Mohammad Azhar Hussain

15:2007 WORKING PAPER

Time, length of accounting periods, equivalence scales, and income definitions

RESEARCH DEPARTMENT OF EMPLOYMENT AND LABOUR MARKET ISSUES

The Sensitivity of Income Polarization

The Sensitivity of Income Polarization

Time, length of accounting periods, equivalence scales, and income

definitions

Mohammad Azhar Hussain

Employment and Labour Market Issues Working Paper 15:2007

The Working Paper Series of The Danish National Institute of Social Research contain interim results of research and preparatory studies. The Working Paper

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The Sensitivity of Income Polarization Time, length of accounting periods, equivalence scales, and income definitions MOHAMMAD AZHAR HUSSAIN The Danish National Centre for Social Research, Herluf Trolles Gade 11, DK-1052 Copenhagen K, Denmark, E-mail: mh@sfi.dk Abstract. This study looks at polarization and its components’ sensitivity to assumptions about equivalence scales, income definition, ethical income distribution parameters, and the income accounting period. A representative sample of Danish individual incomes from 1984 to 2002 is utilised. Results show that polarization has increased over time, regardless of the applied measure, when the last part of the period is compared to the first part of the period. Primary causes being increased inequality (alienation) and faster income growth among high incomes relative to those in the middle of the distribution. Increasing the accounting period confirms the reduction in inequality found for shorter periods, but polarization is virtually unchanged, because income group identification increases. Applying different equivalence scales does not change polarization ranking for different years, but identification ranks are affected. The welfare state considerably reduces income polarization and inequality, but at the expense of some more identification. Key words: Income polarization; sensitivity; equivalence scale; accounting period; bi-polarization JEL classification: D31; D63; I31

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1. Introduction This paper investigates the sensitivity of income polarization, to various assumptions about the underlying incomes. The sensitivity evaluation is with regard to income definitions, equivalence scales, and the accounting period. Also the sensitivity to the applied polarization measure is investigated. This has not been done before, but it might be important, just as in the case of inequality (Buhmann et al., 1988), especially when making comparative polarization analysis. The sensitivity is exemplified with Danish data covering each year from 1984 until 2002, which is the maximum span of years available for analysis. The data are very well suited for income analysis since there is great accuracy in reporting, and since data practically include all income components. The included years represent major changes in macro economic conditions, e.g. booms and recessions, globalization etc. Furthermore, the last part of the period is characterized by increasing income inequality caused by the rising stock market rates and major increases in property prices (Economic Council, 2006). Thus, together with the sensitivity analysis, it can also be investigated whether increased inequality has been followed by polarization of incomes in one of the most equal welfare states. A few number of studies have raised the issue of polarization, especially after it has been shown that movements in polarization may not follow movements in inequality. This need not be so, because increased inequality may take place together with constant or reduced polarization, and likewise inequality may be reduced but at the same time income polarization may be on the increase. For instance income may concentrate around local poles, in which case inequality may be lower, but at the same time increasing polarization, because there is increased (group) identity among the persons around the poles and greater alienation (inequality) between income groups concentrated around different poles. Duclos et al. (2004) calls this the identification-alienation framework (IA). Other theoretical considerations providing the insight includes Foster and Wolfson (1992), and Esteban and Ray (1994). Applications include analysis of China (Zhang and Kanbur, 2001), Spain during economic and political transition (Gradin, 2000), kernel estimates used to investigate the relative size of the middle class in UK (Jenkins, 1995), and cross country analysis (Seshanna and Decornez, 2003; Duclos et al., 2004; Duro, 2005). The next Section presents the approach in Duclos et al. (2004) together with other (bi-) polarization measures. Section 3 deals with data and in Section 4 the different inequality and polarization measures are applied. The last Section summarizes the main conclusions. 2. Polarization measures The identification-alienation framework Social tensions and the like might be caused by greater income distances between individuals combined with greater sense of belonging to an income class. The first has to do with the degree of alienation and the second is the degree of identification in a society. Duclos et al. (2004) use the concepts identification and alienation to develop a polarization measure. Alienation in their approach is a function of the Gini coefficient,

2

and the degree of identification for a person with income x depends on the income density f(x) at that income level. They derive the following polarization measure

)1()( ριαα += aFP)

where (1)

a is average alienation (=two times the Gini coefficient), αι is average identification, and ρ is the normalised covariance between the two, alienation and identification. α represents an ethical parameter denoting the weight given to polarization. 0.25≤α ≤1, and the largest difference, at least theoretically, between inequality and polarization occurs when α =1, which is also the value adopted in this paper1. Increases in either alienation or identification increases polarization. For an income distribution F(y), with incomes normalised by average income y1, y2,…, yn, ordered such that nyyy ≤≤≤ ...21 , this polarization measure can be operationalized by

)()](ˆ[1)(

1i

n

ii yayf

nFP ∑

=

= αα)

(2)

where is the nonparametric kernel estimate of income density at income level y)(ˆiyf i.

Here a Gaussian kernel is applied

∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

n

j

jii h

yyhn

yf1

2

21exp

2111)(ˆπ

(3)

with the bandwidth h equal to

σα ˆ7.410

nh =

(4)

σ̂ is the standard error of the normalised incomes. Alienation, the second part of (2), is defined as

∑−

=

−−−

+=1

1

2)1(2ˆ)(

i

jjii y

ny

nniya μ

(5)

where μ̂ is the mean of the normalised incomes, which is 1. The Gini coefficient is half the size of this measure of alienation. Duclos et al.’s (2004) approach is applied throughout the paper. But the analysis is supplemented with other polarization measures, which are presented next. Bipolarization Foster and Wolfson (1992) develop a polarization measure from aggregate income distribution indices which captures bipolarization2, and Prieto-Rodríguez et al. (2003) restates it in terms of incomes below and above the median:

SGGP WBFW )( −=

G(6)

B is inter group inequality and GW is intra group inequality. S is a simple measure of income skewness (two times the ratio between average and median income). It is clear

1 Some calculations are also carried out for other values of α (available from the author upon request) 2 Using individual incomes, in contrast to aggregate indices, (with an even number of persons) the formula is

2/)1(,321,2/)1(,21,21

2 +>∀+−=+≤∀−−== ∑=

niniwniniwywMn

P ii

n

iiiFW

This is however not applied, because (6) is more informative about what causes polarization changes.

3

from (6) that bipolarization increases with greater distance between the income level of below and above median persons (higher GB). It also increases when people above and/or below the median are more alike (lower GW), e.g. lower inequality caused by intra below-median or intra above-median Pigou-Dalton transfers increases bipolarization. The last factor that might increase bipolarization is when top incomes are further away from the middle of the income distribution (higher S). Using an axiomatic approach, Wang and Tsui (2000) generalize Foster and Wolfson’s polarization measure. They further develop polarization indices, but still from the concepts they call increased spread (IS) and increased bipolarity (IB), which are also the foundations of PFW. Here, the scale invariant version of Wang and Tsui’s bi-polarization index is applied

10,1/

1

1 <<−= ∑=

− rMynPn

i

riWT

(7)

Higher r implies that greater weight is attached to incomes above double median income (yi>2M) and less to incomes below double median income, thus r is an ethical parameter. The closer r is to 1 the closer PWT is to the average relative income distance from the median. Some “polarization measures” explicitly use a definition of the middle class, e.g. includes the proportion of the population in an interval +/- some percentage from the median. E.g., a narrow definition being 75-150 % of median income (Wolfson, 1994), and a broader definition being 60-225 % of median income (Blackburn and Bloom, 1985). 3. Data The applied data are built from administrative registers in Statistics Denmark. It is a longitudinal representative data set of the Danish population for the years 1984-2002. Many of the analyses will concentrate on the first and last observed year. In Section 4.1, a 1% representative sample with around 50.000 observations in each year is utilised. In all other parts of the paper a ½% sample is used because the calculations are very time consuming - but the reduced sample is not a problem since the sample size is still large, and thus standard errors remain very low. The data set contains information on a number of demographic, educational, income, and labour market variables. Top and bottom coding was applied in order to avoid or at least reduce the effect of extremely high incomes, thus incomes above 30 times average income were excluded. Also persons with zero or negative incomes were excluded. This may cause some bias, but on the other hand the excluded cases accounts for less than 1% of the observations in any of the years 1984-2002. Equivalised disposable family income is the applied income concept. This means that all incomes minus all income taxes and mandatory contributions are included. E.g. wages, capital income, rents, labour market contributions, pension contributions, and the value of owner occupied housing is included. The equivalence scale is equal to the square root of the number of household members. An equal intra household distribution of income is assumed, meaning that each family member has the same

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equivalised household income. Although the household is the economic unit, all analyses are based on individuals. In Section 4.4, where polarization’s sensitivity to the income concept is investigated the applied income concept is market income, e.g. income from work or capital income. 4. Results 4.1. Overall development in polarization Duclos et al. (2004) polarization and its three components in each year from 1984 to 2002 are presented in Table 1. Alienation (or the Gini coefficient) is 0.2461 in the last year of analysis, 2002. The identification estimate is 0.7537, and the normalised covariance between alienation and identification is -0.1904. From (1) this means that polarization in Denmark was 0.1502 in 2002. In the first year 1984 polarization was 0.1430, and alienation/Gini and identification was 0.2245 and 0.7580. Identification is thus almost unchanged in the period (varies between 0.7537 and 0.7763), whereas alienation (the Gini coefficient) increased almost 10% (range is 0.2245 to 0.2461), and polarization increased 5% (varied between 0.1430 and 0.1509)3. Thus, inequality (measured by the Gini coefficient) rose much more rapidly than did polarization. The covariance was -0.1598 in 1984, which means it fell 19% over the period4.

3 The Danish polarization estimates for 1992 and 1995 are close to those in Duclos et al., but both at a somewhat lower level. So in the country rankings, Denmark has least polarization in 1995 (instead of 5th rank in Duclos et al.), and third lowest in 1992 (instead of 4th rank). But, just as in Duclos et al.’s Table 1, we also here see an almost unchanged level from 1992 to 1995. 4 Identification and alienation may be constant over some time period, but if the income groups with high identification are less alienated towards other income groups over that same time period (less normalised covariance) then that will reduce polarization. Thus one interpretation of the reduction in ρ over this 19-year period is that high identification groups are less alienated towards others.

5

Table 1. Polarization decomposition for Denmark, 1984-2002 Parameter assumptions:

o Polarization sensitivity parameter, α: 1 o Income concept: Equivalised disposable income o Income averaged over: None (cross-section) o Equivalence scale elasticity, ε: ½

Normalised covariance

Alienation (*½ = Gini)

Identification

Polarization

ρ a αι αP

1984 -0.1598 0.2245 0.7580 0.1430 1985 -0.1692 0.2278 0.7633 0.1445 1986 -0.1691 0.2292 0.7620 0.1451 1987 -0.1757 0.2316 0.7622 0.1455 1988 -0.1731 0.2305 0.7607 0.1450 1989 -0.1762 0.2277 0.7710 0.1446 1990 -0.1781 0.2275 0.7744 0.1448 1991 -0.1812 0.2280 0.7763 0.1450 1992 -0.1855 0.2329 0.7709 0.1462 1993 -0.1832 0.2336 0.7662 0.1462 1994 -0.1849 0.2319 0.7744 0.1464 1995 -0.1863 0.2336 0.7714 0.1466 1996 -0.1862 0.2385 0.7622 0.1479 1997 -0.1932 0.2431 0.7597 0.1490 1998 -0.1936 0.2453 0.7582 0.1500 1999 -0.1924 0.2444 0.7551 0.1490 2000 -0.2007 0.2457 0.7682 0.1509 2001 -0.1985 0.2446 0.7645 0.1499 2002 -0.1904 0.2461 0.7537 0.1502 Note: Standard errors for polarization estimates are not presented, but they are around 0.0005 for polarization in this table, where approximately 50,000 observations per year are used. This is equal to a 1% representative sample of the whole Danish population. In other sections a ½% sample is used, and the standard error then increases to about 0.0007. Standardisation: Alienation, and thus polarization, is multiplied by ½ because that is equal to the Gini coefficient measure of inequality. Software: In most of the analyses the DAD-software was applied, Zhang (2003). From Table 1 we can extract the standard errors of estimates and their means in the 19 year period, and this tells us that alienation’s relative variation (standard error divided by the mean) is 3½ times greater than for identification, and double the size of polarization’s relative variation. This indicates that variation in polarization might partly be explained by variations in alienation. From the estimates in Duclos et al. we can infer that across country relative alienation variation is much greater than polarization variation. Thus, although Denmark has much less income inequality (alienation) than for instance the UK or the USA, polarization in incomes between the three countries may not be that different5. The fear that more unequal societies might be more prone to less cohesion may thus not be 5 In fact inequality is 35 and 43% higher in the UK and the USA, but polarization is only 16 and 7% higher in the same two countries compared to Denmark (calculation based on Duclos et al., 2004, Table III)

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that relevant, because unrest and revolt (Esteban et al, 1999) is more likely the result of polarization. In Denmark there has been increased focus on rising income inequality, but the Danish society does not seem to have become more unstable (politically or socially). A reason may be that the increase in polarization, although statistically very significant, is much less than the earlier documented increase in inequality (alienation). Duclos et al. detects large cross country differences in identification together with polarization differences, but in a given country, e.g. Denmark, this does not seem be the case over time. Decomposition of polarization change Taking natural logarithms in (1) and suppressing the α index changes the relation in year t and some later year t+s into

)1ln(lnlnln tttt aP ρι +++=

)1ln(lnlnln stststst aP ++++ +++= ρι And thus the (relative) change in polarization can be decomposed into the (relative) changes in alienation, identification and (one plus) the normalised correlation coefficient between the two

⇒+−++−+−=− ++++ )1ln()1ln(lnlnlnlnlnln tsttsttsttst aaPP ρριι

)1ln(lnlnln ρι +Δ+Δ+Δ=Δ aP The cumulative relative change (since 1984) in polarization and its three components

are shown in Figure 1 for each year from 1985 until 2002. In the figure the value of the polarization curve in a given year is the sum of the alienation and (1+

(8)

ρ ) bars and the value of the identification curve, cf. (8).

Figure 1 . Decomposition of polarization change as in (8). Denmark.Change from 1984. α =1.

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1+ρ (left axis) Alienation/Gini (left axis) Identification Polarization

It is seen that the total change from 1984 to 2002 is 5% in polarization (right axis) and almost 10% in alienation (left axis), while identification (right axis) has not changed

7

much. Figure 1 clearly shows that alienation (inequality) is indeed the driving force behind the observed change in polarization. In fact simple regressions (see Table A1 in the Appendix) suggests that 98% of the variation in polarization is explained by variation in alienation, and furthermore that a 1% increase in alienation increases polarization by ½% (elasticity of ½). The explanatory power is reduced to 60% when regressing differences in logs to remove trend effects, but the relation is almost unchanged (elasticity of 0.44). Including differences in log per capita GDP increases the explanatory power to 71%, with the sensitivity of inequality still at 0.44, and elasticity with respect to GDP at 0.1. Thus changes in GDP growth rates also significantly affects polarization change, with higher growth rates producing higher income polarization in society. Thus in addition to the classical trade-off between economic equality and economic growth (see e.g. Atkinson, 1996), there also seems to be a trade-off between polarization and growth. In conclusion, no doubt polarization and inequality are theoretically different concepts, but empirically there is a tendency that the two concepts are highly correlated, partly caused by off-setting opposite changes in identification and (more importantly) the normalised correlation between alienation and identification. But differences between polarization and inequality changes may be great in other instances as is illustrated next. 4.2. Polarization’s sensitivity to the length of the accounting period The level of polarization is measured for average incomes based on 1, 2, 3, 4, and 5 consecutive years. The single year is the latest year 2002. The two-year period covers 2001-2002 and the five-year period is the average of incomes in 1998-2002. Only persons with positive incomes in every year from 1998 to 2002 are included, and therefore the sample size is constant over the five-year period.

Figure 2 . Polarization and its components when expanding the accounting period from 1 to 5 years. 1998-2002

0.85

0.90

0.95

1.00

1.05

1.10

1.15

2002 2001-2002 2000-2002 1999-2002 1998-2002

Accounting period

Inde

x: F

irst y

ear v

alue

=1

1+ρ Alienation/Gini Identification Polarization

8

Figure 2 shows the development in polarization, alienation and identification when the accounting period increases from one to five years. Large changes in alienation (inequality) and identification take place, but polarization is almost unaffected. This is indeed an example of significantly different developments in inequality and polarization. Earlier studies show that inequality is reduced the more years we include to calculate income, because temporary fluctuations in income are (partly) neutralised in the averaging process and long run effects of unemployment, sickness, etc. dominates. Thus, temporarily a person with high education and a person with low education may be alike with respect to income, but taking more and more years into account means that the low skilled person will more likely resemble his/her group’s distribution of higher unemployment etc. and so will the highly skilled person. But this is obviously not the case with polarization. The reason for this being that lower alienation (towards other groups) now is off-set by more identification (within groups). This is clear from Figure 3 showing the income distribution for average incomes in 1998-2002 and the income distribution in the single year 2000 (the middle year of the five-year period). These curves represent in (3). )(ˆ

iyf

Figure 3 . Nonparametric kernel income densities from (3)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

Normalised income

Den

sity

Average income 1998-2002 Cross-section income from 2000

The huge reduction in alienation (inequality) when using five-year average incomes in stead of one-year incomes is caused by fewer persons in the tails of the income distribution when using the average, especially at the lower end. But the income density curve for average income (dotted curve) is also for most portions of the upper end below the one-year curve (solid curve). The significant increase in identification is evident from Figure 3, where the middle part of the 1998-2002 average distribution is considerably above the one-year distribution in 2000. The results indicate that the accounting period does not seem to matter much for polarization analysis. On the other hand, it punctures the widespread myth that the

9

distributional challenges are much smaller when we have access to more precise estimates of individual permanent income. Because, this does indeed reduce inequality and poverty risk (Hussain, 2004), but polarization is not reduced. This raises the question of which policies might (at least theoretically) assure a given acceptable level of polarization. A consequence of the results is also that differences in accounting periods in country comparisons may not bias polarization estimates, while identification and inequality levels may most likely be biased. These results are independent of the applied years since qualitatively the same results are found when applying the first five years of the observation window (1984 to 1988); see Figure A1 in the Appendix. 4.3. Polarization’s sensitivity to the choice of equivalence scale Consumption possibilities are estimated by dividing the total disposable income of an economic unit with its equivalence scale. Equivalence scales can be defined in a variety of ways, but the widely applied so-called modified OECD scale (Hagenaars et al., 1994) is very well captured by the equivalence scale formula presented in Buhmann et al.’s (1988) comprehensive survey: εZES = . Here the equivalence scale ES is defined as the size of the economic unit Z in the power ε, which is the elasticity of the equivalence scale with respect to the size of the economic unit. It can also be interpreted as the assumed degree of economies of scale. ε=0 implies that household disposable income is divided by 1, in which case there are maximum economies of scale because an additional household member leaves consumption possibilities unchanged. The other extreme is ε=1, in which case household disposable income is divided by the number of household members. This implies total lack of economies of scale, because the relative change in household size must be accompanied by the same relative change in household disposable income in order to keep consumption possibilities unchanged. Except for this section, ε=½ throughout the paper. In Figure 4, the four concepts in (1) are calculated for different values of ε between 0 and 1 for the first, last, and a middle year (1984, 1993, and 2002) in the analysed period. Looking at Figure 4a, the rank of years with respect to polarization is unchanged no matter which equivalence scale is applied. But for large values of ε polarization of 1984 get close to the level in 1993. Minimum polarization is achieved at about ε=0.45. From Figure 4b it can be seen that ranks are also unchanged regarding alienation, but 1984 and 1993 are again close for large values of the equivalence scale elasticity. Minimum alienation is found at ε around 0.6. In the case of identification in Figure 4c the rank does change with the choice of ε. For low values, 1984-identification is above both 1993 and 2002, but around ε=0.7, 1984-identification is well below 1993 and very near 2002-identification.

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Figure 4. Polarization and its components at different levels of the equivalence scale parameter on the horizontal axis Figure 4a. Polarizat ion

0.141

0.143

0.145

0.147

0.149

0.151

0.153

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1984 1993 2002

Figure 4b. Alienat ion/Gini coeff icient

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1984 1993 2002

Figure 4c. Ident if icat ion

0.61

0.630.65

0.67

0.690.71

0.73

0.750.77

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1984 1993 2002

Figure 4d. Standardised correlat ion

-0.21-0.2

-0.19-0.18-0.17-0.16-0.15-0.14-0.13-0.12-0.11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1984 1993 2002

There seems to be considerable change in Figure 4, but when looking at levels relative to the initial values of polarization components, it is clear that polarization is not that sensitive to the choice of equivalence scale, see Figure 5. Thus, in contrast to inequality (and identification) analysis, the assumptions about economies of scale are less important for polarization measurement.

11

Figure 5 . Equvalence scales and polarization components, 2002. Denmark

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Equivalence scale parameter, ε

Inde

x, v

alue

at ε

equ

al to

0=1

1+ρ Alienation/Gini Identification Polarization

4.4. Polarization’s sensitivity to the definition of income In this section the income definition is changed so that only market income is included, and thus income before transfers and taxes is analysed. In this way, we can get an idea of the effect of the Danish welfare state on polarization and inequality. This is very much counter factual simulations, since dramatically reducing (or abolishing) taxes and transfers most likely affect labour supply and thereby earned market income. Thus, the simulations in this paragraph rather indicate the upper limit of effects on the income polarization. On the other hand, the Danish labour supply elasticity with respect to income taxes is as low as around 0.1 (Frederiksen et al., 2001) - lower for men (0.05) than for women (0.15) - and therefore the counter factual may be more relevant than in other high elasticity economies. The Duclos et al. polarization index and its components are presented in Table 2 when applying equivalised market income.

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Table 2. Polarization decomposition for Denmark using market income, 1984-2002 Parameter assumptions:

o Polarization sensitivity parameter, α: 1 o Income concept: Equivalised market income o Income averaged over: None (cross-section) o Equivalence scale elasticity, ε: ½

Normalised covariance

Alienation (*½ = Gini)

Identification

Polarization

ρ a αι αP 1984 -0.0007 0.4076 0.6284 0.2559 1985 -0.0045 0.4059 0.6289 0.2541 1986 -0.0035 0.4032 0.6267 0.2518 1987 -0.0006 0.4118 0.6509 0.2679 1988 0.0048 0.4185 0.6658 0.2799 1989 0.0064 0.4218 0.6663 0.2828 1990 -0.0051 0.4321 0.6674 0.2869 1991 -0.0055 0.4362 0.6727 0.2918 1992 -0.0020 0.4407 0.6936 0.3051 1993 -0.0056 0.4497 0.6996 0.3129 1994 -0.0071 0.4527 0.7301 0.3281 1995 -0.0056 0.4494 0.7272 0.3250 1996 -0.0118 0.4516 0.7310 0.3262 1997 0.0011 0.4449 0.7503 0.3342 1998 -0.0079 0.4444 0.7195 0.3172 1999 -0.0042 0.4381 0.7097 0.3096 2000 -0.0065 0.4396 0.7097 0.3100 2001 -0.0132 0.4424 0.6989 0.3051 2002 -0.0097 0.4348 0.6745 0.2904 Average over 1984-2002:

Market income -0.0043 0.4329 0.6869 0.2966 Disposable income -0.1830 0.2351 0.7649 0.1468 Difference, % 98 84 -10 102 Note: See Table 1 Market income polarization varies between 0.2518 and 0.3342, whereas disposable income polarization only varies between 0.1430 and 0.1509 during the 19-year period 1984-2002 (Table 1). The welfare state thus halves income polarization in Denmark. Welfare regime induced polarization reductions are furthermore a little higher than inequality reductions. But the Danish welfare state also slightly increases identification. 4.5. Other polarization measures The sensitivity of polarization development is investigated by supplementing the Duclos et al. polarization measure with the Foster-Wolfson bipolarization measure in (6). Like for overall polarization, the bipolarization measure has increased in the 19-year period from 1984 to 2002, see Figure 6. The level was 0.1800 in 1984 and 0.1899 in 2002 representing an increase of 5½%, which is close to the 5% increase in the over all polarization measure. Increased polarization after around 1994 is thus reconfirmed, but this time in the sense that we also, like in other countries (Jenkins, 1997; Peterson and Strobel, 1997), see a reduced middle class. The change in bipolarization can be decomposed by using (6):

13

SGGP WB

FW ln)ln(ln Δ+−Δ=Δ This decomposition is applied in Figure 7, where we include the (approximate relative)

change from 1984 to some later year ending with 2002. Thus, the cumulative change since 1984 is observed.

(9)

The almost constant polarization in the beginning of the period up till 1994 hides underlying movements in the income distribution. Thus, skewness has been rising over the period, but its effect has been almost neutralised by an opposite change in inequality – more particularly inequality among the two groups separated by the median rose more than inequality between them. From 1995 and onwards skewness continued its increase while inequality could not follow, and this caused the increase in bipolarization. Actually in the latest year (and in 1999) skewness and inequality both contributed to more polarization, because inequality between groups rose more than among groups. Figure 7 also suggests that one component (S) already was on the rise (well) before the economic boom in the 1990s, which started around 1993/1994.

Figur 6 . Polarization P α and bipolarization PFW

0.143

0.144

0.145

0.146

0.147

0.148

0.149

0.150

0.151

0.152

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 020.177

0.179

0.181

0.183

0.185

0.187

0.189

0.191

Polarization (left axis) Bipolarization (right axis)

14

Figure 7 . Decomposition of bipolarization change. Denmark. 1984-2002. Change from 1984, see (9)

-0.03

-0.02-0.01

0.00

0.010.02

0.03

0.040.05

0.06

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

Between/within inequallity diff. Skewness Bipolarization

Using different values of the ethical parameter r in PWT gives the same results, namely that Wang and Tsui bipolarization has increased leaving a smaller middle class behind, see Figure 8, which uses normalised measures. The level of polarization when r=0.1, 0.5, and 0.9 was 0.9843, 0.5162, and 0.3251 in 1984, and 0.9849, 0.5351, and 0.3606 respectively in 2002. Thus, level and changes in polarization heavily depend on the chosen value of r (see also Figure A2 in the Appendix). Low r is associated with slow polarization changes. But the ranking of years with respect to polarization seems to hold regardless of r’s level, except for low values of r, where PWT is (only very slightly) below the 1984-level (in 1985 and 1989-1991). The increase in polarization is almost 4% when using the middle value of r, and about ½% when r=0.1 and 9% when r=0.9.

15

Figure 8 . Development in Wang and Tsui's (2000) bipolarization measure, Denmark, 1984-2002

0.98

1.00

1.02

1.04

1.06

1.08

1.10

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02

Inde

x 19

84=1

r=0.1 r=0.5 r=0.9

Table 3 presents an overview of the development of additional “measures” of polarization over the period 1984 to 2002. The first four measures are those presented for comparison in Wolfson (1994), using the same population share and income range definitions. These four measures are all based on rather arbitrary definitions of the concept of polarization. Table 3. Other measures applied to income polarization, 1984 and 2002

Population share in range

of median income Range of income covering middle population/median

75-150% 60-225% 40-60% 30-70% 20-80% 1984 0.6265 0.8507 18.89 40.05 65.53 2002 0.6026 0.8543 19.80 42.17 67.93 Change -0.0239 0.0036 0.91 2.12 2.41 Change, % -3.8 0.4 4.8 5.3 3.7

The first result of the population share measure reports the fraction of individuals with incomes between 75 and 150% of the median. The number of people constituting this definition of middle class has evidently decreased (-3.8%) during the observed period in accordance with the previously described measures. Extending the definition of the middle class excessively to including incomes between 60 and 225 percent of the median results in an increase of the middle class by 0.4% (reduced polarization) - this does not support the former conclusion, but does not indicate a strong case against previous results either. The income range measures are all consistent with the hypothesis of an increased polarization in the distribution of Danish incomes during the years in question. Thus, the span of income made up of the difference between, for instance, the 6th and 4th decile, has increased relative to the median. The same goes for the other reported

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income spans, indicating a shrinking middle class (polarization increases of 3.7-4.8% in 1984-2002). It seems as if almost all the non-axiomatic approaches do well compared to the approaches in Duclos et al. (2004), Foster and Wolfson (1992), and Wang and Tsui (2000), e.g. they correctly predict increasing polarization from 1984-2001. But this may merely be a coincidence, and using other time periods might reduce their predictive power. 5. Concluding comments Polarization has increased about 5% from 1984 to 2002 when applying either Duclos et al.’s (2004) measure, or the Foster and Wolfson (1992) bipolarization measure. This increase has been caused by more alienation (inequality), and faster income growth among high incomes relative to the middle. Wang and Tsui’s (2000) measure also indicates an increase in the period, but with a little less increase than the two other indices, when using a middle value of the ethical parameter. The application of additional “measures” indicates an almost unambiguous increase in polarization irrespective of the preferred approach to measurement. Different income accounting periods were tried (between 1 and 5 years). When the income accounting period is extended we find reductions in inequality, but polarization is virtually unchanged, which is because identification also increases, and thus off-sets the decrease in inequality. Applying different equivalence scales does not change the polarization ranking for different years, but identification ranks are affected. Thus, especially for identification one must be careful when comparing e.g. different countries. The welfare state markedly reduces polarization via much lower inequality, but also slightly increases identification. In future work it might be relevant to let alienation be represented by other inequality measures than the Gini coefficient. Furthermore, it might also be relevant to identify which mechanisms (e.g. demography, labour market career and education) at the individual or group level contributes to the observed increases in polarization.

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References Atkinson, A.B.: Seeking to explain the distribution of income, In J. Hills (ed.) New Inequalities: The changing distribution of income and wealth in the United Kingdom, Cambridge University Press, Cambridge, 1996 Blackburn, M.L. and Bloom, D.E.: What is happening to the Middle Class?, American Demographics 7 (1985), 18-25 Buhmann, B., Rainwater, L., Schmaus, G. and Smeeding, T.M.: Equivalence Scales, Well-Being, Inequality, and Poverty: Sensitivity Estimates across Ten Countries Using the Luxembourg Income Study (LIS) Database, Review of Income and Wealth 34 (1988), 115-42 Duclos, J.-Y., Esteban, J. and Ray, D.: Polarization: Concepts, Measurement, Estimation, Econometrica 72 (2004), 1737-72 Duro, J.A.: Another Look to Income Polarization across Countries, Journal of Policy Modeling 27 (2005), 1001-07 Economic Council: Dansk Økonomi, efterår 2006 (Danish Economy, autumn 2006), Copenhagen Esteban, J. and Ray, D.: On the Measurement of Polarization, Econometrica 62 (1994), 819-51 Foster, J.E. and Wolfson, M.C.: Polarization and The Decline of the Middle Class: Canada and the U.S., mimeo, Vanderbilt University, 1992 Frederiksen, A., Graversen, E.K. and Smith, N.: Overtime Work, Dual Job Holding and Taxation. Forthcoming Working Paper. CIM and CLS, Department of Economics, Aarhus School of Business, Aarhus, 2001 Gradin, C.: Polarization by Sub-populations in Spain, 1973-91, Review of Income and Wealth 46 (2000), 457-74 Hagenaars, A., de Vos, K. and Zaidi, M.A.: Poverty Statistics in the Late 1980s: Research Based on Micro-data, Office for Official Publications of the European Communities, Luxembourg, 1994 Hussain, M.A.: Børnefattigdom i Danmark 2002. Tema: Fattigdommens dynamik (Child Poverty in Denmark 2002. Theme: The Dynamics of Poverty), the Danish National Institute of Social Research, and Save the Children, Denmark, 2004 Jenkins, S.P.: Did the Middle Class Shrink during the 1980s? UK Evidence from Kernel Density Estimates, Economics Letters 49 (1995), 407-13

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Peterson, W.C. and Strobel, F.R.: Class Warfare and Middle Class Decline in America, Journal of Income Distribution 7 (1997), 175-201 Prieto-Rodríguez, J., Rodríguez, J.G. and Salas, R.: Polarization Characterization of Inequality-Neutral Tax Reforms, Economics Bulletin 4 (2003), 1-7 Seshanna, S. and Decornez, S.: Income Polarization and Inequality across Countries: An Empirical Study, Journal of Policy Modeling 25 (2003), 335-58 Wang, Y.-Q. and Tsui, K.-Y.: Polarization Orderings and New Classes of Polarization Indices, Journal of Public Economic Theory 2 (2000), 349-63 Wolfson, M.C.: When Inequalities Diverge, American Economic Review 84 (1994), 353-58 Zhang, Q.: DAD, an innovative tool for income distribution analysis, Journal of Economic Inequality 1 (2003), 281–284 Zhang, X. and Kanbur, R.: What Difference Do Polarisation Measures Make? An Application to China, Journal of Development Studies 37 (2001), 85-98

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Appendix Table A1. Simple regressions (OLS) of polarization on alienation and GDP per capita. Denmark, 1984-2002. (SE in italics) Response variable:

Nat. log of PFW Change in

nat. log of PFW

Explanatory variable(s), levels of: Nat. log of alienation/ Gini 0.499 0.397 0.019 0.039 Nat. log of GDP per capita 0.015 0.005 Explanatory variable(s), changes in: Nat. log of alienation/ Gini 0.442 0.438 0.090 0.080 Nat. log of GDP per capita 0.098 0.042

R2 0.977 0.985 0.601 0.708 n 19 19 18 18

Figure A1 . Polarization and its components when expanding the accounting period from 1 to 5 years, 1984-1988

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1988 1987-1988 1986-1988 1985-1988 1984-1988

Accounting period

Inde

x, F

irst y

ear v

alue

=1

1+ρ Alienation/Gini Identification Polarization

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Figure A2 . Wang and Tsui's (2000) bipolarization measure PWT for different values of the ethical parameter r

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r

P WT

1984 1993 2002

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