Swings in the Economic Support Ratio and Income Inequality by Sang-Hyop Lee and Andrew Mason 1

Transcription

1 Swings in the Economic Suort Ratio and Income Inequality by Sang-Hyo Lee and Andrew Mason 1 Draft May 3, 2002 When oulations are young, income inequality deends on the distribution of earnings and wealth among woring-age adults, marriage atterns, marital sorting, female labor force articiation, and childbearing atterns. The income of the elderly and the systems that rovide them with economic security are relatively unimortant. This is a simle matter of numbers. If only one in ten adults is elderly, overall levels of inequality are not much influenced by their economic circumstances. That is not to say that the elderly are economically disadvantaged in young (develoing) societies. Most low-income countries do not have comrehensive, state-sonsored systems that rovide suort to the elderly. In many countries, financial systems are so under-develoed that accumulating wealth for old age is not a realistic otion. But in many develoing countries, the etended family is an effective institution for roviding suort, financial and otherwise, to elderly who can no longer suort themselves. As oulations have begun to age, overall levels of inequality deend to a much greater etent on the economic circumstances of the elderly and the suort systems that eist to maintain their standards of living. Many of the eisting systems that rovide suort to the elderly may not be sustainable when one in three adults, or more, are elderly instead of one in ten. The vulnerability of ublic ensions rograms has received a great deal of attention in recent years, but the family suort system is vulnerable to the same forces. Given the seed of aging in many relatively low-income countries, it is liely that the economic status of the elderly and the overall level of inequality will deend as much on how aging affects the family suort system as how it effects formal, state-sonsored suort systems. The effect of aging on inequality has been the subject of a number of studies. Lam (1997) resents what has become the standard methodology for analyzing the effects on inequality of changes in age comosition and discusses many of the issues that arise. The standard aroach is to estimate age rofiles of the mean and variance of income, the log of income, or consumtion. The variance for the oulation is calculated holding the age rofiles constant and varying the roortions of the oulation in each age grou. In some cases, Lam and Levison (1992) for eamle, the method is alied to the earnings of individuals. In other instances, the method is alied to households using rofiles and oulation weights estimated by the age of the household head. Two recent eamles are 1 Presented at the 2002 Annual Meeting of the Poulation Association of America, Atlanta, GA, May The authors are Assistant Professor of Economics, University of Hawaii at Manoa and Professor of Economics, University of Hawaii at Manoa and Senior Fellow, East-West Center, resectively. Our thans to Turro Wongaren and Jonghyu Kim for their assistance. For additional information lease contact leesang@hawaii.edu or amason@hawaii.edu. The authors gratefully acnowledge the suort of the Rocefeller Foundation. 1

2 Schultz (1997) study of the variance of household income er adult in Taiwan, using the same data emloyed here and Deaton and Pason s (1997) analysis of consumtion for the US, Great Britain, Thailand, and Taiwan, using the same data for Taiwan as we emloy here. In both the Schultz and the Deaton and Pason studies, oulation aging is found to increase the level of inequality rimarily because the variance of income or consumtion for the elderly is greater than the variance of income or consumtion for the non-elderly. Analyses of the imact of changing age structure on household variables imlicitly assume that changes in age structure will not affect the age rofiles of the variables in question. In a oulation where multi-generation households are revalent, however, changes in age structure may have a large effect on the demograhic characteristics of households and the economic variables of interest. If the number of seniors increases relative to woring-age adults, at least one of three imortant household characteristics must change: the roortion of seniors living in etended households, the roortion of woring-age adults living in etended households, and/or the number of seniors er woring-age adult within etended households. Each of these changes has otentially imortant imlications for income inequality. A rise in the roortion of seniors living in etended households would, other things equal, reduce inequality by increasing the etent to which incomes are ooled. A rise in the roortion of woring-age adults living in etended households would have a similar effect. The imact of a change in the number of seniors er woring-age adult on income inequality within etended families deends on the characteristics of the income distributions of woring-age adults and seniors who form etended households. Changes in age structure have more than a comositional effect. They induce changes in the family suort system with otentially imortant imlications for income inequality. In this aer we resent a simle theoretical structure that The objective of this aer is to eamine the imact of aging on income inequality when the etended family is an imortant art of the suort system. We show that under a variety of circumstances oulation aging will lead to lower income inequality.... [add basic findings here] BACKGROUND The family offers an efficient institution for roducing, consuming, and redistributing resources among family members and across generations. In all societies, the family is the rimary institution through which resources are transferred to children from roductive members of the oulation (arents). As children age and become roductive, resources may begin to flow from children to their arents (Lee 2000). The etent and direction of resource flows between rime-age adults and their elderly arents is erhas more comle. Decisions by the family may be governed by altruism (Becer 1981). The 2

3 etended family may offer an efficient institution for echange. Elderly arents may care for their grandchildren in echange for financial suort. Or, adult children may care for the elderly arents in return for a bequest. Or, the family may rovide a system by which family members ool riss that they face in various asects of their daily lives (Kotlioff and Siva (1981)). The etended family may offer scale economies in consumtion that allows members to achieve a higher standard of living (McElroy and Horney (1981)). Of course, the roles are not mutually eclusive nor is the list rovided here ehaustive. Fulfilling these roles does not require co-residence. Family members can live in indeendent households and echange goods and services (money or time), but coresidence rovides an efficient means for carrying out these transactions. If the transactions are large and frequent and if they involve time, family members may choose to live together throughout their lives. If the transfers tend to be eisodic or confined to a articular eriod during the lifecycle, family members may choose to vary their living arrangements deending on the current circumstances. One of the emirical features of living arrangements in Taiwan is that for many seniors co-residence is not a ermanent feature. The roortion of seniors living with their adult children increases with the age of those seniors. In the absence of anel data, however, we cannot tell whether the decision by seniors to live with their children is a ermanent one or whether it is common for seniors to change living arrangements frequently. Some seniors may rotate their residence from one child to another so that they are ermanently in an etended household, but their children are not. In any event, the rise among seniors in etended living arrangements as they age is consistent with the view that the formation of etended families is a resonse to changing circumstances associated with aging, i.e., the riss associated with growing old. The riss faced by the elderly tae a number of forms. Older adults face substantial financial riss. Three that directly affect financial wealth seem articularly imortant: remature forced retirement, investment ris due to fluctuations in financial marets, and longevity ris (the ris of living longer than eected and, hence, outliving ones resources). Older adults also face riss on the consumtion side of which unanticiated health care eenditure looms large. As older adults age, they eerience a succession of shocs that reduce their financial resources, and for some, to uneectedly low levels. The financial hardshi they eerience may induce transfers from their children and ultimately lead to coresidence. 2 In the absence of uncertainty, there is no obvious reason why the economic situation of the elderly would decline as they age, so long as the elderly were effective lifecycle lanners. 2 Not all riss are downside riss. Investment ris may lead to an increase as well as to a decline in financial resources in any eriod. Even here, however, if elderly face reeated shocs the roortion whose wealth dros below any given level (say the level for indeendent living) will increase over time (with age). 3

4 Financial ris is not the only uncertainty faced by the elderly. Indeed, it may not be the most imortant factor that leads to increased co-residence. The elderly face riss that may also greatly affect the value they attach to ersonal attention or time inuts from their children. Two asects seem articularly imortant the first is health riss. The elderly face health crises that influence the need for ersonal care. Because ersonal care from strangers is a oor substitute for ersonal care from children and because ersonal care requires direct contact, the efficiency gains from co-residence may be esecially large. The second event faced by the elderly that influences their demand for attention from their children is the death of their souse. A souse may be a source of comanionshi and a source of ersonal care. The loss then will lead to increased demand for comanionshi and ersonal care from children. In addition, the loss of a souse may also influence other calculations of the cost and benefits of co-residence. The economies to be gained may be substantially larger when a single erson household is absorbed rather than a two-erson household. The family is not the only means of insuring against these riss. Both commercial and social insurance can lay an imortant role. Wealth can be annuitized by articiating in emloyer-based defined benefit retirement rograms, by urchasing commercial annuities, or by articiating in ublic ension rograms. These rograms rotect the elderly from both investment ris and longevity ris. Health insurance, either rivate insurance or the ublic rovision of health care, reduces the financial riss associated with illness. Moreover, individuals can self-insure by accumulating more wealth during their woring years. The availability of ris-sharing alternatives to the family increases as societies develo and as growth in the number of elderly increases the demand for commercial and social insurance. Pension rograms develo and are etended to growing numbers of worers emloyed in the ublic sector and by larger rivate firms. Comrehensive health insurance becomes increasingly available and may be etended to retirees. Publicly funded health rograms meet the health care needs of the elderly. But even in the most economically advanced economies, the elderly face riss that are difficult or imossible to insure. The maret for annuities is thin and the rice is so high that few elderly can rotect themselves against investment and longevity ris in the absence of comrehensive social insurance. 3 Many of the other riss described are essentially uninsurable. Thus, there is little reason to thin that the role of the family, and intergenerational coresidence, will shrin to nothing as Asian societies develo. Even if the world faced by the elderly were one of comlete certainty, income differences across generations would eist and etended living arrangements would be an efficient way for family members to transfer resources from low to high income members. These transfer may be an altruistic resonse to income differences that inevitably arise within a family because of differences in natural ability, effort, or luc. They may be a systematic resonse to the lifecycle roblem. As an alternative to saving, 3 Annuities are subject to adverse selection, i.e., only those who eect to live a long life urchase annuities. This drives u the rice to levels that are unattractive to individuals who do not eect to live to an unusually old age. 4

5 arents may invest in their children with the understanding that they will receive suort in their old age. Or transfers and etended living arrangements may arise simly because macroeconomic forces have influenced rates of economic growth, creating large intergenerational differences in lifetime income. This latter henomenon is certainly a feature of Taiwan, the subject of our emirical analysis and one of the fastest growing economies in the world since MODEL Assume that individuals in a one-se oulation live for u to 3g eriods where g is the length of a generation. Individuals become adults, marry, give birth, and consider establishing searate households at age g. At age 2g they become seniors, face declining roductivity, retirement, and elevated health riss. The focus of this analysis is entirely on the adult oulation, those aged g and older. We refer to members of the oulation g or older, but younger than 2g, as non-senior adults or non-seniors and to individuals aged 2g or older as seniors. Families consist of non-senior adults and their surviving arents. 4 A family cohort consists of all non-seniors born during the same eriod and their surviving arents. 5 There are two tyes of households: etended households, consisting of at least one member belonging to each generation; and nuclear households, consisting either of seniors or non-seniors, but not both. The income of an individual is designated by Y and is eogenous. When individuals form households they are assumed to fully ool their income. Hence, we do not consider any intrahousehold distributional issues. The oulation of non-seniors is designated as Ka ( ) where g a< 2g and the oulation aged Pa ( + g) are the arents of those aged a. The total oulation (of 2g adults) is given by M = ( Ka ( ) + Pa ( + g)). At any oint in time, the old-age a= g deendency ratio for non-senior adults aged a is given by: Da ( ) = Na ( + g)/ Na ( ) = sa ( + g)/( f( a+ gsa )( )). (1.1) where f( a+ g) is the fertility rate of arents aged a+g and sa ( ) and sa ( + g) are the survival rates for the oulations aged a and a+g. 6 The deendency ratio within etended households D( a ) is equal to P( a+ g)/ K( a), where the subscrit is used to distinguish etended households and n to distinguish nuclear households. 4 For simlicity we assume that children do not die before their arents and that the oulation is closed. 5 Given the simlifying assumtions the adult oulation can be searated into family cohorts that are mutually eclusive and ehaustive. 6 To define these concets more recisely, f(a) is the mean number of children ever born, s(a) is the roortion surviving to age a, and s(a+g) is the roortion of seniors surviving from age g to age a+g. We assume that the fertility rate and survival rates are indeendent. 5

6 Let ( a ) be the roortion of non-seniors aged a living in etended households and ( a+ g) be the roortion of seniors aged a+g living in etended households. For each family cohort, the following relationshi must hold: ( a+ g) = d ( a ) ( a) for a<2g. (1.2) where: d ( a) = D ( a)/ Da ( ) (1.3) is the deendency ratio in etended households relative to the general deendency ratio. Equation (1.2) reresents a demograhic constraint that catures the fundamental tradeoff between rates of co-residence among non-senior adults and the relative concentration of seniors within etended households faced by each family cohort. Given the roortion of seniors living with their adult children, ( a+ g), the roortion of non-senior adults living in etended households may be low (high), but only if those who live with their arents live with relatively many (few) of them. The choice that is made among these alternatives has clear imlications for income inequality for each family cohort. The demograhic constraint and its general bearing on the relationshi between living arrangements and inequality is illustrated by Figure 1. The demograhic constraint is shown in the uer anel, with two cases considered. In one case, (a+g) high, a high roortion of seniors are living with their adult children. The roortion of nonsenior adults living with their arents is X1 and d1 is the deendency ratio of etended households relative to the general oulation. The corresonding level of inequality is given by I1 in the lower anel of Figure 1. Note that the figure is drawn so that inequality increases as we move down the scale in the lower anel. Suose that old-age burden were shared more equally among non-senior adults. This would be achieved by moving to the right along (a+g) high. Such a change in living arrangements would lead to a reduction in inequality, reresented by the (a+g) high line in the lower anel, as the roortion of individuals ooling resources in the oulation increased. 7 Figure 1. A Model of Co-residence and Inequality A downward shift in the roortion of seniors living with their children is reresented by the shift to the (a+g) low lines shown in both of the uer and lower anels. Inequality must be greater when the roortion of seniors living in etended households declines, given the roortion of non-senior adults living in etended 7 This is an aiomatic roerty of measures of income inequality. If income is transferred from a higher income individual to a lower income individual, income is said to be more equally distributed. Assuming that income is fully ooled within etended households, any increase in membershi involves a transfer from the higher income members to the lower income members and, thus, an decline in income inequality. 6

7 households, again because of the unambiguous effects of income ooling on income inequality. Suose that society eeriences oulation aging, i.e., a rise in the deendency ratio for the general oulation, D(a). How is that reflected in Figure 1? 8 There is a range of ossible outcomes, all of which involve different ossible outcomes with resect to d( a ), ( a ), and ( a+ g). One ossibility is that the deendency ratio within etended households increases by an equal ercentage, leaving d( a ), ( a ), and ( a+ g) at their original level. Another ossibility is that neither the deendency ratio within etended families nor does the roortion of seniors living with their children change, but d( a ) would then fall and the roortion of non-seniors living in etended households would rise. A third olar case is that the deendency ratio within etended families and the roortion of non-seniors living in etended families would remain constant, and the roortion of seniors living in etended families would decline. Of course, what we might eect is that some combination of all three would occur in resonse to aging, i.e., the deendency ratio in etended household would rise by less than the deendency ratio for the oulation, the roortion of seniors living in etended households would decline, and the roortion of non-seniors living in etended households would rise. Given these range of ossibilities how will inequality be affected by aging when etended living arrangements constitute an imortant art of the economic suort system? A Primer on Aging, Etended Living Arrangements, and Inequality Consider the simlest case. Suose that a cohort 9 consists of F families and M members. The number of non-senior adults is designated as K, the number of seniors by P, and the members of the oulation and household i by M and M i. Each family consists of one non-senior adult and one senior. Families may form X etended households consisting of one senior and one non-senior adult or N nuclear households consisting of either a single senior or a single non-senior. The income of a non-senior adults is designated by Y and the income of senior by Y. The er adult income of etended households is Y and of nuclear households is Y n. The variance of any variable is reresented using the oerator V( ), for eamle, VY ( ) and VY ( ). Given this information we can readily calculate inequality, as measured by the variance in income, in two olar cases. If the cohort consists entirely of nuclear households, the variance in er caita household income is given by: 8 Figure 1 is constructed to illustrate the imlications of alternative living arrangements given the oulation age structure, not the imlications of changes in age structure. There is no reason to eect that relationshi between demograhics and inequality reresented by the lower anel in Figure 1 would be invariant with resect to changes in age structure. 9 We define the family cohort as consisting of all members of a single birth cohort (non-senior adults) and their arents (seniors). To ease analysis we assume that all siblings are members of the same birth cohort. 7

8 VY ( ) mvy ( ) mvy ( ) mm ( Y Y ) 2 = + + (1.4) where m is the roortion of non-seniors in the (adult) oulation and roortion of seniors in the oulation. m is the This can be contrasted with the case where all individuals live in etended households. This is ossible only if the number of seniors and non-seniors are equal given our assumtion that etended households consist of one senior and one non-senior. Under these restrictive assumtions, the variance in er caita household income is: VY ( ) = 0.5(1 + ρ% ) VY ( ), where ρ% Y Y Y Y 1 = Mi( Yi Y)( Yi Y)/ VY ( ). M i (1.5) If the income of non-senior adults and their arents were indeendent, the variance of er caita income would be reduced by one-half if the etended family system entirely relaced the nuclear family system. If the covariance between the incomes of children and arents were negative, the imact of establishing an etended family system would be even greater. The covariance is undoubtedly ositive, however, and the greater the covariance the smaller the imact of establishing an etended family system. In the etreme, i.e., the correlation between the income of arents and children were erfect ( ρ % = 1), then the variance and covariance are equal, there is no reduction in inequality Y Y achieved by establishing etended households. Otherwise, oulations consisting of etended household will always have lower variance than oulations consisting entirely of nuclear households because of the effects of ooling. This case is secial and simle because the decision to live in etended households is not influenced by the incomes of arents and children. Liewise, the relationshi between inequality and the roortion living in etended households is simle if the assignment of families to the etended status is random or indeendent of the incomes of seniors and non-seniors and there are equal numbers of seniors and nonseniors. In this case, if we tae m as the roortion of ersons living in etended households, then the variance of er caita household income is: VY ( ) = mvy ( ) + (1 m ) VY ( ). (1.6) n This case is simle, in art, because the simlifying assumtions insure that the er caita income for nuclear and etended households are identical and that the variance of er caita income in etended and nuclear households and the covariance of income for etended households are indeendent of the roortion of households that are etended. Under these conditions, the variance in income is linear in the roortion of ersons living in etended households. 8

9 In the absence of etended households, the effect of changes in age structure is readily analyzed using equation (1.4) or more general formulation used by Lam (2001) and others. Changes in age structure influence the overall variance of income because both the variance of income and mean income vary with age. Thus, if the oulation is more heavily concentrated in age grous with high variance or with a mean income that differs substantially from the mean income of other age grous, overall inequality will be greater. The first two right-hand-side terms in equation (1.4) cature differences in the variance in income between senior and non-senior oulations. If the variance in income of seniors is less than the variance in income of non-seniors, which is tyically the case, a rise in the senior oulation will deress inequality. The third term catures the effect on inequality of the differences in the mean income of seniors and non-seniors. Any change in age structure that results in a more uniform distribution will lead to greater inequality. Given the current situation in any oulation, m > m, an increase in the roortion senior will increase the difference in means effect. Thus, in the absence of etended households, the imact of aging on inequality is uncertain and will deend on the secifics of the income distribution of the senior and non-senior oulations. One of the difficulties that must be confronted in analyzing the relationshi between aging and inequality in the resence of etended households is that changes in age structure must influence either the roortion of the oulation living in etended households, the age comosition of etended households, or both. Given our assumtion that etended households consist of eactly one senior and one non-senior, changes in age structure must affect the roortion living in etended households. We can consider the imlications of changing age structure for inequality in the resence of etended living arrangements by relaing our assumtion that there are equal numbers of seniors and non-seniors, while retaining the assumtion that co-residence decisions are indeendent of income. Under these conditions, the variance in er caita income is given by: 1 2 ( ) = 1 ( ) + 2 ( ) + 3( ), where VY bvy bvy b Y Y b = m ( m + 0.5(1 + ρ% ) m ), b = m ( m + 0.5(1 + ρ% ) m ), and 2 3 n n Y Y n n Y Y b = mm m + + % m + mm m 2 n n ( n 0.5(1 ρ ) ) n ( n 0.5). Y Y (1.7) The variables m n and m n are the roortion of nuclear household members who are non-senior adults and seniors, resectively. The first two terms on the right-hand-side determine the weight of non-senior adults and seniors in the determination of overall inequality. In the etreme case of the roortion nuclear being 1, the weights are simly the roortion of the oulation that are seniors and non-seniors. But as the roortion nuclear declines and the roortion etended increases, the first two coefficients and inequality decline. The third term in equation (1.7) catures the imact on inequality of the differences in the mean income of senior and non-senior nuclear households. 9

10 Equation (1.7) is an incomlete characterization of the relationshi between age structure and inequality. The coefficients vary with the roortion of seniors and nonseniors living in etended households. As the relative number of seniors and non-seniors changes, the roortion of one or both grous living in etended households must change. 10 Let s consider two olar cases. In the first we hold the roortion of seniors living in etended households constant; in the second case we hold the roortion of nonseniors living in etended households constant. If the roortion of seniors living in etended households is constant, a rise in the roortion of non-seniors in the oulation leads to a rise in the roortion of non-seniors in nuclear households and a rise in the roortion of the oulation living in nuclear households. Suose, however, that the roortion of non-seniors living in etended households is fied. Then a rise in the roortion of non-seniors leads to a rise in the roortion of seniors living in nuclear households, but a decline in the roortion of the oulation living in nuclear households. In the first case, the roortion of the oulation living in etended households is equal to m = 2m where is the roortion of seniors living in etended households and m is the roortion of seniors in the adult oulation. The roortion of the seniors in the nuclear household oulation is m (1 ) /(1 2 n = m m ). If the roortion of non-seniors living in etended households is fied, then the roortion of the adult oulation living in etended households is given by m = 2m and the roortion of nuclear household members who are non-senior adults is m = (1 ) m /(1 2 m ). As is true in the absence of etended households, whether aging leads to an increase or a reduction in inequality cannot be determined on a riori grounds. Again, the relationshi deends on the characteristics of the income distributions of seniors and non-seniors and the correlation between the incomes of etended family members. If the difference in mean incomes is sufficiently large, an increase in the roortion elderly can lead to a rise in income inequality. However, using observed values for Taiwan in for illustrative uroses and assuming that the correlation between the income of seniors and non-seniors is 0.25, an increase in the roortion seniors leads to a monotonic decline in income inequality, as measured by the variance, in either of the two olar cases (Figure 2). The effect of inequality on aging is greater if the roortion of seniors living in etended households does not decline in resonse to the increase in their numbers. This is the case because the roortion living in etended households necessarily increases under these circumstances. The relationshi between age and inequality in Taiwan in 1992 is dominated by the fact that the variance of income for the senior oulation is much less than the variance of income for the non-senior oulation. This is more imortant than the decline in the roortion of the oulation living in etended households that occurs when the roortion of non-seniors is held constant and the reduction in overall inequality that occurs because seniors have substantially lower income than non-seniors. n 10 Given our assumtion that the age structure within etended households does not change. 11 For non-senior adults the variance and mean income of all ersons aged are used. Values for seniors are based on those 60 and older. 10

11 Figure 2. Inequality and aging (combined two figures). The effect of the roortion living in etended households is also illustrated in Figure 2. The level of inequality is consistently reduced by an increase in the roortion living in etended households given the roortion seniors. There is little interaction between the roortion living in etended households and aging in 1992 Taiwan. The effect of aging on inequality is essentially indeendent of the roortion of non-seniors living in etended households. There is a negative interaction between aging and the roortion of seniors living in etended households. An increase in the roortion of seniors leads to a greater reduction in inequality if a higher ercentage of seniors are living in etended households. To summarize, even given the eceedingly simle models emloyed here little can be said on a riori grounds about the effects of aging in the resence of etended living arrangements. We can say that (1) inequality is reduced if seniors live with their adult children and that (2) the imact is diminished if income is highly correlated across generations. Neither of these are startling conclusions. We cannot say with certainty how oulation aging will influence overall levels of inequality. Given one set of arameters based on Taiwan, aging leads to a decline in inequality irresective of the etent of intergenerational co-residence. Also, given a articular set of arameters we find that the effects of oulation aging on inequality are greater if a high roortion of seniors live in etended households (and the roortion living in etended households does not change in resonse to the aging). Two imortant considerations beyond those discussed above influence how changes in age structure and living arrangements will influence income inequality. The first is the manner in which changes in age structure influence living arrangements. The model resented above shows that the roortion can either rise or decline as oulations age. An additional ossibility that is not elicitly considered is that the age comosition of households may change. The second consideration is that co-residence decisions are influenced by income. If, for eamle, altruism governs decision-maing, then living arrangements will more effectively ool income and reduce inequality to a greater etent than suggested by the simle models considered here. 12 Some Additional Comlications: Variation in the Comosition of Etended Households Our rimer on inequality imosed some simlifying assumtions so that we could focus on a few ey issues. Here we rela these assumtions allowing the membershi of nuclear and etended households to vary and maing no assumtions about the relationshi between income and co-residence decisions. This allows us to consider how income inequality will be influenced by changes in the suort ratio within etended households, a demograhic resonse to oulation aging. Etending the model comes at 12 An additional consideration is that survival may be influenced by one s own income and the income of other family members. Thus the mean incomes of seniors and non-seniors living in etended households may differ systematically from the means incomes of those living in nuclear households. 11

12 a cost, however. Changes in the comosition of etended households inevitably must influence the comosition of nuclear households. A comrehensive assessment of the affect on inequality requires an understanding of the effects of changes in both nuclear and etended households. This is an issue we intend address further, but have not yet done so. We continue with our focus on inequality within a cohort of families consisting of non-senior adults aged a and their arents. 13 For a family cohort aged a the variance of er adult household income is a weighted sum of the variance of the income of nuclear and etended households and the squared difference between the mean incomes of nuclear and etended households. The weights are the roortion of the family members living in etended and nuclear households: VY ( ) mvy ( ) mvy ( ) mm ( Y Y ) 2 = + n n + n n (1.8) This is a simle etension of equation (1.6) above. The variance in er adult income of nuclear families is given by: where: VY ( ) mvy ( ) mvy ( ) mm ( Y Y ) 2 n = n n + n n + n n n n (1.9) VY = K Y Y 2 ( n ) i( ni n), and i VY = PY Y 2 ( n ) i( ni n). i where K i and P i are the number of non-seniors and seniors in nuclear household i, Y i is the average income er adult in nuclear household i, and Ynis the mean income of all age a nuclear households. The weights in equation (1.9) are the roortions of nuclear household members who are non-seniors and seniors. The variance in er adult income for etended households is: = VY ( ) mvy ( ) mvy ( ) ( Y Y ) V( m ) + 2( mycmy ( ) + my CmY ( ) + CYY ( )) (1.10) where: 13 To simlify notation we dro a from all notation. 12

13 VY = mk Y Y K 2 ( ) i i( i ) / i CmY = K m m Y Y K 2 ( ) i( i )( i ) / i V m = M m m M 2 ( ) i( i ) / i CYY ( ) = M ( my my )( my my )/ M, and i i i i i i m = K / M. i i i The terms for seniors in equation (1.10) are obtained by relacing K with P and m, in the terms for non-seniors. m with Much of the comleity in equation (1.10) arises because of variation in m, the suort ratio, across households. We can abstract from this comleity for the moment by considering a secial case in which the suort ratio is the same for all etended households. Equation (1.10) simlifies to: VY ( ) = mvy ( ) + mvy ( ) + 2 CYY ( ) (1.11) where the variance and co-variance terms are weighted by the number of household members but not by the suort ratio. As the suort ratio ranges from 0 to 1, the variance in income for etended households ranges from the variance in income for the senior sub-unit to the variance in income for the non-senior sub-unit. For intermediate values of the suort ratio, the variance of etended households will decline relative to a simle weighted average of the variances of the two sub-units. The effect of changes in the suort ratio deends also on the covariance between the income of non-senior adults and their arents. The relationshi is illustrated in Figure 3 using values obtained for all age grous combined in Given these values the variance in etended households reaches a minimum when there are about 0.3 non-senior adults er household member as comared with the actual value for 1992 of The variance in income for etended households is comared with the variance in income that we would observe if these households were to establish nuclear households. The ga between the two series reaches a maimum for a suort ratio at 0.5 and is symmetric with resect to the suort ratio. It is always the case that the greatest reduction in the variance of income occurs when there are equal numbers of non-senior adults and senior living in etended households. Figure 3. Variance of income and the suort ratio. Once we abandon our simlifying assumtion (that the old age suort ratio is the same in all etended households), the variance in etended household income also deends on the variance in the suort ratio, the third additive term in equation (1.10), 13

14 and the covariance between the suort ratio and the average income of non-senior adult members and the average income of the arents. In general, if the variance in the suort ratio is large, the variance in household income in etended households will be greater. If the covariance between the suort ratio and the income of non-senior adults is negative, the variance of income will be lower because higher income non-senior adults are bearing a heavier deendency burden. Liewise, if the covariance between m and the income of arents is negative, the variance of household income will be lower because low income adults are living in households where the suort ratio is relatively high. LIVING ARRANGEMENT, INEQUALITY, AND AGING IN TAIWAN In some resects Taiwan is an ideal subject for this research. Demograhic and economic changes there have been very raid. In the early 1950s, Taiwan had barely begun its demograhic transition and its eole were quite oor. By 1999, er caita GNP had reached $13,250, life eectancy at birth was 78 for females and 72 for males and the total fertility rate 1.6 births er woman. The oulation is beginning to eerience significant aging. In 2000 an estimated 8.6 ercent of the oulation was 65 and older. Income is very equally distributed the Gini coefficient is Its levels of educational attainment are very high gross enrollment ratios for secondary school are 101 for females and 98 for males. About 60% of the oulation levels in urban areas (ADB 2001). Thus, in five decades Taiwan transformed itself into one of Asia s most advanced economies. Only Jaan, Hong Kong, and Singaore can boast of a higher standard of living. DATA We use the Survey of Family Income and Eenditure in Taiwan (FIES, also nown as the Survey of Personal Income Distribution in Taiwan until 1993). The FIES was first conducted in 1964 and, then, every other year until Since then, the survey has been conducted annually and data are available for the 1976 and subsequent surveys. For technical reasons, we have confined our analysis to surveys conducted in 1978 and later until The number of household surveyed has varied over time, but the samle size is more than sufficient for our uroses. In 1998, about 0.4 ercent of all households (14,031 households and 52,610 individuals) were covered. These are not anel data, but reeated cross-sections. There are two features of the FIES that are imortant to the analysis resented below. First, the FIES has a comlete households roster with age, se, relationshi to head, and other individual characteristics of household members. The household roster is used to artition households into grous of individuals belonging to the same generation. For eamle, the head, souse of the head, or sibling of the head would belong to one generation. The mother, father, aunt, or uncle of the head would belong to another generation. All individuals who are related to the head are assigned to a generation. Etended households are defined as households consisting of two or more generations in which at least one member is 30 years of age or older. The great majority of etended 14

15 households consist of only two adult generations. If there are more, we collase the oldest generations into a single grou. Thus, all etended households consist of two generations of adults. Each of these generations is assigned a head, who is the individual with the greatest earnings within the generation. If no member has income or if two members are tied for the highest income, the youngest member is the head of the subunit. The age of this individual is used as the age of each generation within etended households. The second feature of the FIES is that household income is assigned to members of the household. Although there is a residual category for income that cannot be assigned to an individual, this category is rarely used. Consequently, we can calculate income characteristics searately for the non-senior and senior generation within etended households. Income is measured by total current income ecluding dereciation. We analyze income er adult. All means and variances are weighted by the number of adults in the household or sub-unit. 15 Seniors consist of all those who are 60 years of age or older. Non-seniors consist of those who are The generation length is assumed to be 30 years, which is very close to the average difference between generations in Taiwan that have not yet been subject to high rates of mortality. 16 Age 30 is used to measure adults for a variety of reasons. Marriage and childbearing are relatively late in Taiwan. Many of those who are in their 20s are still in school, have not yet married, have not yet entered the labor force or are woring full-time. Given our focus on the family as a suort system for seniors, we have used a very conservative definition of adult. RESULTS Presentation of the emirical results is organized around one ey issue: Has aging led to greater income inequality in Taiwan? To answer this question requires that we elore a number of subsidiary issues: 1) the effect of aging on the deendency ratio within etended households and on the roortions of seniors, non-seniors, and adults living in etended households; 2) the effect of changes in the etended household deendency ratio on the variance of etended household income; 3) the effect of changes in the roortion of senior nuclear households on the variance on nuclear household income; 4) the effect of changes in the roortion of seniors on the ga between the mean incomes of seniors and non-seniors. 15 For a discussion of some of the issues that arise in measuring income inequality see Lam 1997 or Schultz The age difference between generations declines when older members of the senior generation are subject to higher mortality rates. 15

16 Answering these questions allows us to reach a more comrehensive answer to whether or not aging is a source of greater income inequality using equation (1.8). Living arrangements and aging Taiwan eerienced substantial oulation aging between 1978 and 1998, although it is still early in the aging rocess as comared with Jaan, for eamle. The senior oulation (60+) as a ercentage of the adult oulation (30+) increased from 14.5 ercent in 1978, to 20.5 ercent in 1988, and 25.7 ercent in For family cohorts the increase in the deendency ratio between 1978 and 1998 ranged from a rise of 67 ercent for year old to an increase of 234 ercent for year olds and 251 ercent for years. 17 The deendency ratio within etended households increased during the same eriod, but by much less than the deendency ratio for the general oulation (Figure 4). For etended households the rise in the deendency ratio ranged from 1.4 ercent for the year-old grou to a high of 11.7 ercent for the year-old grou. The elasticity of the etended household deendency ratio with resect to the family deendency ratio ranged from a low of to a high of 0.005, well below an elasticity of 1 the value consistent with constant roortion of seniors and non-seniors living in etended households (see equation (1.2)). Figure 4. Deendency ratios, etended household and general oulations, by age grous, Because the deendency ratio in the general oulation increased so much more raidly than within etended households, the roortion of seniors living in etended households must have declined and/or the roortion of non-seniors living in etended households must have increased between 1978 and Both haened, but the resonse of non-seniors was much stronger (Figure 5). The roortion of those aged living in etended households increased from 19 ercent to 30 ercent between 1978 and 1998 while the roortion of seniors living in etended households droed from 57 ercent to 53 ercent. As noted above, the effect of aging on the roortion of the adult oulation living in etended households deends on the relative strength of the dro in the roortion of seniors living in etended households and the rise in the roortion of nonseniors living in etended households. In Taiwan, the rise in the roortion of nonseniors dominated roducing a rise in the roortion of adults living in etended households of 11 ercentage oints, from 24.3 ercent in 1978 to 35.5 ercent in The deendency ratio for a family cohort is calculated as M( a+ g)/ M( a) using five-year age grous and a value of 30 for g. 16

17 [Figure 5. Proortion of seniors, non-seniors, and adults living in etended households, ] It is unliely that the changes in living arrangements eerienced in Taiwan were a resonse entirely to changes in age structure. Taiwan was eeriencing raid social and economic develoment during much of this eriod, with otentially imortant imlications for the resonse of living arrangements to changes in age structure. 18 Of course, we can not rule out that the roortion living in etended households would have risen even more in resonse to changes in age structure had other circumstances remained the same in Taiwan. It is a striing feature of Taiwan s demograhy that the roortion living in etended households increased so substantially. The Effect of Aging on the Distribution of Income among Etended Households The rise in the deendency ratio in Taiwan has led to an increase in the deendency ratio within etended households, albeit one that aears to be quite small. Nonetheless, inequality among etended households will change. To assess the effect we mae several simlifying assumtions. We assume that the mean and variance of the incomes of seniors and non-seniors living in etended households is unaffected by the change in the etended households deendency ratio; that the covariance between the income of seniors and non-seniors living in etended households is unaffected; and that the variance of the deendency ratio and the covariance with the income and seniors and non-seniors is unaffected by changes in the deendency ratio. The only effect we assess is comositional, that the mean and variance in income will reflect the greater weight of seniors in etended households. The comositional effect of the change is easily assessed by simulation. We increase the number of seniors living in etended households by 10 ercent in every etended household in our samles and recalculate the variance of er caita income of etended households. The results of this eercise alied to data for 1978, 1988, and 1998 dislayed in Table 1 show that an increase in the deendency ratio consistently led to a decline in the variance in er caita household income for etended households. The imact of a change in the deendency ratio is substantial. A ten ercent increase in the deendency ratio roduced a decline in the variance of income that ranged from 6.2 to 11.4 ercent deending on the year and the age grou. Although the variance in income for etended households aears to be sensitive to aging, the changes in the deendency ratio in etended households was relatively modest as elained above. Between 1978 and 1998, the deendency ratio increased by roughly 10 ercent for those aged 35-49, but by much less for the other grous. Why does aging among etended households lead to a decline in the variance of er caita income? As discussed in more detail above, the effect of changes in age 18 Lee and Mason (2002) we consider this issue in more detail. 17

18 Table 1. Effect on VY ( ) ten ercent increase in M, Taiwan, 1978, 1988, and Year Age M M Y Y Y m m VY ( ) grou change Before After Before After Before After comosition deend on the mean and variance of income for seniors and non-seniors within the family cohort. The mean incomes of seniors living in etended households 18