Sources of inequality - a formal analysis

In this section I shall present some rather stylized thought experiments in an attempt to highlight the conditions under which public pensions pro-duce high or low levels of overall inequality in conjunction with private income sources. What are the more speeific mechanisms through which variation in the size of public pensions and their distributive profile could affect the overallievel of inequality found among pensioner households?

The aim is to identify a set of criticallinks in the causal chain leading from changes (variation) in public pensions to changes (variation) in the over-all level of income inequality among pensioners. Disaggregating the research question in this way will help to sharpen the focus on the rele-vant theoretical issues and to increase the empirical leverage over the more general relationship between institutional characteristics of pension systems and finaloutcome in terms of the degree of income inequality among pensioners.⁹⁶

For the sake of simplieity Ishall assurne, in the following, that the income package of the elderly consists of two types of components only:

96 The possibilities and limits of a comparative approach to the study of distributive out-comes will be diseussed more in detail in Chapter 4 below.

public transfers and income from "private" sources. I also have to abstract from all details about the workings of these two components except for their scope/ generosity and the degree to which they are concentrated among the population of retirees. In other words, I am interested in the consequences that might follow if public and private components are more or less generous and more or less concentrated among the relevant population. Variation in aspects like the incidence of pensions among dif-ferent house hold types and the time-path of retirement income will be totally ignored.

In order to proceed, I need a method for evaluating how public and private income components interact to produce a certain level of inequal-ity in total income. Here I shall take the so-called naturai decomposition of the Gini coefficient as a point of departure. It devises a very simple for-mula for decomposing Gini inequality of total income according to the contribution made by each of a number of individual income components.

Like other methods of its kind, the natural decomposition of the Gini index is completely static. It takes the distribution of the overall income package as given and it ignores behavioral responses or any possible dependency of one income component upon the size and distribution of others (Shorrocks, 1982). One should therefore be very careful when using this and similar methods to make causal claims and counterfactual arguments (see Shorrocks, 1982; 1988; Podder, 1993). I shall show, how-ever, that with due respect for its severe lirnitations, the method does pro-vide a conceptual framework for discussing the outcome of hypothetical changes in the mix of public and private retirement provision.

The natural decomposition of the Gini index

Lerman and Yitzhaki (985) have sugge sted a very helpful way to write the natural decomposition of the Gini index that is applicable to any par-titioning of a total income package in k separate components. According to their formula (see below), the contribution made by each income com-ponent (Qk) to overall Gini inequality is a product of three factors: the Gini coefficient for the component itself (Gk), the share of the component in the overall income package (Sk) and finally the so-called Gini correla-tion between the component and the overall income package (Rk). ⁹⁷

97 The Gini correlation is a hybrid between the more familiar measures of correlation, Pearson's rand Spearman's rank-correlation coefficient. For some further explanation of the Gini corre1ation see footnote 277 in Chapter 7.

However, it must be emphasized that, as a general rule, one cannot infer anything about the change in total inequality that will follow from non-trivial changes in the k'th component by just looking at Qk and its further breakdown according to the formula above. Substantial changes in one component will almost always affect the assessment of the contribution made by the other components - partly because their share in the total income package will be modified and partly because the ranking of observations according to total income is liable to change - and then all the R-terms will change toO.⁹⁸

With this in mind, Ishall apply the tools provided by the decomposi-tion formula to discuss the consequences of the hypothetical introducdecomposi-tion of new public pension benefits as well as of changes in the size and con-centration of existing components in the income package of the retired.

First Ishall consider the very simple case where an equally distributed,

"universal pension" is introduced or an already existing one is expanded, and Ishall briefly compare it with the effect of an income-tested benefit.

Then Ishall go on to discuss the effect of the introduction of social insur-ance benefits that are supposed to be directly proportional to pre-retire-ment income levels.

The impact of a uniform incame transfer

As we saw in the previous chapter, a number of OECD countries offer flat-rate benefits to every resident beyond a certain age as part of the public pension system. Such benefits are the dosest real-life approxirna-tion to what I here consider as a "uniform" income transfer.⁹⁹

98 Lerman and Yitzhaki (1985) and Podder (1993) have used the natural deeomposition to devise a formula for the marginal elasticity of the overall Gini eoefficient with re-speet to proportional ehanges in individual components. However, the fonnula is only valid in general for infinitely small changes that do not cause any re-ranking of eases.

99 In practical empirical analyses, even flat-rate unlversal pensions will not always turn out to be exactly uniformly distributed among the income units (the Gini coefficient is not exactly zero). Most countries employ a residenee test that excludes some sections of the resident population in the relevant age-groups from receiving full benefits. An-other important reason concerns the treatment of individuals in different household settings. Any benefit system has an built-in equivalence scale, deciding whether the pension for a married person should be equal to or less than the pension benefit going to a single person, and this will not necessarily be in agreement with the equivalenee scale used by the researcher.

It is particularly simple to analyze the effect on total inequality of changes in such a uniform income component. As long as one is only manipulating its size (or introdueing such a transfer from scratch), and assuming that the distribution and absolute size of the remaining income components are unaffected (no behavioral responses or other second-order effects), the change in total inequality is extremely simple to calculate.

The Gini coeffieient for a uniform transfer is zero by definition, so its own contribution to overall inequality is also zero. According to the natu-raI decomposition formula, total inequality will in this situation be equal to the contribution made by the rernaining income package taken together. The remaining income package might consist of private compo-nents only or a combination of private compocompo-nents and public soeial insurance pensions. The only thing that matters is the Gini coefficient for these companents taken together and their share in total income (one minus the share taken up by the uniform benefit).lOO The size of the uni-form component affects total inequality only through its impact on the share commanded by the remaining income package. The absolute reduction in overall inequality is simply given by the size of the incre-ment (measured as a proportion of the new income package) multiplied with the Gini coefficient for the remaining components. If a uniform income transfer is introduced and it takes up 10 percent of the new, expanded incame package, then total inequality will decrease by exactly 10 percent.

Thus the equalizing potential of a uniform benefit is indisputable.

Increasing its size will always lead to less inequality, and as long as we ignore any behavioral responses, the reduction in inequality will be exactly proportional to the relative size of the increment with respect to the entire expanded income package. However, one should at the same time note that, in order to achieve very substantial reductions in overall inequality from a given initiallevel, the necessary (increase in the) size of a uniform income transfer will soon become overwhelming as measured against the size of the "old" income package.¹⁰¹

100 Since the ranking of observations in terms of total income will be entirely determined by the remaining income package, its R-term must necessarily be equal to 1, and hence it can be ignored.

101 In order to accomplish a 50 percent reduction in overall inequality the uniform transfer must take up 50 percent of the new income package, and thus be exactly equal ir1 size to the entire "old" income package.

101

Of course, an income-tested benefit would clearly be more cost effi-dent under these completely static conditions. To demonstrate this in terms of a Gini-decomposition, we can imagine that an income-tested benefit has been carefully constructecl so that it cloes not cause any re-ranking in the distribution of total income, and so that it at the same time consistently pays out higher amounts the further down people are posi-tioned in the overall income distribution. Thus the benefit has aperfeet negative Gini correlation with the overall income package (the R-term is equal to -1). This admittedly very stylized benefit will on its own be responsible for a negative contribution to total inequality that is equal to its share in the income package multiplied by its Gini coeffident. 102 This effect comes in addition to the equalizing effect that the uniform transfer also has, by causing a reduction in the share taken up by the remaining income package.

Before leaving the discussion of uniform and means-tested benefits, let me briefly consider how behavioral responses could enter the picture. lf antidpated in advance, changes in the pattern of public pensions might lead to changes in the savings behavior of the individuals concerned and/

or in the behavior of relevant collective actors like ernployers and unions, and therefore the possibility of behavioral (and institutional) responses must be taken into account (this is the topic of Sections 3.4-3.5 below).

Generally speaking, behavioral responses (and second-order effects more generally) could have very different implications for the level of income inequality that will finally obtain, depending on whose income is affected (rich or poor) and in what direction (a decrease or an inerease in income from alternative sources).

Let us here assurne that the expansion of a uniform income transfer will to some extent be offset by a reduction in other (private) income compo-nents.¹⁰³ lf this reduction were simply proportional, that is, if it did not affect the Gini coefficient for the private income components taken together, it would just have the effect of further strengthening the equaliz-ing effect of the uniform income transfer. The reason is, of course, that the share of the remaining income package becomes even smaller, and there-fore its contribution to overall inequality further declines as compared to the static example above. However, the assumption of a proportional effect

102 To the extent that a means-tested benefit does lead to same re-ranking and its negative correlation with the overall income package is less than perfeet, the equalizing effect will be weaker.

103 This assumption is by no means unquestionable, as we shall see in Section 3.4 below.

is quite heroic. ^It is more likely that the relative reduction in private income sources will be weak among high-income strata, and the consequence would be that the Gini coefficient for the remaining income package would tend to inerease. In .the language of Gini-decomposition, the net result depends on the relative strength of these two effects: the relative reduction in the share taken up by private components, and a possible relative increase in their Gini inequality. Of course, the two effects would exactly balanee each other, if everybody experienced an absolute reduction in their income from private sources, that was exactly equal to the size of the universal benefit. This is not a very plausible outcome either, simply because members of the low-income strata might not initiaUy have enough private income to offset the uniform transfer. ¹⁰⁴ In practice much will depend on the level of inequality obtaining in the status quo. The higher the initial level of inequality, the less likely it is that the introduction of a uniform income transfer will induce an increase in the concentration of pri-vate sources that is anywhere near strong enough to offset the equalizing effect ..

For typical means-tested benefits there are more compelling reasons to expect that offsetting behavioral responses will materialize and that they will tend to be concentrated in the lower and middle part of the income distribution. Like the uniform pension, an income-tested benefit has a pureincome effect that might lead people, who expect to receive this benefit, to acquire less private retirement income than they would other-wise have done. Secondly, and probably more importantly, the implicit taxation of income from other sources that goes with income-testing, gives rise to a kind substitution effect as weU. In the income ranges where the benefit is leveled off (usuaUy the lower and middle part of the income distribution), the marginal gain from additional private in come is cur-tailed, creating an incentive against the acquisition of private retirement income. Neither of these effects are likely to involve high-income strata (pre- and post retirement) as people belonging to these strata do not expect to receive the benefit anyway and as they are typicaUy above the income range affected by the tapering off of means-tested benefits. If it is correct to assurne that behavioral responses to means-tested pensions are likely to be strong and concentrated in the lower half of the income dis-tribution, then Gini inequality for the remaining income package can be

104 Below Ishall argue that a significant fraction of retirees can be expected to end up with very littIe priyate income irrespective of the institutional setting.

expected to change systematically in the direction of more inequality. So the net result, as compared to a uniform benefit of a similar size, might be substantially less favorable than you would expect from the static com-parison. Theory and empirical evidence as to the extent and character of behavioral responses is needed in order to foresee the final result.

The impact of social insurance benefits

How might inequality among pensioner households be affected by the presenee of social insurance benefits that vary among the recipient households in direct proportion to pre-retirement income levels?

Usually social insurance pensions are built up over a long time span, but Ishall continue to ignore the time dimension and simply assume that a state of maturation has been reached instantaneously. In the status quo (or the counterfactual situation), I stipulate there to be a uniform public pen-sion of some size, while income from private sources eau ses a certain level of inequality to prevail in the distribution of total income. In the status quo the distribution of private income will unilaterally decide the ranking of individuals in terms of their level of total income, so there must be a per-feet Gini correlation between the private component and the entire income package (~ = 1). Therefore, the Gini coefficient for total income must equal the Gini coefficient for the income from private income sources multiplied by their share in the total income package (G = G_{p .} SJ. If we for instance assume that the Gini coefficient for the private income compo-nent is 0.6 and its share in the total income package is 50 percent, then the Gini coefficient for the entire income package will be equal to 0.3.

If we further assume that the new social insurance pension is perfectly rank-correlated with private income and hence with the prevailing income distribution among the retired, then the conditions under which the change is equalizing or dis-equalizing can be stated with precision. Ine-quality in total income will decrease if the Gini coefficient for the social insurance benefits, Gi' is smaUer than the Gini coefficient for the "old"

income distribution, and it will inerease if it is bigger. In the numerical example, the critical value for the concentration of the social insurance benefit is 0.3. We can even go on to estimate the change in the overall Gini coefficient, which will be given by the expression (Gi - G) . S/, where G is the "old" Gini coefficient and S/ is the share taken up by the new transfer in the expanded income package. 105 In other words, the new Gini coefficient, G', will equal G + (Gi - G) . S/.106 Suppose, for instance, that the Gini coefficient for the pre-retirement income distribution and hence

the sodal insuranee benefit is 0.2, and that it takes up 20 pereent of the new ineome paekage; then total inequality will go down from 0.3 to 0.28.

So far it has been established that the introduetion of a social insur-anee seheme with a eertain level of inequality bu ilt into its benefit strue-ture can be equalizing, under completely static circumstances - Le., with-out any behavioral responses. For this result to oceur, the new sodal insurance benefits must be less strongly concentrated than the existing income paekage. The more inequality there is at the outset, through the interaction of a large share and a high eoncentration of private income sources, the more likely and the stronger the equalizing effect. In this case it was assumed that in the status quo the eoncentration of private retirement income was three times stronger than the concentration of pre-retire ment incornes, and it was assumed that the existing income package was split in half by private income and a uniform income transfer.

In the following Ishall consider the consequence of modifying the underlying assumptions in three different direetions (one at a time). What is the result of dropping the assumption about aperfeet correlation with the existing income paekage? What happens if the sodal insurance benefit is not just added to the existing income paekage but indirectly "financed"

by reductions in the uniform pension offered in the status quo? As a third and final point Ishall consider the irnpact of behavioral responses.

For technical reasons I have assumed that the sodal insurance trans-fers are perfectly rank-correlated with the private sources of retirement income. It is not unrealistic to expeet a fairly strong positive correlation, since the propensity to acquire private means of income provision in retirement is related to some of the same factors that usually govern the distribution of social insurance benefits: income and employment histo-ries of the redpient households. However, we should not necessarily expeet the correlation to be perfeet. One must expect the distribution of private income to (rank-)correlate less perfeetly with income levels prior

For technical reasons I have assumed that the sodal insurance trans-fers are perfectly rank-correlated with the private sources of retirement income. It is not unrealistic to expeet a fairly strong positive correlation, since the propensity to acquire private means of income provision in retirement is related to some of the same factors that usually govern the distribution of social insurance benefits: income and employment histo-ries of the redpient households. However, we should not necessarily expeet the correlation to be perfeet. One must expect the distribution of private income to (rank-)correlate less perfeetly with income levels prior