individuals that give a number to each person (from 1 to n.) The modality 1
in the second column represents the USA and the modality 2 Japan. The third
column deals with the growth rate of the American and Japanese real wages
between 1960 and 1996. This first example is done in order to get used to the
tool.

**DOWNLOAD / THE USE**

** **

__I. Arranging the Data__

Put your data in a new Excel
sheet and select them as in the following table:

Individuals’ Codes |
Groups |
Incomes |

1 |
1 |
100 |

2 |
1 |
90 |

3 |
1 |
250 |

… |
… |
… |

102 |
5 |
50 |

103 |
5 |
20 |

Column 1 represents the individuals codes that give an index to each person
(for instance from 1 to 103 for a sample of 103 individuals).

Column 2 deals with the different groups of the population. The individuals
belonging to the same subpopulation must be put together. In the table example
there are 5 groups.

Column 3 represents the wages of the individuals belonging to the groups of
column 2.

** **

__II. Code 1: Dagum’s Gini
Decomposition and the Additive Decompositions of three Generalized Entropy
Inequality Indices __

To obtain the results:

1) Download the software then open the file
"Dagum" in Excel ("yes" to activate the macro);

2) Download the data, open the file
"donnees" in Excel, type "Alt+F8" select
"Dagum.xls!CalculDagum" then "execute" and put the number
of groups "2" then "OK".

3) You should obtain the following results: Download the results. Follow the theoretical decomposition to make relevant
interpretations.

__III. Code 2: the ____(α,β)-Decomposition__

This program is a generalization of the previous
one. This new version focuses only on the pair-based inequality measures that
are weakly decomposable in Ebert’s (2010) sense. The (α,β)-decomposition
is an adaptation of Dagum’s (1997a, 1997b) decomposition to all the weakly
decomposable measures and permits to include a parameter of **inequality aversion** denoted by α as
well as a parameter of **sensibility
towards transvariation** β into the calculation of the various
components. A first attempt of generalization had been proposed by Chameni
(2011) and had been programmed by Fattouma Souissi and Pauline Mornet of the
University of Montpellier 1 (LAMETA PhD students).

This new program is inspired from theoretical researches:
Chameni (2006, 2011), Mussard and Terraza (2009) and Ebert (2010). Besides
allowing to capture inequality aversion, the (α,β)-decomposition
permits to decompose the Gini index (obtained when α=1, forall
β≥1), as well as the coefficient of variation squared (when α=2,
forall β≥1). So, this generalization brings out a link between the
Gini index and an entropic measure [see Chameni, 2006, 2011].

Following Ebert (2010), the main axioms PC
[resp. PD], PP, NM, SM are respected as far as α>0 [resp.
α≥1]. Furthermore for any α≥2 the principle of strong
diminishing transfers is also satisfied by such inequality measures [see Mornet,
P. Zoli, C. Mussard, S. Sadefo-Kamdem
J., Seyte, F. and M. Terraza, 2013].

To apply the decomposition:

1) Download the software then open the file "(alpha, beta)-decomposition
(en)" in Excel ("yes" to activate the macro);

2) Download the data, open the file "data" in Excel and
copy the information in “sheet2”. Type "Alt+F8" and select: “The_2_parameters_weak_decomposition"
then "Execute".

Put the number of groups,

and your sensitivity parameters (any positive real values),

then "OK".

3) You should obtain the following
results: Download the results. Follow the theoretical papers^{ } to make relevant interpretations.

**References:**

a) Ebert, U. (2010), “The
Decomposition of Inequality Reconsidered: Weakly Decomposable Measures”, *Mathematical Social Sciences* **6 **(2), 94-103.

b) Chameni Nembua C. (2011), “A
generalisation of the Gini coefficient: Measuring economic inequality”, *Mimeo*.

c) Mussard, S. et Terraza M. (2009), “ La
décomposition du coefficient de Gini et des mesures dérivées de l'entropie :
les enseignements d'une comparaison”, *Recherches
Economiques de Louvain* **75 **(2),
151-181.

d) Chameni Nembua C. (2006),”Linking Gini to Entropy: Measuring Inequality
by an interpersonal class of indices”, *Economics
Bulletin* **4 **(5), 1-9.

**OTHER RELATED
SOURCES**

* You can obtain the same results more quickly than an Excel macro using a
Gauss program that can contain more than 64,000 observations:

By Michele Costa, University of Bologna: Gini.g

* With SAS : used in Koubi, M. Mussard, S., F.
Seyte et M. Terraza (2005):

By Malik Koubi,
INSEE: Gini-SAS

* If your incomes are composed of several income sources (wages + income taxes
+ transfers + etc.), you may try the GAUSS Gini multi-decomposition (a non
generalized program), used in Mussard (2006):

By Stéphane Mussard: g-revenu.g

* If your incomes are composed of several income sources (wages + income
taxes + transfers + etc.) and several partitions of groups, use the GAUSS Gini
multi-decomposition in multi-levels, used in __Mussard__ S., Pi-Alprein M.-N., Seyte F.
and Terraza M. (2006):

By Stéphane Mussard: g-revenu-2.g

**Contacts**

**LICENCE**

** **

THESE PRODUCTS ARE PROVIDED FOR FREE ON AN 'AS IS' BASIS, WITHOUT ANY
WARRANTIES OR CONDITIONS. NEITHER STEPHANE MUSSARD, FRANCOISE SEYTE, MICHEL
TERRAZA, PAULINE MORNET AND SOCREES, NOR THE LAMETA AND THE UNIVERSTY OF
MONTPELLIER 1 SHALL HAVE ANY LIABILITY FOR ANY INDIRECT, INCIDENTAL, SPECIAL,
OR CONSEQUENTIAL DAMAGES WHATSOEVER, INCLUDING, BUT NOT LIMITED TO, LOSS OF
REVENUE OR PROFIT, LOST OR DAMAGED DATA OR OTHER COMMERCIAL OR ECONOMIC LOSS.