INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY

Transcription

1 INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY VANI K. BOROOAH University of Ulster JOHN CREEDY University of Melbourne This paper incorporates poverty persistence in a measure of aggregate poverty over two-periods by decomposing the Foster et al. (1984) class of poverty measures into those temporarily in poverty and those in poverty in both periods. An additional weight is added to the permanent component in forming an aggregate poverty measure over both periods; this weight re ects the degree of povertypersistence aversion. The effect on aggregate poverty of mobility between permanent and temporary poverty is found to be unambiguous only in the case of the headcount poverty measure. Simulations are used to investigate the relationship between poverty and mobility. The effects of two different types of mobility (random proportional income changes and a systematic regression towards or away from the median) are isolated. I. Introduction The theory of poverty measurement is usually cast in a single-period framework within which the aggregate level of poverty is determined by incomes in that period. However, there is increasing concern with issues relating to the persistence of poverty; see Bane and Elwood (1986), Mof t (1992), Stevens (1994), Headey and Krause (1994) and Ravallion (1996). This concern is associated with the discussion of `welfare-dependency' and of the growth of the `underclass', which are based upon a distinction between the temporarily and the permanently poor. It is not clear how the length of time that individuals spend below the poverty line should in uence judgements about aggregate levels of poverty. The primary purpose of this paper is to provide, within a two-period context, an analytical framework by means of which povertypersistence might be incorporated into an index of aggregate poverty. In a two-period model, developed in Section II, aggregate poverty over both periods may be written as a weighted average of poverty in each of the periods. This allows a decomposition of aggregate multiperiod poverty in terms of the poverty of persons who are temporarily poor (poor in one period but not in the other) and the poverty of those who are permanently poor (poor in both periods). In calculating multi-period poverty, a higher weight can be attached to the poverty of the latter group by means of a coef cient which represents the degree of aversion to `poverty-persistence'. It should be stressed that all poverty measures are `non-welfarist'; that is, they are based solely on individuals' incomes (or a measure of resources) rather than welfare levels, and the # Blackwell Publishers Ltd., 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden MA 02148, USA and the University of Adelaide and Flinders University of South Australia.

2 1998 INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY 37 poverty line is imposed by the judge (rather than being a parameter of individuals' utility functions). 1 A major analytical result, derived in Section II, is that transfers of persons from temporary to permanent poverty lead, even with positive aversion to poverty-persistence, to an unambiguous increase in aggregate multi-period poverty only when poverty is measured by the head-count ratio. When more complex measures are deployed an increase in numbers in permanent poverty leads, within the context of a given number of total poor, to an unambiguous increase in aggregate poverty only under very restrictive assumptions. Consequently, simulation experiments were conducted in order to shed light on the relationship between mobility, temporary and permanent poverty and the level of aggregate poverty. These experiments are described in Section III. Brief conclusions and discussion of related issues are in Section IV. II. Poverty Decomposition in atwo Period Context a) A decomposition of aggregate poverty Consider a framework in which there are two time periods (t ˆ 1, 2) and N persons in each time period, their incomes being arranged in ascending order as: y 1t, y 2t... y Nt.Ifz 1 and z 2 are the poverty lines for periods 1 and 2, then S 1 ˆfjjy j1 < z 1 g and S 2 ˆfjjy j2 < z 2 g are the sets of poor persons in periods 1 and 2 respectively. The sets S 1 and S 2 may be written as S 1 ˆ (S 1 \ S 2 ) [ (S 1 \ S 2 ) ˆ A [ C (1) S 2 ˆ (S 1 \ S 2 ) [ (S 1 \ S 2 ) ˆ B [ C (2) where S 1 and S 2 are the converse sets of S 1 and S 2 respectively and A, B, C are disjoint sets. Using the class of poverty measures de ned by Foster et al. (1984), the measure of poverty in each period, P t (t ˆ 1, 2) is given by P t ˆ 1 X z t y á jt N z t (3) j2s t Let P T t and P P t denote respectively the poverty measures for period t including those who are poor only in period t and those who are poor in both periods. Then the above decompositions (1) and (3) can be used to show, for example, that Hence, in general P 1 ˆ 1 N X z 1 y á 1t 1 X z 1 N j2a j2c z 1 y 1t z 1 á ˆ P T 1 PP 1 (4) P t ˆ P T t P P t (5) 1 This means, for example, that there is no role for the individual welfare metric which might otherwise provide an avenue for introducing an aversion to poverty persistence on the part of individuals. Attempts to include a poverty line in individual utility functions include Lewis and Ulph (1988) and Creedy (1997). For a discussion of the uses of both inequality (using welfarist measures) and poverty (using non-welfarist measures) in a social evaluation function see Atkinson (1987).

3 38 AUSTRALIAN ECONOMIC PAPERS MARCH An aggregate measure of poverty over the two periods, H, may be de ned in terms of a weighted average of the P t s with weights w t so that H ˆ Ó t w t P t. Using (5) this becomes H ˆ X2 w t P T t X2 w t P P t (6) This illustrates that such an aggregate measure gives the same weight to permanent and temporary poverty in each period. A degree of `persistence-aversion' may be introduced using the coef cient ã < 1, so that H is re-de ned as H ˆ X2 w t P T t ã X2 w t P P t (7) Hence, when ã. 1 the aggregate poverty of the permanently poor is given a greater weight than that of the temporarily poor. The persistence-aversion parameter echoes the use of an inequality-aversion parameter by Atkinson (1970). Poverty aversion on the part of the policy maker may stem from two sources. It may stem from equity considerations through a feeling that it is unfair to suffer persistently from poverty. Alternatively, it may stem from ef ciency considerations related to the belief that persistent poverty impedes the ef cient functioning of society and involves resources being diverted to harmful social consequences. Much of the debate about a `social underclass' re ects such concerns; see Smith (1992). b) Effects of mobility on aggregate poverty In order to examine the effects of mobility on the aggregate poverty measure, consider rst the head count measure where á ˆ 0. De ne M t as the number of poor persons in period t. Denote by M A, M B and M C respectively the number of persons who are poor in: period 1 but not in period 2; in period 2 but not in period 1; both periods. Then M 1 ˆ M A M C and M 2 ˆ M B M C. De ne v t ˆ M C =M t as the proportion of the total poor, in period t, who are `permanently' poor (poor in both periods) and let m t ˆ M t =N be the proportion in poverty in period t. Then H ˆ X2 w t (1 v t )m t ã X2 w t v t m t (8) An increase in the number of people who are permanently poor, with the number of poor in each period, M t, unchanged, produces a change in H given H ˆ (ã 1) X2 m t w t C M t M C ˆ 0ifã ˆ 1 M C. 0ifã. 1: In the case where á ˆ 1, the mean poverty gaps are relevant. Let g t denote the mean poverty gap, expressed as proportions of the poverty line, of persons who are poor in only period t, and g C t the mean poverty gap of persons in set C (poor in both periods) calculated using period t's incomes. Then aggregate poverty is

4 1998 INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY 39 H ˆ X2 w t (1 v t )m t g t ã X2 w t v t m t g C t (10) In the case where, in each period, the incomes of all poor persons are equal, it can be shown H=@M C. 0ifãˆ1 H=@M C. 0ifã. 1. However, in the general case, when poor incomes are unequal, it is not possible unambiguously to attach a sign H=@M C since the change in the mean poverty gaps consequent upon a change in the number of permanently poor persons, M C, depends upon the incomes of poor persons transferring between permanent and temporary poverty, relative to the incomes of poor persons not making such a transfer; for details see Borooah and Creedy (1994). For example, suppose the distribution of income is such that the permanently poor have the lowest incomes so that incomes arranged in ascending order are, for period t, givenby y 1,t,..., y MC,t, y M C 1, t,..., y M t. Then g C t. g t. Suppose the (M C 1)th person transfers in period 1 from temporary to permanent poverty; that is such a person who was poor in period 1 only is now also poor in period 2. Then g1 C=M C and g 1 =M C are both negative because the mean incomes, in period 1, of both the permanently and the temporarily poor both rise. On the other hand, with a xed number, M 2, of poor persons in period 2, the increase in M C in period 2 (which is the result of a person previously poor in period 1 entering poverty in period 2) is accommodated by a reduction in M B, through a non-poor person in period 1 escaping poverty in period 2 as well. If this entry and escape in period 2 occurs at the highest end of the income distribution (in period 2) of, respectively, the permanently and the temporarily poor, then g 2 =M C. 0 and g2 C =M C, 0. Different scenarios for transfer between permanent and temporary poverty will yield different signs for the relevant partial derivatives. Although it is not possible to predict the direction of change in H consequent upon a change in M C, conditions under which change in the value of ã, the persistence aversion parameter, affect the magnitude of change, whatever the direction of change might be, can be derived. It can be shown that suf cient conditions H=@M C )=@ã to be positive g t =@M C, g C t =M C for t ˆ 1, 2. In other words, provided that the change in the mean poverty gaps (expressed as a proportion of the poverty line) of the permanently poor, consequent upon a transfer between numbers of permanent and temporary poverty, is suf ciently small (informally, the percentage change in the mean poverty gap ratios of the permanently poor) an increase in the value of ã will attenuate the magnitude of the change in H consequent upon the above transfer. When á ˆ 2, the poverty measures, P T t and P P t are P T t ˆ (1 v t )m t fg 2 t (1 g2 t )r2 t g (11) P P t ˆ v t m t f(g C t )2 (1 (g C t )2 )r 2 C, t g (12) where r t is the coef cient of variation of income of those poor in period t only, and r C,t is the coef cient of variation of period t's incomes of those poor in both periods. The aggregate poverty measure in this case can be obtained by substituting equations (11) and (12) into equation (7) As with the earlier case when á ˆ 1, it is not possible unambiguously to H=@M C. However, it is again possible to obtain conditions under which the size of the change is magni ed through changes in the value of ã, the persistence-aversion coef cient. A suf cient condition H=@M C )=@ã to be positive can be obtained as in the case when á ˆ 1. Provided, as a consequence of a transfer of persons between the temporarily and permanently poor, the changes to the poverty gap ratios and the inequality coef cients

5 40 AUSTRALIAN ECONOMIC PAPERS MARCH of the two groups are suf ciently small, an increase in the persistence-aversion parameter ã magni es the effects of this transfer on changes to aggregate poverty; for details see Borooah and Creedy (1994). III. Income MobilityandPoverty The previous section showed that the effects of income mobility on permanent and aggregate poverty are very complex, particularly for poverty measures which depend on the average poverty gap and the dispersion of income among the poor. The present section uses a model of income mobility in order to obtain some comparative static results using simulation methods. An advantage of the model is that it distinguishes between two basic types of relative mobility: income changes can have both a systematic inequality increasing or reducing component, and a stochastic component. Let y it denote, as before, individual i's income in period t and let ì t denote the arithmetic mean of logarithms of income in period t; then m t ˆ exp (ì t ) is the geometric mean. The distribution of income in period 1 is given exogenously, and the mobility process generating period 2's incomes is written as â y i2 ˆ yi1 exp (ì 2 u i ) (13) m 1 where u i is a random variable assumed to be distributed as N(0, ó 2 u ). This represents the stochastic component of income changes. The parameter â measures the degree of `Galtonian' regression towards, or away from, the geometric mean. It in uences the extent to which those with relatively higher incomes experience a relatively lower percentage change from year one to year two. If â ˆ 1 there is no regression and mobility simply involves each individual receiving a random proportionate change; for further discussion of this type of model, see Creedy (1985, pp. 33±39). Equation (13) can be written as log y i2 ì 2 ˆ â(log y i1 ì 1 ) ì i (14) so that taking variances gives the relationship between the variance of logarithms, ó 2 t,of income in each period as ó 2 2 ˆ â2 ó 2 1 ó 2 u (15) Hence income inequality, as measured by the variance of logarithms, is constant if ó 2 u ˆ ó 2 1 (1 â2 ). Using the assumption that income in the rst period is lognormally distributed as Ë(ì 1, ó 2 1 ), a simulated set of y i1s can be obtained with random numbers, v i, drawn from an N(0, 1) distribution, using y i1 ˆ exp (ì 1 v i ó u ). Equation (13) can be applied for assumptions about â, ó 2 u, and ì 2 in order to obtain the corresponding y i2 s. The results reported below assume that ì 1 ˆ ì 2 ˆ 9:5 and ó 2 1 ˆ 0:5. Furthermore the poverty lines in each period are set equal to The income distribution in the rst period is assumed to be xed, so the standard poverty measures applying to that period are constant whatever the mobility process. However, the temporary and permanent components P1 T and PP 1 for that period vary as the mobility process varies. Changes in the parameters â and ó 2 u have different effects on the income distribution of the second period. Increases in either â or ó 2 u unequivocally increase the dispersion of income in the second period. However, it is not immediately obvious how such

6 1998 INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY 41 increases will affect the various components of poverty. This is because of the way in which the two components of relative income mobility interact. The links between the mobility processes, which have a clear effect on income dispersion in the second period, and the various poverty measures are complicated by the role of the covariance between incomes in the two periods and the various integrals required. The effects on the poverty measures of changing the mobility process can, however, be examined numerically. Inspection of results for a variety of values of â and ó 2 u suggests that a speci cation with poverty expressed as a linear function of â and ó 2 u, along with an interaction term, âó 2 u, provides a good approximation. Results are shown in Table I for each of the three poverty measures, depending on the value of á. The table shows regression results for two different dependent variables; these are the standard poverty measure for the second period, P 2, and the permanent component of aggregate poverty, Ó t w t P P t. These results are based on simulated populations, with N ˆ 2000, for variations in ó 2 u from 0 to 0.6 in steps of 0.1 and variations in â from 0.4 to 1.6 in steps of 0.2. Hence the number of `observations' used in the regressions is 49 in each case. These variations in the mobility parameters are very wide, as can be seen by considering the implied values of ó 2 2, the variance of log-income in period 2. The estimated `t' values are given in parentheses. De ne the coef cients on â, ó 2 u and the interaction term, âó 2 u, for the regression involving P 2 as ä 1, ä 2 and ä 3 respectively, and the corresponding coef cients in the regression involving the permanent component of aggregate poverty as è 1, è 2 and è 3. Then the changes in mobility parameters which leave P 2 unchanged are given, after total differentiation, by dâ dó 2 ˆ ä 2 ä 3 â u P 2 ä 1 ä 3 ó 2 (16) u Table I shows that for each poverty measure the value of ä 3 is negative, so the sign of the marginal rate of substitution in (16) is not unequivocal. However, substitution using the results from Table I shows that in the case where á ˆ 0 (the head count poverty measure) and â is relatively large, dâ=dó 2 u jp 2 is positive; otherwise it is negative. Hence, if there is substantial regression away from the mean, a further increase in â must be matched by an increase in ó 2 u in order to keep P 2 constant. But if there is regression towards the mean, an increase in â (implying a reduction in the degree of regression) must be matched by a decrease in ó 2 u, for the proportion in poverty in the second period to remain constant. The change in mobility parameters which leaves aggregate poverty unchanged is seen to be dâ dó 2 u H ˆ w 2ä 2 (ã 1)è 2 â(w 2 ä 3 (ã 1)è 3 ) w 2 ä 1 (ã 1)è 1 ã 2 (w 2 ä 3 (ã 1)è 3 ) When ã 1, corresponding to no aversion to permanent poverty, then it is clear that dâ=dó 2 u j P 2 ˆ dâ=dó 2 u j H. The difference between the two marginal rates of substitution increases as ã increases. Except for high values of â and the head count poverty measure, the substitution of values from Table I gives the result that dâ=dó 2 u j H is also negative. The absolute value of this marginal rate is less than that for which P 2 is constant. Hence changes which involve a reduction in â, which compensate for an increase in ó 2 u such that P 2 (and hence a conventional aggregate measure of poverty) is constant, result in a decrease in aggregate poverty when ã. 1. Alternatively, an increase in â, compensated by a fall in ó 2 u to keep P 2 constant, results in an increase in aggregate poverty when ã. 1. These results are consistent with what would be intuitively expected of regression away from the (17)

7 42 AUSTRALIAN ECONOMIC PAPERS MARCH Regressions of poverty on mobility compo- Table I nents Coef cient and á Poverty in 2 P 2 Permanent poverty w 1 P P 1 w 2 P P 2 á ˆ 0 constant (0.204) (2.869) â (23.779) (13.199) ó 2 u (17.481) (2.707) âó 2 u ( ) ( 4.333) R á ˆ 1 constant ( ) ( 2.255) â (83.061) (38.787) ó 2 u (45.105) (8.121) âó 2 u ( ) ( 8.314) R á ˆ 2 constant ( ) ( ) â (33.414) ( ) ó 2 u (14.228) (22.248) âó 2 u ( 4.143) ( ) R mean. It is therefore possible for a change in mobility characteristics to have no effect on the time pro le of a conventional single period poverty measure, yet to increase permanent poverty. This makes it important for the design of poverty alleviation policies to establish the precise pattern of income mobility and changes in that pattern over time. IV. Conclusions The aim of this paper has been to incorporate poverty persistence in a measure of aggregate poverty over two-periods. This was achieved by decomposing the Foster et al. (1984) class of poverty measures into two components, covering those temporarily in poverty and those in poverty in both periods. An additional weight was then added to the permanent component in forming an aggregate poverty measure over both periods; this weight re ects the degree of poverty-persistence aversion of the policy maker. It was found that the effect on the aggregate poverty measure of mobility between the categories of permanent and temporary poverty are not unambiguous (except in the special case of the headcount poverty

8 1998 INCOME MOBILITY, TEMPORARY AND PERMANENT POVERTY 43 measure). Simulations were then used, based on a process of income mobility, in order to investigate the relationship between poverty and mobility. The effects of two different types of mobility (random proportional income changes and a systematic regression towards or away from the median) were isolated. A parallel with analysis of poverty persistence is provided by the attitude of policy makers towards long-term unemployment. Most governments exhibit what may be termed, using the language of this paper, `duration aversion' in the sense that they regard the continued joblessness of persons a problem in its own right, though the degree of concern may differ across governments. However, conventional measures of unemployment, as with conventional measures of poverty, represent a snap-shot of labour force status and ignore the history of individual employment experience. There is increasing concern about the adequacy of conventional measures of unemployment; see, for example, Shorrocks (1993). In the context of poverty, Ravallion (1988) has considered the effect of income risk (variability) on poverty. The question he asked, that is closest to the present study, is: under what conditions does an increase in the riskiness of an income stream lead to an increase in poverty? He is concerned with the expected value of poverty, rather than with income mobility per se. The present analysis is explicitly concerned with the role of mobility in helping some people to escape poverty while causing others to enter poverty. It considers how the measurement of poverty is affected by the existence of an aversion to poverty persistence, when there is income mobility. It identi es the conditions under which an increase in the number of permanently poor, with a concomitant decrease in the numbers temporarily poor, leads to an increase in the value of the aggregate poverty index in the face of persistence aversion. But Ravallion identi ed the conditions under which an increase in risk leads to an increase in expected poverty. Available evidence suggests that poverty persistence is an important phenomenon, so that there is a clear need to allow for such persistence in measuring aggregate poverty. For example, given the observed correlation between unemployment and poverty, the increasing extent of long term unemployment in many countries suggests that poverty persistence may also be increasing. The link with unemployment also suggests that there may well be a regional dimension to aggregate poverty, given an aversion to poverty persistence. The results demonstrate the need to have information about the process of income dynamic, in order to design appropriate policies to alleviate poverty. R EFERENCES Atkinson, A.B. (1970), `On the Measurement of Inequality', Journal of Economic Theory, vol. 2, pp. 244±263. Atkinson, A.B. (1987), `On the Measurement of Poverty', Econometrica, vol. 55, pp. 749±764. Bane, M.J. and Ellwood, D. (1986), `Slipping Into and Out of Poverty: The Dynamics of Spells', Journal of Human Resources, vol. 21, pp. 1±23. Borooah, V.K. and Creedy, J. (1994), `The Temporary Versus the Permanently Poor: Measuring Poverty in a Two Period Context', University of Melbourne Department of Economics Research Paper, no Creedy, J. (1985), Dynamics of Income Distribution (Oxford: Basil Blackwell). Creedy, J. (1997), `Labour Supply and Social Welfare when Utility Depends on a Threshold Consumption Level', Economic Record, vol. 73, pp. 159±168. Foster, J.E. (1984), `On Economic Poverty: A Survey of Aggregate Measures', Advances in Econometrics, vol. 3, pp. 215±51.

9 44 AUSTRALIAN ECONOMIC PAPERS MARCH Headey, B. and Krause, P. (1994), `Inequalities of Income, Health and Happiness: The Strati cation Paradigm and Alternatives', in B. Bradbury (ed), Social Policy Research Centre Report and Proceedings, pp. 133±172 (Sydney: University of New South Wales). Lewis, G.W. and Ulph, D.T. (1988), `Poverty, Inequality and Welfare', Economic Journal, vol. 98, pp. 117±131. Mof tt, R. (1992), `Incentive Effects of the US Welfare System: A Review', Journal of Economic Literature, vol. 30, pp. 1±61. Ravallion, M. (1988), `Expected Poverty under Risk-induced Welfare Variability', Economic Journal, vol. 98, pp. 1171±1182. Ravallion, M. (1996), Issues in Measuring and Modelling Poverty', Economic Journal, vol. 106, pp. 1328±1343. Shorrocks, A.F. (1993), 'On the Measurement of Unemployment', University of Essex Department of Economics Discussion Paper, no Smith, D. (ed) (1992), Understanding the Underclass (London: Policy Studies Institute). Stevens, A.H. (1994), `The Dynamics of Poverty: Updating Bane and Ellwood', American Economic Review, Papers and Proceedings, vol. 84, pp. 34±37.