Is Income Mobility Socially Desirable? Antonio Abatemarco 1 a Dipartimento
di Teoria e Storia dell’Economia Pubblica, Universit` a di Napoli “Federico II”, Via Cinthia - C.sso Monte S’Angelo - 80126 Napoli (I),
[email protected]. Working Paper
Abstract In welfare economics the relevance of income mobility has been often highlighted in order to allow for the measurement of equality of opportunity. In this paper we observe that intra and intergenerational income mobility have very few in common and, in particular, only the latter concerns with equality of opportunity. In contrast with existing literature we focus on the former not the latter, as in social welfare analysis it cannot be transcended that equity and efficiency evolves in a spatio-temporal dimension. If a moral and impartial observer is assumed to be sympathetic to individual interests in a random framework, especially to the loss of individual well-being due to uncertainty, then income mobility is both source of uncertainty (ex-ante) and inequality effect (ex-post). In this context we show that income mobility does not need to be necessarily either socially neutral or socially desirable ex-ante, instead, it might be well the case that social undesirability occurs ex-ante. Key words: welfare, mobility, inequality, intragenerational
1
Introduction
It has been often observed that income mobility is a multi-faceted concept (Fields and Ok, 1996). This fact is a crucial starting point for any research activity oriented to such an economic phenomenon. Unfortunately, the relevance of income mobility’s information for several research interests has been often source of confusion. In this paper we start from a general overview about the main objectives and 1
I’m in debt with Prof. M. Fleurbaey, Prof. F. Stroffolini and Prof. D. Van de Gaer for very useful discussions. Errors are mine.
interests which may be associated to income mobility. These preliminary observations are compulsory in order to allow for the identification of the main reference field for this paper. In such a framework, we focus on both ex-ante and ex-post social evaluation of intragenerational mobility. We consider main existing approaches to the definition of the ex-ante welfare function. As it will be clearly observed, the crucial point turns out to be the aggregation of individuals’ welfare. With this objective in mind, two different implementations of the social welfare function (Bergson, 1938) are considered. In the social welfare approach it is shown that uncertainty 2 has very strong theoretical implications and it can be exploited (measured) in order to understand if income mobility can be regarded as socially desirable or not. As it will be discussed section 2, such a dilemma makes sense for the intra not intergenerational context. Adding ex-ante considerations to the traditional ex-post ones, we show that income mobility is not required to produce the same qualitative and quantitative effects over different income distributions. That is, given two societies, the same mobility transformation may be welfare improving for one society, not the other. For instance, initial inequality may be well affecting the desirability of income mobility. Also, given an impartial observer, it may be the case that populations differ in their subjective characteristics. 3 A strong evidence about heterogeneity comes from Alesina et al. (2001), where it is clearly showed that Americans tend to associate poverty to low effort, while Europeans tend to associate it to unluckiness. There is no need to suppose that one or the other population is more rational than the other, instead, the focus may be better on main sources of heterogeneity among these two economic systems. In other words, it may be the case that even if Germany and United States strongly differ in terms of income mobility, both keep a welfare optimal mobility structure. We also discuss the theoretical and philosophical background behind the introduction of uncertainty within a social welfare function. In particular, it is observed that the pure SDM approach might not suit the definition of welfare ex-ante under uncertainty. In this context, uncertainty can be properly handled through the implementation of the impartiality requirement in Harsanyi (1955). That is, impartiality is regarded as an ethical value judgement. In line with Harsanyi (1977), we assume that the impartial observer is sympathetic to individual interests ex-ante. The main consequence behind this assumption is that the degree of aversion to uncertainty belongs to the private sphere of
2
In this paper we stick with the term uncertainty even if the main framework would suggest the term risk as more appropriate (Ellsberg, 1961). The main rationale behind this decision is that, basically, individuals face distinct lotteries in the reality, whose probabilities are not expected to be objectively defined. 3 Individual subjective characteristics are indirectly involved when the idea of impartiality is implemented in Harsanyi’s sense.
2
individuals, i.e. it is not a social choice. 4 From these observations and from a basic re-interpretation of main existing contributions in this field (Atkinson, 1983; Dardanoni, 1993; Hammond, 1983), we move to the definition of a methodology for the evaluation of social desirability of income mobility effects. This paper is organized as follows. In section 2 the main field of interest of this paper is established. Main contributions are re-interpreted in such a way to highlight their importance for social desirability evaluations. In section 3 the main foundations for social welfare analysis in presence of uncertainty are discussed. Also, main theoretical and analytical difficulties are highlighted. In section 4 we propose an analytical procedure in order to investigate quantitatively social desirability ex-ante and ex-post of income mobility by the use of the social abbreviated welfare function. Basic properties of socially relevant indices are discussed in section 5. Section 6 concludes.
2
A multi-faceted concept
In dealing with income mobility there are several definitions which cannot be escaped, because they strongly affect main research objectives. First of all, it is opportune to distinguish between intra and intergenerational income mobility. Intragenerational income mobility measures changes in income over time (usually yearly) for the same income unit. Intergenerational income mobility, instead, measures the change in income among two different income units (usually father and son). Mainly, the latter is oriented to the recognition of the the role played by equality of opportunity. 5 One of feasible approaches to this topic consists of assuming that intergenerational mobility information can be exploited as a proxy for effort (O’Neill et al., 2002). A rich income unit today coming from a rich family is expected to be characterized by lower effort than a rich income unit today coming from a poor family. Then, most of existing theories of justice would say that mobility is always welfare improving (monotonicity). In addition, it may be assumed without loss of generality that optimal intergenerational mobility consists of perfect mobility, in the sense that, whatever the origin (income class), each individual has equal opportunities to move to each destinations in the second period (Prais, 1955). 4
As it has been observed later on, this fact does not exclude radically the feasibility of the a pure social choice approach, even if more articulated definitions of uncertainty may be required. 5 This concept is strictly related to Sen’s contribution. It is not inequality itself which really matters, but inequality of basic capabilities. Similarly, it is opportune to distinguish among inequality ex-ante and inequality ex-post. The former concerns with equality of opportunity, while the latter may require a proper evaluation of abilities, effort and capabilities.
3
From the other side, it is almost clear that intragenerational mobility may not be aimed to evaluating equality of opportunity (Van de gaer et al., 2001). Instead, it is the elaboration of dynamically consistent evaluation of equity and efficiency that really matters. This issue is as much relevant as the previous one. “... a given extent of income inequality in a rigid system in which each family stays in the same position in each period may be more a cause for concern than the same degree of inequality due to great mobility”, (Friedman, 1962). This is particularly evident in poverty analysis, where the real concern is with poverty persistence more than poverty itself. In this context it would be at least risky to claim that intragenerational income mobility is always welfare improving, and, as a result, it might be opportune investigating social desirability of income mobility. 6 Within the same literature on measurement, many researchers have been interested in the definition of an efficient mobility measure, where efficiency may refer both to normative or statistical characteristics (Shorrocks, 1978a; Chakravarty et al., 1985; Conlisk, 1990). 7 Others have been investigating income mobility effects when mobility issues are involved in welfare analysis (Atkinson, 1983; Dardanoni, 1993; Gottschalk and Spalaore, 2002). It’s straightforward that measuring income mobility is not equivalent to measuring social welfare in a dynamic scenario. In the last approach, the “irrelevance” of exchange mobility for social welfare orderings has been the main source for the apparent abandonment. 8 In this paper we are mostly concerned with the effects of intragenerational income mobility for social welfare in the random scenario. The basic starting point is that intragenerational mobility is not directly relevant for social welfare purposes, but it is crucial for a proper evaluation of equity and efficiency in a dynamic sense. 9 Income mobility is source of uncertainty and, as a result, it might cause a welfare loss ex-ante. Also, income mobility (especially exchange mobility) allows for inequality reduction in multi-period analysis, and, as a result, it is welfare improving ex-post. Even if the field of interest for this paper has been strongly restricted, it is still necessary to to draw an additional separating line that is already implicit in the last sentence. Let’s focus for simplicity to the case of two periods. Given 6
Jarvis and Jenkins (1998) highlights the fact that in the existing literature there is still no agreement about “goodness” or “badness” of income mobility. 7 Within this literature it may be argued that there exists a strong difference among welfaristic and statistical measurement. In the first case, mobility measures are defined from the variation of welfare due to “isolated” income mobility’s changes. The second approach, instead, focuses since the beginning on the definition of measures by which main desired properties are not violated. 8 Exchange mobility refers to switching in income positions. Structural mobility is null when there is no change in the income distribution over time. On the irrelevance of exchange income mobility see Atkinson (1983) and Dardanoni (1993). 9 From now on with the term mobility we intend intragenerational mobility.
4
a homogeneous population N := {1, ..., n}, we indicate with xt , xt+1 ∈ V I ), where W and W I indicate respectively welfare under actual income mobility and welfare under the hypothesis of perfect immobility. 38
4.1 Ex-post Restricting our attention to exchange mobility it must be the case that mobility produces one straightforward effect over ex-post welfare, that is an inequality effect. In this context, the income mobility effect over welfare can be measured using inequality indices and comparing actual inequality over aggregated periods with aggregated inequality in absence of income mobility, i.e. I(xt,t+1 ) − I(xt,t ) where xt,t+1 = xt + xt+1 and xt,t = xt + xt . Notice that in a two-periods economy and assuming homogeneity of degree 0 of the inequality index, I(xt,t+1 ) − I(xt,t ) = I(xt,t+1 ) − I(xt ) where the right hand side is just the income mobility effect assumed in Shorrocks (1978b). If only inequality is ex-post affected by income mobility, then the social effect can be equivalently measured in terms of welfare as U (xt,t+1 ) − U (xt,t ), where the last term consists of ex-post welfare in the case of perfect immobility. Notice that we are referring to social preferences, that is, the SDM is not sympathetic to individual interests ex-post, i.e. moral observer. Recalling V (µ, I) = 12 µ(1 − I) the measure of social desirability of income mobility ex-post can be simply defined as Dp = µt,t+1 (It,t − It,t+1 ) (17) 35
In line with the existing literature, we focus on mean-independent orderings, even if it is worth noticing that homogeneity of degree one as well as specific intermediate responses to changes in mean lead to a unique definition of -V and -I too. 36 Notice that such specification of - comes directly from the generalized Lorenz V dominance (Shorrocks, 1983), i.e. second-order stochastic dominance. 37 We just replace the original Gini’s index with Atkinson’s one. Two notations: the latter is mostly introduced for simplicity, even if the use of the Gini would be more rigorous, ii) the use of Atkinson’s index satisfies the three requirements above and all implicit conditions discussed in Kondor (1975). 38 This methodology is similar to the welfaristic measurement of inequality and mobility (Chakravarty et al., 1985; Ruiz-Castillo, 2000).
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(neglecting the irrelevant multiplicative constant 1/2) which given only exchange mobility can be re-written as d(ε)
d(ε)
d(ε)
d(ε)
Dp = xt,t+1 − xt,t = xt,t+1 − 2xt d(ε)
(18)
d(ε)
where xt,t+1 , xt are respectively the equally distributed equivalent income calculated over (xt +xt+1 ) and xt . Given constant mean income µ and mobility of the exchange kind, it must be the case that xdt,t is the minimum value of the welfare indicator in a two period economy, that is, Dp is non-negative. Summing up, even if the abbreviated social welfare function is implemented instead of the Atkinson’s welfare function (2), the effect of income mobility is qualitatively the same, i.e. exchange income mobility can be only welfare improving ex-post. More complex is the definition of ex-ante social desirability of income mobility. In line with the discussion above, it is opportune distinguishing between the uncertainty and the expected inequality effect. In addition, there are at least three approaches which might be invoked.
4.2 Ex-ante 4.2.1 Impartial observer Let’s assume that individuals have homothetic preferences with degree of relative risk aversion λ as estimated from empirical data and xd(λ) is the empirically estimated certain equivalent income. In addition, there exists an observer, who evaluates income at time t+1 under uncertainty. Also, he/she does not impose his/her own preferences (E[U (·)]), but he/she is sympathetic to individual interests over time, that is, he/she - ex-ante - refers to individual preferences (E[ui (·)]). Given identical individual subjective preferP d(λ) ences W = U (X) = N1 i u(xi,t , xi,t+1 ). In line with Dardanoni (1993), additive separability over time can be assumed without loss of generality, that is d(λ) d(λ) u(xt , xt+1 ) = u(xt ) + u(xt+1 ). 39 In this framework, exchange mobility is necessarily irrelevant, as it has been already shown by Dardanoni (1993). Indeed, W − WI =
X
πi,t u(xdλ i,t+1 ) −
X
πi,t u(xi,t )
(19)
i
i
P
P
I where by construction i πi,t u(xdλ i πi,t+1 u(xi,t+1 ), that is, W = W . i,t+1 ) = This result is not surprising. Even if additive separability can be justified by invoking subjective, not ethical preferences, in this framework a perfect 39
Here additive separability is assumed over subjective individual preferences, not ethical ones. Also, the discount factor has been assumed 1 for simplicity.
15
compensation arises between the uncertainty and inequality effect as discussed in section 2.
4.2.2 Pure SDM Let’s assume that the SDM has homothetic preferences and he/she is averse to both inequality and uncertainty. 40 The pure SDM approach is not concerned at all with time-averaged inequality as equally distributed equivalent incomes are taken with respect to each period and state. Recalling (1) and assuming additivity over equally distributed equivalent incomes in each social state, 41 then X πs dε 1−λ W = (xdε (20) t + xt+1,s ) 1 − λ s where πs can be determined from the estimated transition matrix (conditioned joint density function), that is, πs =
n Y
πs˜|i
(21)
i=1
where there are nn 42 possible social states (s) and s˜ associates a specific j (income position at time t+1) to each i (income position at time t) depending on the combination s (social state). 43 We say that combination s belongs to the set of combinations Ω if it is obtained from a transition matrix of the exchange kind (πt = πt+1 ). Once again, the mobility effect can be defined with respect to the case of perfect immobility, that is W − W I . However, the definition of an ordering may be not straightforward in this context. Both ε, λ > 0 are socially defined, that is, they indicate respectively aversion to inequality and uncertainty of the SDM. In addition, if a social abbreviated evaluation function has to be defined then it cannot be transcended that it is concerned with a real lottery problem (Es [·]). In this sense, the ordering has to be defined recurring to second order stochastic dominance. In the existing literature, it is well-known that second order stochastic dominance and Generalized Lorenz dominance (Shorrocks, 1983) are equivalent. 44 However, once again for simplicity, we refer to V = µ(1 − I) with I indicating 40
As we will argue later on, aversion to uncertainty is not straightforward in this context. 41 It’s worth observing that in (20) it is not additivity the source of loss of interest in time-averaged inequality, but separability among money measures of welfare at time t and t+1. 42 n indicates the number of income classes. 43 That is, in (21), s ˜ is not required to be the same for each i. 44 In finance, the Gini’s has been often adopted as a measure of dispersion (Eftekhari and Satchell, 2000).
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Atkinson’s index. It can be shown that a DH = xdλ t(dε),t+1(dε) − xt(dε),t(dε)
(22)
with 1 Ã ! 1 Ã ! 1 1−λ 1−λ 1−ε 1−ε X X 1−ε s 1−ε π π x + xdλ = π x s i,t i,t t(dε),t+1(dε) i,t i,t+1 s i i X
(23)
a Then, DH represents a measure of social desirability of income mobility. It can be both positive or negative. However, in line with Hammond (1983), it cannot be transcended that many inconsistencies arise in this approach. The SDM is not immediately concerned with uncertainty over income fluctuations, instead λ would indicate the social degree of aversion to uncertainty over different income vectors at time t+1. That is, the SDM is not concerned at all with who get what, instead, he/she is concerned with his/her lottery over possible income vectors at time t+1. Also, there is no interest in time-averaged inequality, which, as it has been observed in section 2, represents the crucial information when dealing with intragenerational income mobility.
4.2.3 Mixed approach In this approach we define λ as the degree of aversion to income uncertainty for each individual subjective preferences and ε as the degree of aversion to timeaveraged inequality as calculated over lifetime certainty equivalent incomes. Recalling (13) and considering the corresponding abbreviated social welfare function as before in a two periods economy, the income mobility effect can be defined as V − V I = µt,t+1(dλ) (1 − It,t+1(dλ) ) − µt,t (1 − It,t )
(24)
dε DCa = xdε t,t+1(dλ) − xt,t
(25)
by which with xdε t,t+1(dλ) indicates the equally distributed equivalent income (ε) taken over the certain equivalent income (λ), i.e.
xdε t,t+1(dλ) =
X i
πi,t xi,t +
X j
πj|i x1−λ j,t+1
1 1−λ
1 1−ε 1−ε
(26)
By easy algebraic calculations DCa = (µt,t+1(dλ) − µt,t )(1 − It,t+1(dλ) ) + µt,t (It,t − It,t+1(dλ) ) 17
(27)
where the first factor on the RHS is the uncertainty effect (change in mean income given constant inequality), while the latter is the ex-ante time-averaged inequality effect (change in inequality given constant mean income). Then, the overall ex-ante effect of mobility, DCa , can be decomposed in uncertainty and inequality effect, DCu = (µt,t+1(dλ) − µt,t ) DCi =
à dε x
xdε t,t+1(dλ) µt,t+1(dλ)
(28)
!
t,t+1(dλ)
µt,t+1(dλ)
xdε − t,t µt,t µt,t
(29)
It’s straightforward that DCu ≤ 0 and DCi ≥ 0, i.e. income mobility causes a negative effect over the cake ex-ante and a positive effect over ex-ante timeaveraged inequality. In addition, it’ worth observing that, if income at time t is involved too, (9) and (14) are not equivalent any further at λ = ε. This highlights that the ex-ante mobility effect is not necessarily null at λ = ε. 45
5
Indices
In the previous section it has been rapidly observed that by the use of the abbreviated social welfare function the impartial observer’s approach leads to social irrelevance of exchange income mobility. This result is not likely to occur in the pure SDM approach, but the interpretation of the basic procedure would be not straightforward at all. Things turn out to be extremely different when the mixed approach is preferred. Indeed, the interpretation of basic results is immediate and it might be well the case that DCa 6= 0. The main assumption behind the mixed approach is that utilitarianism and social choice approach cannot fully deal with welfare comparisons when both inequality and uncertainty aversion is involved. The increasing interest in the social choice approach to welfare economics has been mostly justified by the need for ethical not subjective foundation for inequality aversion. However, we argue that under uncertainty conditions it might be misleading assuming that uncertainty belongs to the sphere of social choices. Instead, aversion to uncertainty can be assumed only with respect to subjective preferences. In this context, in order to answer our intial question, there are two indices which deserve particular attention, i.e. Dp and DCa . In particular, we refer to the corresponding relative measures. Ex-post p DR =
dε xdε t,t+1 − xt,t µt,t+1
45
(30)
It is trivial that if income at time t is considered, then the mixed and impartial observer approach coincide only if λ = ε = 0.
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p that is equivalent to Shorrocks (1978b). Also, DR ∈ [0, 1]. Ex-ante dε xdε t,t+1(dλ) − xt,t a DR = µt,t+1 a where DR ∈ [−∞, 1].
(31)
Proposition 5.1 a R 0 if and only if DR
µt,t+1 − µt,t+1(dλ) It,t − It,t+1(dλ) Q µt,t+1 1 − It,t+1(dλ)
(32)
Proof 5.1 Straightforward from (31). In other words, mobility is ex-ante socially desirable if and only if the relative welfare loss in terms of uncertainty is lower than the relative welfare gain in terms of expected time-averaged equality. Both relative indices have very straightforward interpretations. The former indicates the proportion of reduction in inequality ex-post due to income mobility, which in our framework is equivalent to the proportion of welfare gains ex-post due to income mobility. The latter, instead, indicates the difference among re-scaled time-averaged equality in presence of income mobility and time-averaged equality under the assumption of perfect immobility. a From section 5, the decomposition of DR in terms of inequality and uncertainty effect leads to the following relative indices, !
Ã
xdε xdε µt,t+1(dλ) − µt,t t,t+1(dλ) t,t+1(dλ) − = (1 − It,t+1(dλ) ) = µt,t µt,t µt,t+1(dλ) dε dε xt,t+1(dλ) xt,t i DR = It,t − It,t+1(dλ) = − µt,t+1(dλ) µt,t
u DR
(33) (34)
whose interpretations are straightforward once again. All the indices discussed in this section are relative, replication invariant and symmetric. In addition, the following table summarizes the possible behavior of each index at specific transition mechanism PI
PE
PM
p DR
0
>0
It
a DR
0
Q0
It
u DR
0
0
It
Table I: PI, PE and PM indicate resp. perfect immobility, perfect equality of opportunity and perfect mobility.
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a The possibility for DR to be negative at PE depends on the joint density function and the two parameters of aversion (inequality and uncertainty). a Given xt , xt+1 ∈
i
X
(xdε )1−ε +
i
P
X
x1−ε =2 i
i
X
(xdε )1−ε >
i
X
(xi + xi )1−ε
i
P
= i (xdε )1−ε . For 0 < ε < 1 we get a concave homoBy construction i x1−ε i thetic transformation, so that every translation is never more affecting for the RHS, i.e. X
(xi + xdε )1−ε ≥
i
X
X
i
i
(xdε + xdε )1−ε =
(xi + xi )1−ε
In order to avoid any misunderstanding, Proposition (5.2) cannot be generalized out of the case πj|i = k ∀ i, j. 46 In general, Proposition (5.1) applies.
6
Conclusive remarks
We now go back to our initial question: is income mobility socially desirable? In the existing literature it would not be very easy to find an answer to such a question. In Jarvis and Jenkins (1998) it is clearly highlighted that there is still much to do in order to find an answer. In this paper we have tried to highlight why income mobility can be definitely regarded as a multi-faceted concept. In particular, we have been focusing on 46
When focusing on exchange income mobility, PE (πj|i = k ∀ i, j) represents a particular case because it cancel out the dependency of the certain equivalent xdλ with respect to the initial income distribution.
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intragenerational exchange mobility, that is, the change in income over time for the same income unit, excluding the possibility for structural changes in the income distribution. In this context, it has been shown that there are at least two different approaches which deserve particular attention: ex-ante and ex-post. In the traditional ex-post framework there are no doubts about consistency of the SDM approach in evaluating social desirability of income mobility. In this context, income mobility can be definitely labelled as “good”. This statement is strictly related to the use of ethical preferences by which the assumption of additive separability would be highly misleading (Atkinson, 1983). In the ex-ante framework, instead, things turn out to be more complex. Both the SDM and the utilitarian approach do not suit the definition of welfare ex-ante under uncertainty conditions. In this sense, we have proposed a mixed approach, by which both uncertainty and inequality are treated for their basic nature. In this sense, inequality aversion belongs to the sphere of SDM’s choices, while aversion to uncertainty better suits individuals’ subjective preferences. In this context, income mobility is not necessarily socially desirable, that is, the uncertainty effect might be overwhelming with respect to the ex-ante time-averaged inequality effect. In line, with the ex-post approach, income mobility is still welfare maximizing at perfect immobility, which might be labelled as More’s society, where minimum uncertainty comes at minimum time-averaged inequality. The possibility of a negative ex-ante social desirability of income mobility is not without any implication for answering our initial question. If the overall social desirability of income mobility might be defined as the aggregation of ex-ante and ex-post effects, the possibility of negative desirability ex-ante highlights that assuming goodness of income mobility a priori would be highly misleading. In addition, policy recommendations would strongly differ dependa ing on the sign of DR . Indeed, an increase in exchange income mobility (flexia bility in the earnings market) may be recommended for a society with DR > 0, a while the same recommendation is not generally valid if DR < 0. Future research interest will be paid to the possibility for aggregation of exante and ex-post mobility effects as well as to the generalization of the two periods-economy.
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