- Email: [email protected]
Regional differences and the components of growth and inequality change
Ekonomics Letters North-Holland
295
25 (1987) 295-298
REGIONAL DIFFERENCES AND THE COMPONENTS AND INEQUALITY CHANGE Z.M. BERREBI
and Jacques
OF GROWTH
SILBER
Bar Iian University, 52100 Ramat Gan, Israel Received
4 September
1987
Inequality change can be decomposed simply in terms of per capita income and population changes, using the Gini index. An illustration based on U.S. data for the two intercensus periods 1960-1970 and 1970-1980 is given. This decomposition analysis is also applied to the study of changes in a welfare index originally proposed by Sen.
1. Introduction One way to measure income inequality is to use the Theil index which is derived from the concept of entropy. This index is particularly useful when one tries to break down the inequality into various components. Theil and Sorooshian (1979) for example, have shown how the change in U.S. regional inequality could be decomposed in terms of per capita income changes and population changes. The purpose of the present note is to indicate that a simple decomposition may also be obtained when using the Gini index of inequality, since the latter may be expressed in a convenient matrix form. The use of the Gini index could be preferred for two reasons: firstly, it is related to a convenient graphical representation, the Lorenz curve; secondly, it can be easily translated in terms of social welfare as was suggested by Sen (1974). An illustration is given based on the two U.S. intercensus periods, 1960-1970 and 1970-1980.
2. Regional inequality and the Gini index In a recent paper the authors of the present note have shown [Berrebi Gini index of inequality IG could be written as IGC
ts i=l
(n-i> I[~_
and Silber (1985)] that the
(i-1)
n
n
1’
where s, is the proportion of the total income earned by the individual which has the i th rank in the income distribution, assuming that n is the number of individuals and that s, rs,...
r.si2
Expression
. . . 2s,. (1) may also be written
(2) [Silber (1986)] as
Zo = e’[ D - D’]s,
0165-1765/87/%3.50
(3) 0 1987, Elsevier Science Publishers
B.V. (North-Holland)
Z.M. Berrebi, J. Silber / Regional differences, components
296
of growth, inequality change
where e is a column vector of n elements which are all equal to l/n (e’ being the corresponding line vector), s is a column vector of n elements being respectively equal to si, s2,. . . , s, and D is an n by n lower triangular matrix defined as 1 1
0 1
... ...
0 0
0 0 (4)
1
1
...
1
1
whereas D’ is the transpose of D. If one defines a new matrix G as G=D-D’,
(5)
so that its elements
g,j are equal to 0 when i = j, to - 1 if j > i and to + 1 if j < i, one derives that
I, = e’Gs.
(6)
Expression (6) can be applied to the study of regional inequalities in the distribution of incomes the U.S.A. If the within-region inequality is ignored, it can be seen that (6) will be written as I,=f’Gs,
in
(7)
f, }, f, being the share of state i in the total population of the U.S. corresponding where f= {fi,..., to the share s, of state i in the total U.S. income and f’ being the transpose of f.
3. Population and per capita income components
of growth and inequality
change
Let Y,,, be the per capita income of state i at time t and f,,, the population share of state i at time t. The growth rate g in the U.S. per capita income between periods t and t + 1 may be written as
1+ g =
5fl,t+lYl,r+l I?f;,tY;,,.
i=l
(8)
i=l
One may define in a similar way growth rates which are computed either by assuming that no change occurred in the size of each state’s population (gr) or that the per capita income of each state did not vary (gv). In the first case, the growth in U.S. per capita income is to be attributed solely to variations in the per capita income of each state; in the latter case, to changes in the size of the population of each state. Using (7) one may now compute an index AG of inequality change which will be defined as
AG = (.h:1%+11) - (.I%,).
(9)
Here also it is possible to define an index of inequality change AG, which would be computed by assuming that no change occurred in the size of each state’s population (e.g., if t is the base year, one vector would assume that f,: 1 = f, and that st+ 1 would be computed on the basis of the population f,) or an index AG, computed by assuming no variation in the per capita income of each state (Y,,,+1 = y,,,Vi).
Z.M. Berrebi, J. Silber / Regional differences, components
4. Population and per capita income components
297
of growth, inequality change
of change in welfare
It has been often assumed that an index of the welfare of a society should depend not only on the level but also on the distribution of income. Sen (1974) for example, has proven that a social welfare function satisfying four axioms which he called weighting equity, limit equality, ordinal information and independent monotonicity would rank the set of distributions of a given total income Y in the same way as the negative of the Gini coefficient of the respective distributions. It is, therefore, possible to define a welfare index W, at time t as
where J, is the average per capita income at time t and Io., is the Gini inequality The growth rate g, in the welfare index W, will then be defined as
1
+
gw =
Yr+t(l-
index
~ci,,+,>
x(1 -&i.,>
at time t.
(11)
.
Here also it is possible to compute capita income and not the population reverse being true when one computes
a welfare growth rate g,, which would assume that the per size of each state varied between periods t and t + 1, the a welfare growth rate gwu.
5. Application: The U.S.A. during the periods 1960-1970
and 1970-1980
Table 1 gives the U.S. population and U.S. personal income per capita in constant (1972) dollars for the years 1960, 1970 and 1980. It presents also estimates of the inter-states Gini inequality index and of the welfare index proposed by Sen (1974) for each of these census years. In table 2 two sets of results are presented. Firstly, the inequality change indices AG, AC, and AG, are computed for both periods 1960-1970 and 1970-1980. Secondly, growth rates of both the U.S. per capita income and the welfare index proposed by Sen are given for each of the periods. The average intercensus and per capita income data have been used respectively as weights in the ‘constant population’ and ‘constant per capita income’ computations. The results indicate clearly that whereas during the first intercensus period, variations in the population size and in the per capita income of each state had a similar impact on the U.S. per capita income growth rate and on the inter-state inequality change, during the second period (1970-1980) changes in the population size of the states would have led to a negative U.S. per capita income
Table 1 U.S. population
and income
data at various
censuses.
U.S. personal
income per capita in constant (1972) dollars U.S. resident population ( x 1000) a Gini index of inequality (inter-state inequality) U.S. welfare index a Sources:
Statistical
Abstract
of the USA, 1981 and 1985.
a
1960
1970
1980
3,070 179,323 0.107
4,265 203,303 0.082 3,917
5,322 226,546 0.065 4,974
2,142
Z.M. Berrebi, J. Silber / Regional differences, components of growth, inequality change
298 Table 2 Inter-census
growth
rates and inequality
changes. 1960-1970
1970-1980
- 0.025 - 0.012 - 0.013
- 0.016 0.023 - 0.040
Growth rates In U.S. per capita income (g) at ‘constant state population’ (gn) at ‘constant state per capita income’ (gu)
0.389 0.384 0.004
0.248 0.254 - 0.005
In U.S. werfare index ( gw) at ‘constant state population’ (gwp) at ‘constant state per capita income’ (g,,)
0.428 0.404 0.017
0.270 0.308 - 0.028
Total inequality change ( AG) Inequality change due to population change (AC,) Inequality change due to per capita income change (AC,)
growth rate and to an increase in inter-state inequality if there had not been a compensating effect of the variation in the per capita income of each state. Table 2 indicates also that the growth rate in Sen’s welfare index was higher during both intercensus periods than that of the U.S. per capita income. Such a result was expected given that during both periods inter-state inequality decreased as U.S. per capita income increased. Similarly, one may notice that when the per capita income of the states is kept constant, the resulting welfare growth rate G,, during the second intercensus period is even more negative than that of the U.S. per capita income since inter-state inequality would have increased at a time when the U.S. per capita income would decrease. These results as a whole are quite similar to those based on the Theil index which were presented by Theil and Sorooshian (1979) for the 1970-1977 period. The contribution of the present study is that it shows that a simple decomposition of inequality change into income and population components may also be obtained when using the Gini index. The use of the latter index may sometimes be preferred because it is related to a convenient graphical representation, the Lorenz curve, or because it can be translated in terms of social welfare comparisons as suggested by Sen (1974).
References Berrebi, Z.M. and J. Silber, 1985, Income inequality indices and deprivation: A generalization, Quarterly Journal of Economics 99, 807-810. Sen, A., 1974, Informational bases of alternative welfare approaches, Journal of Public Economics 3, 387-403. Silber, J., 1986, Factor components, income classes and the computation of the Gini index of inequality, MRG working paper no. 8641 (University of Southern California, Los Angeles, CA). Statistical Abstract of the USA, 1981 and 1985 (U.S. Bureau of the Census, Washington, DC). Theil, H. and C. Sorooshian, 1979, Components of the change in regional inequality, Economics Letters 4, 191-193.
295
25 (1987) 295-298
REGIONAL DIFFERENCES AND THE COMPONENTS AND INEQUALITY CHANGE Z.M. BERREBI
and Jacques
OF GROWTH
SILBER
Bar Iian University, 52100 Ramat Gan, Israel Received
4 September
1987
Inequality change can be decomposed simply in terms of per capita income and population changes, using the Gini index. An illustration based on U.S. data for the two intercensus periods 1960-1970 and 1970-1980 is given. This decomposition analysis is also applied to the study of changes in a welfare index originally proposed by Sen.
1. Introduction One way to measure income inequality is to use the Theil index which is derived from the concept of entropy. This index is particularly useful when one tries to break down the inequality into various components. Theil and Sorooshian (1979) for example, have shown how the change in U.S. regional inequality could be decomposed in terms of per capita income changes and population changes. The purpose of the present note is to indicate that a simple decomposition may also be obtained when using the Gini index of inequality, since the latter may be expressed in a convenient matrix form. The use of the Gini index could be preferred for two reasons: firstly, it is related to a convenient graphical representation, the Lorenz curve; secondly, it can be easily translated in terms of social welfare as was suggested by Sen (1974). An illustration is given based on the two U.S. intercensus periods, 1960-1970 and 1970-1980.
2. Regional inequality and the Gini index In a recent paper the authors of the present note have shown [Berrebi Gini index of inequality IG could be written as IGC
ts i=l
(n-i> I[~_
and Silber (1985)] that the
(i-1)
n
n
1’
where s, is the proportion of the total income earned by the individual which has the i th rank in the income distribution, assuming that n is the number of individuals and that s, rs,...
r.si2
Expression
. . . 2s,. (1) may also be written
(2) [Silber (1986)] as
Zo = e’[ D - D’]s,
0165-1765/87/%3.50
(3) 0 1987, Elsevier Science Publishers
B.V. (North-Holland)
Z.M. Berrebi, J. Silber / Regional differences, components
296
of growth, inequality change
where e is a column vector of n elements which are all equal to l/n (e’ being the corresponding line vector), s is a column vector of n elements being respectively equal to si, s2,. . . , s, and D is an n by n lower triangular matrix defined as 1 1
0 1
... ...
0 0
0 0 (4)
1
1
...
1
1
whereas D’ is the transpose of D. If one defines a new matrix G as G=D-D’,
(5)
so that its elements
g,j are equal to 0 when i = j, to - 1 if j > i and to + 1 if j < i, one derives that
I, = e’Gs.
(6)
Expression (6) can be applied to the study of regional inequalities in the distribution of incomes the U.S.A. If the within-region inequality is ignored, it can be seen that (6) will be written as I,=f’Gs,
in
(7)
f, }, f, being the share of state i in the total population of the U.S. corresponding where f= {fi,..., to the share s, of state i in the total U.S. income and f’ being the transpose of f.
3. Population and per capita income components
of growth and inequality
change
Let Y,,, be the per capita income of state i at time t and f,,, the population share of state i at time t. The growth rate g in the U.S. per capita income between periods t and t + 1 may be written as
1+ g =
5fl,t+lYl,r+l I?f;,tY;,,.
i=l
(8)
i=l
One may define in a similar way growth rates which are computed either by assuming that no change occurred in the size of each state’s population (gr) or that the per capita income of each state did not vary (gv). In the first case, the growth in U.S. per capita income is to be attributed solely to variations in the per capita income of each state; in the latter case, to changes in the size of the population of each state. Using (7) one may now compute an index AG of inequality change which will be defined as
AG = (.h:1%+11) - (.I%,).
(9)
Here also it is possible to define an index of inequality change AG, which would be computed by assuming that no change occurred in the size of each state’s population (e.g., if t is the base year, one vector would assume that f,: 1 = f, and that st+ 1 would be computed on the basis of the population f,) or an index AG, computed by assuming no variation in the per capita income of each state (Y,,,+1 = y,,,Vi).
Z.M. Berrebi, J. Silber / Regional differences, components
4. Population and per capita income components
297
of growth, inequality change
of change in welfare
It has been often assumed that an index of the welfare of a society should depend not only on the level but also on the distribution of income. Sen (1974) for example, has proven that a social welfare function satisfying four axioms which he called weighting equity, limit equality, ordinal information and independent monotonicity would rank the set of distributions of a given total income Y in the same way as the negative of the Gini coefficient of the respective distributions. It is, therefore, possible to define a welfare index W, at time t as
where J, is the average per capita income at time t and Io., is the Gini inequality The growth rate g, in the welfare index W, will then be defined as
1
+
gw =
Yr+t(l-
index
~ci,,+,>
x(1 -&i.,>
at time t.
(11)
.
Here also it is possible to compute capita income and not the population reverse being true when one computes
a welfare growth rate g,, which would assume that the per size of each state varied between periods t and t + 1, the a welfare growth rate gwu.
5. Application: The U.S.A. during the periods 1960-1970
and 1970-1980
Table 1 gives the U.S. population and U.S. personal income per capita in constant (1972) dollars for the years 1960, 1970 and 1980. It presents also estimates of the inter-states Gini inequality index and of the welfare index proposed by Sen (1974) for each of these census years. In table 2 two sets of results are presented. Firstly, the inequality change indices AG, AC, and AG, are computed for both periods 1960-1970 and 1970-1980. Secondly, growth rates of both the U.S. per capita income and the welfare index proposed by Sen are given for each of the periods. The average intercensus and per capita income data have been used respectively as weights in the ‘constant population’ and ‘constant per capita income’ computations. The results indicate clearly that whereas during the first intercensus period, variations in the population size and in the per capita income of each state had a similar impact on the U.S. per capita income growth rate and on the inter-state inequality change, during the second period (1970-1980) changes in the population size of the states would have led to a negative U.S. per capita income
Table 1 U.S. population
and income
data at various
censuses.
U.S. personal
income per capita in constant (1972) dollars U.S. resident population ( x 1000) a Gini index of inequality (inter-state inequality) U.S. welfare index a Sources:
Statistical
Abstract
of the USA, 1981 and 1985.
a
1960
1970
1980
3,070 179,323 0.107
4,265 203,303 0.082 3,917
5,322 226,546 0.065 4,974
2,142
Z.M. Berrebi, J. Silber / Regional differences, components of growth, inequality change
298 Table 2 Inter-census
growth
rates and inequality
changes. 1960-1970
1970-1980
- 0.025 - 0.012 - 0.013
- 0.016 0.023 - 0.040
Growth rates In U.S. per capita income (g) at ‘constant state population’ (gn) at ‘constant state per capita income’ (gu)
0.389 0.384 0.004
0.248 0.254 - 0.005
In U.S. werfare index ( gw) at ‘constant state population’ (gwp) at ‘constant state per capita income’ (g,,)
0.428 0.404 0.017
0.270 0.308 - 0.028
Total inequality change ( AG) Inequality change due to population change (AC,) Inequality change due to per capita income change (AC,)
growth rate and to an increase in inter-state inequality if there had not been a compensating effect of the variation in the per capita income of each state. Table 2 indicates also that the growth rate in Sen’s welfare index was higher during both intercensus periods than that of the U.S. per capita income. Such a result was expected given that during both periods inter-state inequality decreased as U.S. per capita income increased. Similarly, one may notice that when the per capita income of the states is kept constant, the resulting welfare growth rate G,, during the second intercensus period is even more negative than that of the U.S. per capita income since inter-state inequality would have increased at a time when the U.S. per capita income would decrease. These results as a whole are quite similar to those based on the Theil index which were presented by Theil and Sorooshian (1979) for the 1970-1977 period. The contribution of the present study is that it shows that a simple decomposition of inequality change into income and population components may also be obtained when using the Gini index. The use of the latter index may sometimes be preferred because it is related to a convenient graphical representation, the Lorenz curve, or because it can be translated in terms of social welfare comparisons as suggested by Sen (1974).
References Berrebi, Z.M. and J. Silber, 1985, Income inequality indices and deprivation: A generalization, Quarterly Journal of Economics 99, 807-810. Sen, A., 1974, Informational bases of alternative welfare approaches, Journal of Public Economics 3, 387-403. Silber, J., 1986, Factor components, income classes and the computation of the Gini index of inequality, MRG working paper no. 8641 (University of Southern California, Los Angeles, CA). Statistical Abstract of the USA, 1981 and 1985 (U.S. Bureau of the Census, Washington, DC). Theil, H. and C. Sorooshian, 1979, Components of the change in regional inequality, Economics Letters 4, 191-193.