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Nasty, brutish, and shorter?
Economics Letters 79 (2003) 117–123 www.elsevier.com / locate / econbase
Nasty, brutish, and shorter? Fertility, mortality, and dynastic longevity: US cohorts since 1873 Russell D. Murphy Jr.* Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Received 6 May 2002; accepted 9 October 2002
Abstract Dynastic models are widely used in economics, but the extent of dynastic behavior is unclear. I calculate a simple measure of dynastic success, the probability of dynastic extinction. Extinction probabilities were roughly 50% through the early 20th Century, fell over the next two decades to roughly 14%, and increased to 28% by the 1954 cohort. 2002 Elsevier Science B.V. All rights reserved. Keywords: Fertility; Dynastic extinction JEL classification: J11; J13
1. Introduction One of the most basic questions in the economics of the family is a simple one: ‘Why have children?’ Malthus (1817) posited that man is marked by: . . . the constant tendency in all animated life to increase beyond the nourishment prepared for it (p. 1)
Becker and Barro (1988) suppose that individuals are altruistic toward their children and act to maximize a dynastic utility function. Individuals make fertility and savings decisions recursively considering their descendents’ utility. The notion that individuals are altruistically linked across generations has been influential in the economics of the family, and also in the literatures on * Tel.: 11-540-231-4537. E-mail address: [email protected] (R.D. Murphy Jr.). 0165-1765 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1765(02)00295-1
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inequality Loury (1981), public finance, and macroeconomics Chamley (1981). It seems intuitively plausible and is widely accepted; among theoretical biologists, discussion of natural selection typically supposes an interest in self-perpetuation: Our conclusion . . . is that individual parents . . . try to maximize the number of surviving children that they have . . . Dawkins (1976, p. 131)
I use a simple metric, the probability of dynastic extinction, to measure individuals’ dynastic behavior. Over the past century, Americans’ dynastic success has often been limited and has varied widely. Dynastic extinction probabilities were roughly 50% for cohorts born in the late 19th Century, fell steadily to 14% for cohorts born in the early 1930s, but have since increased. For cohorts born in 1954 (the last for which we have lifetime fertility data), the estimated dynastic failure rate is more than 25%. Dynastic extinction is a useful measure because fertility is stochastic from the individual’s perspective, and more fundamentally, because the fertility behavior and success of one’s descendants is stochastic. For cohorts born in the late 19th Century, close to one quarter did not survive to child bearing age. For the most recent cohorts of American women, 17% had no children. Potential dynastic heads would need to consider these realities when making consumption, investment, and fertility decisions. Economists may care about dynastic longevity for several reasons. Understanding linkages between generations is important in assessing policy effects such as Barro (1974)’s Ricardian equivalence. Second, interpretation of estimated inter-generational income correlations Solon (1992) may be different in a world in which dynastic failure rates are 25–50% relative to a world in which all family lines persist indefinitely. Finally, understanding how individuals’ choices generate these patterns may help us to better understand dynastic models such as Becker and Barro (1988).
2. Fertility and mortality Fertility is measured by cumulative births of women by birth cohort to age 44. Survival is measured by the likelihood of a cohort’s survival to the ages at which births 1–7 are assumed to take place.1 Together, they determine the distribution of procreating descendants from generation to generation. Fig. 1 shows the time path of US cohort fertility. Lifetime fertility is relatively high, 3.73 births per woman, for the 1873 cohort. Over the next 82 years, there are three distinct periods of fertility change: declining for the cohorts born from 1873 to 1908, climbing for those born 1909–1933, and falling again for 1934–1954. In general, the cohort distributions by parity shifted back and forth over the three different periods (Table 1). For the late 19th and early 20th Century cohorts, low levels of child bearing became more likely, while high levels became less so. For these cohorts, the big change is a decline of 0.15 in the likelihood of having 7 1 children; another 0.07 of these cohorts shift out of the 5–6 parity group and 1 The fertility data are from Heuser and NICHD (1976) and National Center for Health Statistics (2001). Survival likelihoods are interpolations from data in National Center for Health Statistics, Public Health Service (1996). The data used are available at www.econ.vt.edu / |rdmurphy / research.html along with a detailed description of sources.
R.D. Murphy / Economics Letters 79 (2003) 117–123
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Fig. 1. Lifetime fertility.
almost all of this probability mass shifts to the 1–2 child group. For the second group of cohorts, those born from 1909 to 1933, fertility increases. Interestingly, as the probability of having a small number of children declines, there is also a large reduction (0.13) in the probability of remaining childless. The fertility decline of the first period is primarily a decline in the probability of the highest parity outcomes coupled with an increase in small families; the increase of the second period comes from a relatively large fraction moving out of both the childless and the small (1–2) family groups. The experience of the most recent cohorts is slightly different. There were few individuals left in the 7 1 group (0.07) and even most of this probability mass moves to the left (20.06). There are also declines in the 5–6 (20.11) and 3–4 (20.13) groups. Most of the offsetting increases are in small families (1–2). Childlessness increases substantially ( 1 0.09), although not by as much as it fell over the 1909–1933 cohorts.
3. Dynastic longevity The extinction probability of a dynasty can be calculated as an application of branching processes; see Whittle (1992) and Chu (1998). Let p 5 h pi j Ki50 be the probability distribution over 0,1, . . . ,K immediate descendants (where p1 ± 1). If p also describes the distribution in future generations, the extinction probability r is the smallest root of z 5 G(z), where G(z) 5 o Kj 50 z j pj is the probability generating function for p. Fig. 2 presents estimates of the probability of dynastic extinction for cohorts born between 1873 and 1954. For each cohort, the dynastic extinction probability is based on the distribution of women in Table 1 Changes in cohort fertility patterns Period
1873–1908 1909–1933 1934–1954
Change in average number of children 21.469 10.956 21.234
Change in probability that [ children5x 0
1–2
3–4
5–6
71
10.011 20.132 10.086
10.207 20.131 10.216
10.002 10.174 20.131
20.070 10.074 20.112
20.150 10.015 20.059
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Fig. 2. Probable dynastic failure rates.
the cohort by parity and the likelihood of survival to child-bearing ages at each birth parity.2 The calculation assumes that, for each cohort, the current fertility and mortality patterns would hold for all future generations. We might interpret the calculated extinction probability as that which might be expected by a cohort member who uses the cohort’s experiences as a guide to the likely experiences of his or her dynasty. Estimated cohort extinction probabilities range from a maximum of 0.51 to a minimum of 0.14. For the first group of cohorts (1873–1908), the calculated extinction probabilities were relatively steady, between 0.46 and 0.51; over the second period (cohorts 1909–1933), extinction probabilities fell sharply from 0.46 to 0.15. During the last period (cohorts 1934–1954), the probability of dynastic failure almost doubled to 0.28. Failure rates are not necessarily indicative of dynastic intentions, but the magnitudes of and changes in the extinction probabilities are interesting. Fertility declined steadily for the cohorts in the 1873–1908 period, so the relatively constant likelihood of dynastic extinction arises from a decline in mortality. The probability of surviving to age 22 was 0.76 for the 1873 cohort, 0.88 for the 1908 cohort, and 0.95 for the 1930 cohort. Further gains were necessarily limited. For the later cohorts, mortality declines were limited; the fertility declines in Fig. 1 are such that dynastic failure becomes increasingly likely. However, it is not simply low fertility that drives this increase in extinction probabilities; what matters is the distribution of fertility outcomes. If, for instance, the probability of having one child (which survives to adulthood) is one, then the extinction probability is zero. For the 1954 cohort, expected lifetime fertility is 1.97, not far from the ‘sustainable’ level, but fertility outcomes are sufficiently varied, including a 0.17 probability of childlessness, that the likely extinction probability is 0.28.
3.1. Racial differences? I have limited data on lifetime fertility of women with different demographic characteristics, but can differentiate between cohorts of white women and those of women of other races. Fig. 3 presents the calculated extinction probabilities for white women, while Fig. 4 presents those for women of other races. 2
Based on the median ages at births 1–7 in 1959 (National Center for Health Statistics, 2001), I assume that dynasties have 0–7 births and births 1–7 take place at ages 22, 24, 27, 29, 30, 31, and 32, respectively, for all cohorts.
R.D. Murphy / Economics Letters 79 (2003) 117–123
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Fig. 3. Probable dynastic failure rates: white Americans.
The calculated extinction probabilities for both groups are similar through the early 1900s; mainly, the extinction rates are high, roughly 0.50. The next cohorts of both groups, born 1907–1933, experience steadily declining extinction probabilities. By the early 1930s, extinction probabilities had fallen, for both racial groups, to roughly 0.14–0.20. Post 1933, however, the calculated probabilities follow very different paths. While extinction probabilities for whites increase to more than 0.30, for other racial groups, the extinction probabilities continue to fall, reaching a low of just over 0.05 for the 1954 cohort. The low extinction probabilities are driven by the low childlessness rates for the later cohorts of women of other races. While childlessness rates are 0.16–0.17 for white women of cohorts in the 1950s, the rates are 0.01–0.02 for women of other races in the same cohorts (National Center for Health Statistics, 2001).
3.2. Other demographic differences? For later cohorts, I can use the Current Population Survey 3 (CPS) to generate estimates of the fertility patterns of different groups. Table 2 presents estimates of the probability of dynastic
Fig. 4. Probable dynastic failure rates: Americans of other races. 3
These estimates are based on an extract from the June 1998 CPS Fertility Supplement; all estimates were generated using PWSSWGT as weights.
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Table 2 Probability of dynastic extinction, 1954 cohort Education level
Probability
Less than HS HS or GED Some college Assoc. degree Bachelor degree Graduate school
0.184 0.216 0.265 0.317 0.406 0.619
extinction for education sub-groups of the 1954 cohort. Across groups, the extinction probabilities vary from 0.18 to 0.62. It seems likely that the opportunity costs of women’s time rise sufficiently with human capital investments to generate significant increases in dynastic extinction likelihoods.
4. Conclusions These estimates of dynastic extinction probabilities are based on fertility and mortality patterns of cohorts and assume that dynasties consist of individuals whose experiences will be characterized by cohort fertility and mortality probabilities. This is clearly unrealistic; among other things, it assumes that although current behavior presumably arose out of individual maximizing behavior, future behavior will be largely mechanistic. Nonetheless, the calculations are informative. If I am unsure of the likely experiences of my progeny, current patterns are perhaps an reasonable forecast. The behavior of the past 82 cohorts suggests that, on average, US men and women have made decisions so that their dynasties were likely to be short lived. We should take care in interpreting the results of dynastic models as well as interpreting the results of measures of inter-generational relationships. The existence of inter-generational links is assured backwards, but not forwards.
Acknowledgements I thank Richard Cothren, Sharun Mukand, Aris Spanos, Nic Tideman, and seminar participants at the Virginia Polytechnic Institute and at the Irish Economics Association annual meetings in Mullingar for helpful discussions and comments. Remaining errors and omissions are, of course, my responsibility.
References Barro, R.J., 1974. Are government bonds net wealth? Journal of Political Economy 82 (6), 1095–1117. Becker, G.S., Barro, R.J., 1988. A reformulation of the economic theory of fertility. Quarterly Journal of Economics CIII (1), 1–25. Chamley, C., 1981. The welfare cost of capital income taxation in a growing economy. Journal of Political Economy 89 (3), 468–496.
R.D. Murphy / Economics Letters 79 (2003) 117–123
123
Chu, C.Y.C., 1998. Population Dynamics: A New Economic Approach. Oxford University Press, New York. Dawkins, R., 1976. The Selfish Gene. Oxford University Press, New York. NICHD, Heuser, R.L., 1976. Fertility tables for birth cohorts by color: United States, 1917–73. DHEW Publication No. (HRA) 76-1152. US Department of Health, Rockville, MD. Loury, G.C., 1981. Intergenerational transfers and the distribution of earnings. Econometrica 49 (4), 843–867. Malthus, T., 1817. An Essay on the Principle of Population, This Edition: 1963. Irwin Paperback Classics in Economics. Richard D. Irwin, Homewood, IL, Selected and edited by Lloyd Reynolds and William Fellner. National Center for Health Statistics, 2001. Vital Statistics of the United States, 1997. Natality, Volume I, CD-ROM. See www.cdc.gov / nchs /. National Center for Health Statistics, Public Health Service, 1996. Vital Statistics of the US 1992. Mortality, Part A, Volume II. US Government Printing Office, Washington, DC. Solon, G.R., 1992. Intergenerational income mobility in the United States. The American Economic Review 82 (3), 393–408. Whittle, P., 1992. Probability Via Expectation, 3rd Edition. Springer Texts in Statistics. Springer, New York.
Nasty, brutish, and shorter? Fertility, mortality, and dynastic longevity: US cohorts since 1873 Russell D. Murphy Jr.* Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Received 6 May 2002; accepted 9 October 2002
Abstract Dynastic models are widely used in economics, but the extent of dynastic behavior is unclear. I calculate a simple measure of dynastic success, the probability of dynastic extinction. Extinction probabilities were roughly 50% through the early 20th Century, fell over the next two decades to roughly 14%, and increased to 28% by the 1954 cohort. 2002 Elsevier Science B.V. All rights reserved. Keywords: Fertility; Dynastic extinction JEL classification: J11; J13
1. Introduction One of the most basic questions in the economics of the family is a simple one: ‘Why have children?’ Malthus (1817) posited that man is marked by: . . . the constant tendency in all animated life to increase beyond the nourishment prepared for it (p. 1)
Becker and Barro (1988) suppose that individuals are altruistic toward their children and act to maximize a dynastic utility function. Individuals make fertility and savings decisions recursively considering their descendents’ utility. The notion that individuals are altruistically linked across generations has been influential in the economics of the family, and also in the literatures on * Tel.: 11-540-231-4537. E-mail address: [email protected] (R.D. Murphy Jr.). 0165-1765 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1765(02)00295-1
118
R.D. Murphy / Economics Letters 79 (2003) 117–123
inequality Loury (1981), public finance, and macroeconomics Chamley (1981). It seems intuitively plausible and is widely accepted; among theoretical biologists, discussion of natural selection typically supposes an interest in self-perpetuation: Our conclusion . . . is that individual parents . . . try to maximize the number of surviving children that they have . . . Dawkins (1976, p. 131)
I use a simple metric, the probability of dynastic extinction, to measure individuals’ dynastic behavior. Over the past century, Americans’ dynastic success has often been limited and has varied widely. Dynastic extinction probabilities were roughly 50% for cohorts born in the late 19th Century, fell steadily to 14% for cohorts born in the early 1930s, but have since increased. For cohorts born in 1954 (the last for which we have lifetime fertility data), the estimated dynastic failure rate is more than 25%. Dynastic extinction is a useful measure because fertility is stochastic from the individual’s perspective, and more fundamentally, because the fertility behavior and success of one’s descendants is stochastic. For cohorts born in the late 19th Century, close to one quarter did not survive to child bearing age. For the most recent cohorts of American women, 17% had no children. Potential dynastic heads would need to consider these realities when making consumption, investment, and fertility decisions. Economists may care about dynastic longevity for several reasons. Understanding linkages between generations is important in assessing policy effects such as Barro (1974)’s Ricardian equivalence. Second, interpretation of estimated inter-generational income correlations Solon (1992) may be different in a world in which dynastic failure rates are 25–50% relative to a world in which all family lines persist indefinitely. Finally, understanding how individuals’ choices generate these patterns may help us to better understand dynastic models such as Becker and Barro (1988).
2. Fertility and mortality Fertility is measured by cumulative births of women by birth cohort to age 44. Survival is measured by the likelihood of a cohort’s survival to the ages at which births 1–7 are assumed to take place.1 Together, they determine the distribution of procreating descendants from generation to generation. Fig. 1 shows the time path of US cohort fertility. Lifetime fertility is relatively high, 3.73 births per woman, for the 1873 cohort. Over the next 82 years, there are three distinct periods of fertility change: declining for the cohorts born from 1873 to 1908, climbing for those born 1909–1933, and falling again for 1934–1954. In general, the cohort distributions by parity shifted back and forth over the three different periods (Table 1). For the late 19th and early 20th Century cohorts, low levels of child bearing became more likely, while high levels became less so. For these cohorts, the big change is a decline of 0.15 in the likelihood of having 7 1 children; another 0.07 of these cohorts shift out of the 5–6 parity group and 1 The fertility data are from Heuser and NICHD (1976) and National Center for Health Statistics (2001). Survival likelihoods are interpolations from data in National Center for Health Statistics, Public Health Service (1996). The data used are available at www.econ.vt.edu / |rdmurphy / research.html along with a detailed description of sources.
R.D. Murphy / Economics Letters 79 (2003) 117–123
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Fig. 1. Lifetime fertility.
almost all of this probability mass shifts to the 1–2 child group. For the second group of cohorts, those born from 1909 to 1933, fertility increases. Interestingly, as the probability of having a small number of children declines, there is also a large reduction (0.13) in the probability of remaining childless. The fertility decline of the first period is primarily a decline in the probability of the highest parity outcomes coupled with an increase in small families; the increase of the second period comes from a relatively large fraction moving out of both the childless and the small (1–2) family groups. The experience of the most recent cohorts is slightly different. There were few individuals left in the 7 1 group (0.07) and even most of this probability mass moves to the left (20.06). There are also declines in the 5–6 (20.11) and 3–4 (20.13) groups. Most of the offsetting increases are in small families (1–2). Childlessness increases substantially ( 1 0.09), although not by as much as it fell over the 1909–1933 cohorts.
3. Dynastic longevity The extinction probability of a dynasty can be calculated as an application of branching processes; see Whittle (1992) and Chu (1998). Let p 5 h pi j Ki50 be the probability distribution over 0,1, . . . ,K immediate descendants (where p1 ± 1). If p also describes the distribution in future generations, the extinction probability r is the smallest root of z 5 G(z), where G(z) 5 o Kj 50 z j pj is the probability generating function for p. Fig. 2 presents estimates of the probability of dynastic extinction for cohorts born between 1873 and 1954. For each cohort, the dynastic extinction probability is based on the distribution of women in Table 1 Changes in cohort fertility patterns Period
1873–1908 1909–1933 1934–1954
Change in average number of children 21.469 10.956 21.234
Change in probability that [ children5x 0
1–2
3–4
5–6
71
10.011 20.132 10.086
10.207 20.131 10.216
10.002 10.174 20.131
20.070 10.074 20.112
20.150 10.015 20.059
120
R.D. Murphy / Economics Letters 79 (2003) 117–123
Fig. 2. Probable dynastic failure rates.
the cohort by parity and the likelihood of survival to child-bearing ages at each birth parity.2 The calculation assumes that, for each cohort, the current fertility and mortality patterns would hold for all future generations. We might interpret the calculated extinction probability as that which might be expected by a cohort member who uses the cohort’s experiences as a guide to the likely experiences of his or her dynasty. Estimated cohort extinction probabilities range from a maximum of 0.51 to a minimum of 0.14. For the first group of cohorts (1873–1908), the calculated extinction probabilities were relatively steady, between 0.46 and 0.51; over the second period (cohorts 1909–1933), extinction probabilities fell sharply from 0.46 to 0.15. During the last period (cohorts 1934–1954), the probability of dynastic failure almost doubled to 0.28. Failure rates are not necessarily indicative of dynastic intentions, but the magnitudes of and changes in the extinction probabilities are interesting. Fertility declined steadily for the cohorts in the 1873–1908 period, so the relatively constant likelihood of dynastic extinction arises from a decline in mortality. The probability of surviving to age 22 was 0.76 for the 1873 cohort, 0.88 for the 1908 cohort, and 0.95 for the 1930 cohort. Further gains were necessarily limited. For the later cohorts, mortality declines were limited; the fertility declines in Fig. 1 are such that dynastic failure becomes increasingly likely. However, it is not simply low fertility that drives this increase in extinction probabilities; what matters is the distribution of fertility outcomes. If, for instance, the probability of having one child (which survives to adulthood) is one, then the extinction probability is zero. For the 1954 cohort, expected lifetime fertility is 1.97, not far from the ‘sustainable’ level, but fertility outcomes are sufficiently varied, including a 0.17 probability of childlessness, that the likely extinction probability is 0.28.
3.1. Racial differences? I have limited data on lifetime fertility of women with different demographic characteristics, but can differentiate between cohorts of white women and those of women of other races. Fig. 3 presents the calculated extinction probabilities for white women, while Fig. 4 presents those for women of other races. 2
Based on the median ages at births 1–7 in 1959 (National Center for Health Statistics, 2001), I assume that dynasties have 0–7 births and births 1–7 take place at ages 22, 24, 27, 29, 30, 31, and 32, respectively, for all cohorts.
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Fig. 3. Probable dynastic failure rates: white Americans.
The calculated extinction probabilities for both groups are similar through the early 1900s; mainly, the extinction rates are high, roughly 0.50. The next cohorts of both groups, born 1907–1933, experience steadily declining extinction probabilities. By the early 1930s, extinction probabilities had fallen, for both racial groups, to roughly 0.14–0.20. Post 1933, however, the calculated probabilities follow very different paths. While extinction probabilities for whites increase to more than 0.30, for other racial groups, the extinction probabilities continue to fall, reaching a low of just over 0.05 for the 1954 cohort. The low extinction probabilities are driven by the low childlessness rates for the later cohorts of women of other races. While childlessness rates are 0.16–0.17 for white women of cohorts in the 1950s, the rates are 0.01–0.02 for women of other races in the same cohorts (National Center for Health Statistics, 2001).
3.2. Other demographic differences? For later cohorts, I can use the Current Population Survey 3 (CPS) to generate estimates of the fertility patterns of different groups. Table 2 presents estimates of the probability of dynastic
Fig. 4. Probable dynastic failure rates: Americans of other races. 3
These estimates are based on an extract from the June 1998 CPS Fertility Supplement; all estimates were generated using PWSSWGT as weights.
R.D. Murphy / Economics Letters 79 (2003) 117–123
122
Table 2 Probability of dynastic extinction, 1954 cohort Education level
Probability
Less than HS HS or GED Some college Assoc. degree Bachelor degree Graduate school
0.184 0.216 0.265 0.317 0.406 0.619
extinction for education sub-groups of the 1954 cohort. Across groups, the extinction probabilities vary from 0.18 to 0.62. It seems likely that the opportunity costs of women’s time rise sufficiently with human capital investments to generate significant increases in dynastic extinction likelihoods.
4. Conclusions These estimates of dynastic extinction probabilities are based on fertility and mortality patterns of cohorts and assume that dynasties consist of individuals whose experiences will be characterized by cohort fertility and mortality probabilities. This is clearly unrealistic; among other things, it assumes that although current behavior presumably arose out of individual maximizing behavior, future behavior will be largely mechanistic. Nonetheless, the calculations are informative. If I am unsure of the likely experiences of my progeny, current patterns are perhaps an reasonable forecast. The behavior of the past 82 cohorts suggests that, on average, US men and women have made decisions so that their dynasties were likely to be short lived. We should take care in interpreting the results of dynastic models as well as interpreting the results of measures of inter-generational relationships. The existence of inter-generational links is assured backwards, but not forwards.
Acknowledgements I thank Richard Cothren, Sharun Mukand, Aris Spanos, Nic Tideman, and seminar participants at the Virginia Polytechnic Institute and at the Irish Economics Association annual meetings in Mullingar for helpful discussions and comments. Remaining errors and omissions are, of course, my responsibility.
References Barro, R.J., 1974. Are government bonds net wealth? Journal of Political Economy 82 (6), 1095–1117. Becker, G.S., Barro, R.J., 1988. A reformulation of the economic theory of fertility. Quarterly Journal of Economics CIII (1), 1–25. Chamley, C., 1981. The welfare cost of capital income taxation in a growing economy. Journal of Political Economy 89 (3), 468–496.
R.D. Murphy / Economics Letters 79 (2003) 117–123
123
Chu, C.Y.C., 1998. Population Dynamics: A New Economic Approach. Oxford University Press, New York. Dawkins, R., 1976. The Selfish Gene. Oxford University Press, New York. NICHD, Heuser, R.L., 1976. Fertility tables for birth cohorts by color: United States, 1917–73. DHEW Publication No. (HRA) 76-1152. US Department of Health, Rockville, MD. Loury, G.C., 1981. Intergenerational transfers and the distribution of earnings. Econometrica 49 (4), 843–867. Malthus, T., 1817. An Essay on the Principle of Population, This Edition: 1963. Irwin Paperback Classics in Economics. Richard D. Irwin, Homewood, IL, Selected and edited by Lloyd Reynolds and William Fellner. National Center for Health Statistics, 2001. Vital Statistics of the United States, 1997. Natality, Volume I, CD-ROM. See www.cdc.gov / nchs /. National Center for Health Statistics, Public Health Service, 1996. Vital Statistics of the US 1992. Mortality, Part A, Volume II. US Government Printing Office, Washington, DC. Solon, G.R., 1992. Intergenerational income mobility in the United States. The American Economic Review 82 (3), 393–408. Whittle, P., 1992. Probability Via Expectation, 3rd Edition. Springer Texts in Statistics. Springer, New York.