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Health and inequality
So<. Ser. Mrd. Vol. 16. pp. 1663 to 1666. 19X7 Printed m Great Bntirm
HEALTH SOME
APPLICATIONS
0277-9536,X2:lYl663-04$03.0010 Pergamon Press Ltd
AND INEQUALITY OF UNCERTAINTY
THEORY
JACQUES SILBER
Department
of Economics,
Bar-Ilan
University,
Ramat-Gan,
Israel
Abstract-This paper applies to the Field of Health results of Stochastic Dominance Theory and expressions originally devised for the measurement of Income Inequality. In particular. use is made of Atkinson’s inequality measure to compare health levels in England and Wales in the past century. It appears that the ‘inequality before death’ is less important today than it was in the past, this improvement being parallel to the increase in the average number of years lived (life expectancy). The analysis indicates also that both improvements were greater for women than for men during the period considered.
I. INTRODUCFION
his well-known Essu_~s in the Theory of Risk-Bearing, Arrow writes “The loss due to illness is only partially the cost of medical care. It also consists of discomfort and loss of productive time during illness, and in more serious cases. death or prolonged deprivation of normal functions. From the point of view of the welfare economics of uncertainty. both losses are risks against which individuals would like to insure” [ 11. This citation emphasizes clearly the role of uncertainty in the field of health economics. However, most of the attention has been paid to the effect of uncertainty in decisions involving medical care. But there are other uncertainty aspects related to health, which are .not less important. Health, for example. may influence the decisions of individuals to invest in their own human capital (e.g. education) and, of course, of the state to contribute to such investments. Such a relation between health and investment decisions implies naturally the use of measures summarizing the state of the health of a country. Among the most popular indexes. one has the life expectancy and the infant mortality. But these are only partial indexes which do not consider the whole probability distribution. However. if one wishes to consider the whole probability distribution and to relate the measure of the health of a country to the profitability of investing in some forms of human capital or. more generally, to the level of welfare of this country. it may be worthwhile to apply to this measurement problem some concepts which have been widely used recently in the finance literature. the criteria of stochastic dominance. The purpose of this paper is precisely to show how the theory of stochastic dominance could be applied to the problem of measuring the health of a country. In a first section we shall review the principles of stochastic dominance theory. in particular their application to the measurement of income inequality and indicate how they could be used to compare life tables. In
This
research Foundation Trustees.
was supported b! a grant from the Ford received through the Israel Foundations
1663
In a second section we propose to apply this technique to the measurement of the extent of mortality, essentially its distribution by age and sex, in-England and Wales, during the past 100 years.
II.
STOCHASTIC
DOMINANCE
COMPARISON
CRITERIA
AND
THE
OF LIFE TABLES
As is well known, the notion of stochastic dominance (SD) has been mainly used to order uncertain options and its importance is related to the fact that the rules which have been developed enable us to compare options even when the investor’s preference (utility function) is only partially known (e.g. only the sign of the first and second derivatives of the utility function are known). Originally, two decision rules, corresponding to two different classes of utility functions, have been devised and these SD criteria have been presented separately, more or less at the same time [2-S]. Let us show what these criteria mean when mortality data are used. A life table gives the probability P, of living up to age X. Since the probability of any investment in human capital depends. among other things, on the expected duration of the return flows, one may want to use the probabilities P, as weights to be attached to the utility derived from the earning flows R, which will be obtained if the individual lives up to age x. What do then the different criteria imply? The first one. called first degree stochastic dominance rule (FSD in short) makes only the assumption that the marginal utility of any return flow is not negative, but. as we shall see. it is very often difficult to derive clear conclusions. If F and G are respectively two different cumulative distributions of the deaths by age. FSD implies that distribution F will be preferred to distribution G if it lies below distribution G for at least
one age group
X. and
does
not
lie above
for any
other age group. The second degree stochastic dominance rule (SSD in short) assumes that the marginal utility of any return flow is not negative and that it does not increase with the amount of the return flow (in other words SSD assumes risk aversion). Using SSD one
JACQUESSILBER
I664
will say that distribution F will be preferred to distribution G if the rule devised for FSD applies to the cumulative distributions of the cumulative distributions F and G of deaths by age. Since very often the cumulative distribution of deaths by age intersect. it is clear that SSD can be used more often than FSD, which excludes such a possibility. The rules which have been presented hitherto are evidently ordinal rules. They do not tell us how much preferable is distribution F to distribution G. But, if we compare the life tables of a given country at two different times tF and tc, the criteria presented will tell us, for example, whether investing in education was more profitable, ceteris paribus, at time tF than at time tG. Similar comparisons could eventually be made at a given point of time if one wishes to compare the probabilities of investing in education in different countries. Such a study has in fact been presented recently
Table 1. Ordering the life tables of England and Wales according to the FSD and SSD criteria
C61.
where E(x) is the actual life expectancy (the actual mean of the distribution of deaths by age). To obtain an extimate of xEQ and of I(u) one may want to use the social welfare function U(x) proposed by Atkinson [7] where
Could one obtain also a cardinal classification of the distribution of deaths by age‘? We shall indicate now that if, besides the average health level of a given country, at a given time t, one wishes to take also into account ‘health inequality’ between individuals, there is a possibility of quantifying the differences between the distributions of deaths by age (here again we emphasize that we have limited our measure of health to a kind of dichotomic criterion distinguishing only between life and death, and not between degrees of good or bad health). The measure we propose is a simple application of Atkinson’s index of income inequality [7]. The idea is that utility is derived from the number of years lived and that there exists an additively separable and symmetric social welfare function of individual lengths of life. In this case, one may want to know what the utility (at the country level) or better the social welfare would be if everyone lived the same number of years. More precisely, one may want to calculate the ‘equally distributed equivalent length of life’ xEQ (‘equivalent length of life’ in short) which would give the same social welfare as the actual distribution of deaths by age. One would then measure the inequality in lengths of life as I(x)
= 1-
XEQ E(x)
Year
First degree SD criterion ordering
1871 1891 1901 1911 1921 1951 1960 1964
8 7 6 5 4 3 2 1
Year 1871 1891 1901 1911 1921 1931 1940 1951 1960 1964
10 9 8 7 6 : 3 2 1
U(x)=a+bs
when
E+ I
U(.u) = In (x)
when
E = 1.
and
It can be shown that such a utility function derives from the assumption of constant relative inequality aversion, i.e. of a constant elasticity E of the marginal utility of the length of life (see Cl] for an exposition of these concepts). With such a utility function, the inequality index can be written as
where d(xi) is the distribution of death by age x. By computing I(x) for a given country, at different points of time, one can obtain a quantititative indication of what was the evolution of the inequality of lengths of life during the period.
Table 2. Infant mortality and life expectancy in England and Wales since 1981
Year
Probability of dying before age one
Ordering of the infant mortality rate
Life expectancy
Ordering of life expectancy
1861 1871 1881 1891 1901 1911 1921 1931 1940 1951 1960 1964
0.160 0.164 0.132 0.158 0.158 0.133 0.082 0.066 0.056 0.030 0.023 0.021
11 12 7 9 9 8 6 5 4 3 2 1
41.8 40.9 45.9 43.8 47.4 51.4 58.0 60.3 61.7 68.4 71.1 71.5
11 12 9 10 8 7 6 5 4 3 2
,
Second degree SD criterron ordering
1
Health and inequality Table 3. Estimation of the Inequality index I(x) in England and Wales in the past century
Year
E = 0.50
1861 1871 1881 1891 1901 1911 1921 1931 1940 1951 1960 1964
0.067 0.068 0.054 0.059 0.057 0.047 0.033 0.025 0.02 1 0.010 0.008 0.007
Inequality index I(x) <=I E = 1.25 E = 1.5 0.62 0.63 0.56 0.60 0.59 0.54 0.35 0.34 0.29 0.09 0.19 0.12
0.096 0.095 0.083 0.091 0.093 0.082 0.059 0.048 0.042 0.022 0.018 0.017
We now turn to an illustration proposed to use.
111.MEASURING
0.44 0.44 0.41 0.44 0.44 0.42 0.34 0.30 0.27 0.18 0.15 0.14
0.81 0.81 0.79 0.81 0.81 0.80 0.75 0.71 0.68 0.71 0.51 0.49
of the technique
THE HEALTH
THE CASE OF ENGLAND
E = 1.75
we
OF A COUNTRY:
AND WALES IN
THE PAST HUNDRED
YEARS
In the previous section we used the basic results of the application to stochastic dominance theory to investment decisions and inequality measurement, to propose an ordinal as well as a cardinal classification of level of health of a country at different times or of different countries at a given point of time. The measurement of health was limited to that of the mortality level. Let us now present the results of an empirical investigation we made by using the life tables established by Preston et ~2. [S], for England and Wales, from 1861 to 1964. at intervals usually equal to 10 years. In Table 1 we present the ordering obtained by using respectively FSD and SSD. As we see the classification obtained is more precise in the sense that more countries can be classified when using SSD which puts stronger restrictions on the utility function In Table 2 we present data for the same years on infant mortality and life expectancy. It appears that
1665
here also one notices decades of regression (1871, 1891) but this does not prevent us from ordering the years and this ordering is, in fact, quite similar when one uses the criterion of infant mortality or that of life expectancy. Whether one uses the latter criteria or those of FSD or SSD, it is clear that the mortality level is much lower nowadays than it was a century ago, a fact which could explain. according to Becker [9], why people today study more, do more on-thejob training or migrate more easily. Such conclusions would relate the ordinal measure of health we propose to the profitability of investing in human capital. However, if one does not wish to tie the measure of the state of health of a country to the possibility of making profitable investments, one can still derive some interesting conclusions from the application of SD theory. Rather than taking a somehow narrow view of the health of a country, we may assume that what is important to a country is not the income people earn, but the number of years they live. This is the basis of the cardinal measures I(x) and xro of the health of a country, that we proposed. In Table 3 we present estimations of I(x) in England and Wales for every decade of the past hundred years, for different values of E, the elasticity of the marginal utility of the length of life. During the whole period there has been a clear improvement in the sense that there are today less differences among people in the number of years they live [l(x) is much higher today than it was a hundred years ago]. But the quantification of this improvement is very different for different values of E. The decrease in I(x) varies from more than 907, for E = 0.50 [l(x) decreasing them from 0.067 to 0.0073 to 40% for E = 1.75 [l(x) decreasing from 0.81 to 0.491. In Table 4 we compare the evolution of the distribution of deaths by age among men and women. This comparison of the value of I(x) for men and women is based on the assumption that it is possible to separate the social welfare derived Tram the number of years lived by men from the social welfare derived from the number of years lived by women. The data of Table 4 indicate that already in 1861, there was less inequality among women, and this difference still exists in the 1950’s and 1960’s. This is also the case for the life expectancy (see Table 4 also) so that from both points of view, that of the average number of years lived and that of the differences between individuals in the
Table 4. Male vs female mortality in England and Wales in the past century
Y’ear
Life expectancy
1861 1871 1881 1891 1901 1911 19’1 1931 1940 1951 1960 1964
40.5 39.2 44.3 41.9 45.3 49.4 55.9 58.2 59.4 65.9 68.2 68.6
Males Inequality index I(x) For E = 1.25 For E = 1 0.65 0.65 0.58 0.62 0.61 0.56 0.43 0.36 0.31 0.18 0.14 0.13
0.10 0.10 0.09 0.10 0.10 0.09 0.06 0.05 0.04 0.03 0.02 0.02
Life expectancy 43.1 42.5 47.5 45.7 49.4 53.4 59.9 62.4 63.9 71.0 74.1 74.8
Females Inequality index I(\-) Fore = 1.25 For E = I 0.60 0.60 0.53 0.57 0.56 0.51 0.39 0.31 0.26 0.15 0.12 0.1 1
0.09 0.09 0.08 0.09 0.09 0.08 0.05 0.04 0.04 0.02 0.02 0.02
JACQUES SILBER
1666
number of years lived, the situation of women is today better than it was a hundred years ago. Although we have analyzed only data concerning the level of mortality in .a given country, we could have made a similar analysis, had we had detailed data, on the number of sick days per year, such data being classified by the length of the period of sickness (classification in deciles or even finer classification). Unfortunately, there are very few reliable data of such a type and, in most countries, a dichotomic life-death distinction is still the most reliable one. IV. SUMMARY
AND CONCLUSIONS
We have tried to indicate in this paper how techniques which were originally devised for the study of other fields of economics like finance or income inequality, could be applied to the field of health economics provided health is seen as influencing the profitability of investment in other forms of human capital or as being the main determinant of the social welfare of a country. Given the shortage in detailed reliable data on the number of sick days per year in various categories of the population, we have proposed, as a first attempt of using the measures proposed, to apply the stochastic dominance criteria and Atkinson’s measure of inequality to life tables describing the level and structure of mortality in England and Wales since 1861. Our analysis has indicated that the inequality
before death is less important today than past. this improvement being parallel to in the average number of years lived (life It appears also that both improvements for women than for men during the sidered.
it was in the the increase expectancy). were greater period con-
REFERENCES 1. Arrow K. J. Essu~s ijr rhe Theor!, yf Risk-Brrrrirly. Elsevier. New York. 1974. 2. Hanoch G. and Levy H. The efficiency analysis of choices involving risk. Rrr. Econ. Stud. 36, 1969. 3. Hadar J. and Russell W. R. Rules for ordering uncertain prospects. Am. Econ. Rec. 59, 1969. risk: I. A 4. Rotschild M. and Stiglitz J. E. Increasing definition. J. Econ. Theory 2, 1970. 5. Rotschild M. and Stiglitz J. E. Increasing risk: II. Its economic consequences. J. Econ. Theorv 3, 1971. 6. Paroush J. and Silber J. Stochastic dominance criteria. mortality, risk and investment in education: a crosscountry comparison. Rex Firum. 2, in press. 7. Atkinson A. B. On the measurement of mequality. J. Econ. Theor! 2. 1970. 8. Preston S.. Keyfitz N. and Schoen R. Cuu.ses uf‘ Dear/i. Life Tables for National Populations. Seminar Press. New York. 1972. 9. Becker G. S. Hurnuri Cupitul. A Throrerrd ~tl ./%pirlcd Anulym with Speciui Referencr ro Eductrrron. National Bureau of Economtc Research, New York. 1964.
HEALTH SOME
APPLICATIONS
0277-9536,X2:lYl663-04$03.0010 Pergamon Press Ltd
AND INEQUALITY OF UNCERTAINTY
THEORY
JACQUES SILBER
Department
of Economics,
Bar-Ilan
University,
Ramat-Gan,
Israel
Abstract-This paper applies to the Field of Health results of Stochastic Dominance Theory and expressions originally devised for the measurement of Income Inequality. In particular. use is made of Atkinson’s inequality measure to compare health levels in England and Wales in the past century. It appears that the ‘inequality before death’ is less important today than it was in the past, this improvement being parallel to the increase in the average number of years lived (life expectancy). The analysis indicates also that both improvements were greater for women than for men during the period considered.
I. INTRODUCFION
his well-known Essu_~s in the Theory of Risk-Bearing, Arrow writes “The loss due to illness is only partially the cost of medical care. It also consists of discomfort and loss of productive time during illness, and in more serious cases. death or prolonged deprivation of normal functions. From the point of view of the welfare economics of uncertainty. both losses are risks against which individuals would like to insure” [ 11. This citation emphasizes clearly the role of uncertainty in the field of health economics. However, most of the attention has been paid to the effect of uncertainty in decisions involving medical care. But there are other uncertainty aspects related to health, which are .not less important. Health, for example. may influence the decisions of individuals to invest in their own human capital (e.g. education) and, of course, of the state to contribute to such investments. Such a relation between health and investment decisions implies naturally the use of measures summarizing the state of the health of a country. Among the most popular indexes. one has the life expectancy and the infant mortality. But these are only partial indexes which do not consider the whole probability distribution. However. if one wishes to consider the whole probability distribution and to relate the measure of the health of a country to the profitability of investing in some forms of human capital or. more generally, to the level of welfare of this country. it may be worthwhile to apply to this measurement problem some concepts which have been widely used recently in the finance literature. the criteria of stochastic dominance. The purpose of this paper is precisely to show how the theory of stochastic dominance could be applied to the problem of measuring the health of a country. In a first section we shall review the principles of stochastic dominance theory. in particular their application to the measurement of income inequality and indicate how they could be used to compare life tables. In
This
research Foundation Trustees.
was supported b! a grant from the Ford received through the Israel Foundations
1663
In a second section we propose to apply this technique to the measurement of the extent of mortality, essentially its distribution by age and sex, in-England and Wales, during the past 100 years.
II.
STOCHASTIC
DOMINANCE
COMPARISON
CRITERIA
AND
THE
OF LIFE TABLES
As is well known, the notion of stochastic dominance (SD) has been mainly used to order uncertain options and its importance is related to the fact that the rules which have been developed enable us to compare options even when the investor’s preference (utility function) is only partially known (e.g. only the sign of the first and second derivatives of the utility function are known). Originally, two decision rules, corresponding to two different classes of utility functions, have been devised and these SD criteria have been presented separately, more or less at the same time [2-S]. Let us show what these criteria mean when mortality data are used. A life table gives the probability P, of living up to age X. Since the probability of any investment in human capital depends. among other things, on the expected duration of the return flows, one may want to use the probabilities P, as weights to be attached to the utility derived from the earning flows R, which will be obtained if the individual lives up to age x. What do then the different criteria imply? The first one. called first degree stochastic dominance rule (FSD in short) makes only the assumption that the marginal utility of any return flow is not negative, but. as we shall see. it is very often difficult to derive clear conclusions. If F and G are respectively two different cumulative distributions of the deaths by age. FSD implies that distribution F will be preferred to distribution G if it lies below distribution G for at least
one age group
X. and
does
not
lie above
for any
other age group. The second degree stochastic dominance rule (SSD in short) assumes that the marginal utility of any return flow is not negative and that it does not increase with the amount of the return flow (in other words SSD assumes risk aversion). Using SSD one
JACQUESSILBER
I664
will say that distribution F will be preferred to distribution G if the rule devised for FSD applies to the cumulative distributions of the cumulative distributions F and G of deaths by age. Since very often the cumulative distribution of deaths by age intersect. it is clear that SSD can be used more often than FSD, which excludes such a possibility. The rules which have been presented hitherto are evidently ordinal rules. They do not tell us how much preferable is distribution F to distribution G. But, if we compare the life tables of a given country at two different times tF and tc, the criteria presented will tell us, for example, whether investing in education was more profitable, ceteris paribus, at time tF than at time tG. Similar comparisons could eventually be made at a given point of time if one wishes to compare the probabilities of investing in education in different countries. Such a study has in fact been presented recently
Table 1. Ordering the life tables of England and Wales according to the FSD and SSD criteria
C61.
where E(x) is the actual life expectancy (the actual mean of the distribution of deaths by age). To obtain an extimate of xEQ and of I(u) one may want to use the social welfare function U(x) proposed by Atkinson [7] where
Could one obtain also a cardinal classification of the distribution of deaths by age‘? We shall indicate now that if, besides the average health level of a given country, at a given time t, one wishes to take also into account ‘health inequality’ between individuals, there is a possibility of quantifying the differences between the distributions of deaths by age (here again we emphasize that we have limited our measure of health to a kind of dichotomic criterion distinguishing only between life and death, and not between degrees of good or bad health). The measure we propose is a simple application of Atkinson’s index of income inequality [7]. The idea is that utility is derived from the number of years lived and that there exists an additively separable and symmetric social welfare function of individual lengths of life. In this case, one may want to know what the utility (at the country level) or better the social welfare would be if everyone lived the same number of years. More precisely, one may want to calculate the ‘equally distributed equivalent length of life’ xEQ (‘equivalent length of life’ in short) which would give the same social welfare as the actual distribution of deaths by age. One would then measure the inequality in lengths of life as I(x)
= 1-
XEQ E(x)
Year
First degree SD criterion ordering
1871 1891 1901 1911 1921 1951 1960 1964
8 7 6 5 4 3 2 1
Year 1871 1891 1901 1911 1921 1931 1940 1951 1960 1964
10 9 8 7 6 : 3 2 1
U(x)=a+bs
when
E+ I
U(.u) = In (x)
when
E = 1.
and
It can be shown that such a utility function derives from the assumption of constant relative inequality aversion, i.e. of a constant elasticity E of the marginal utility of the length of life (see Cl] for an exposition of these concepts). With such a utility function, the inequality index can be written as
where d(xi) is the distribution of death by age x. By computing I(x) for a given country, at different points of time, one can obtain a quantititative indication of what was the evolution of the inequality of lengths of life during the period.
Table 2. Infant mortality and life expectancy in England and Wales since 1981
Year
Probability of dying before age one
Ordering of the infant mortality rate
Life expectancy
Ordering of life expectancy
1861 1871 1881 1891 1901 1911 1921 1931 1940 1951 1960 1964
0.160 0.164 0.132 0.158 0.158 0.133 0.082 0.066 0.056 0.030 0.023 0.021
11 12 7 9 9 8 6 5 4 3 2 1
41.8 40.9 45.9 43.8 47.4 51.4 58.0 60.3 61.7 68.4 71.1 71.5
11 12 9 10 8 7 6 5 4 3 2
,
Second degree SD criterron ordering
1
Health and inequality Table 3. Estimation of the Inequality index I(x) in England and Wales in the past century
Year
E = 0.50
1861 1871 1881 1891 1901 1911 1921 1931 1940 1951 1960 1964
0.067 0.068 0.054 0.059 0.057 0.047 0.033 0.025 0.02 1 0.010 0.008 0.007
Inequality index I(x) <=I E = 1.25 E = 1.5 0.62 0.63 0.56 0.60 0.59 0.54 0.35 0.34 0.29 0.09 0.19 0.12
0.096 0.095 0.083 0.091 0.093 0.082 0.059 0.048 0.042 0.022 0.018 0.017
We now turn to an illustration proposed to use.
111.MEASURING
0.44 0.44 0.41 0.44 0.44 0.42 0.34 0.30 0.27 0.18 0.15 0.14
0.81 0.81 0.79 0.81 0.81 0.80 0.75 0.71 0.68 0.71 0.51 0.49
of the technique
THE HEALTH
THE CASE OF ENGLAND
E = 1.75
we
OF A COUNTRY:
AND WALES IN
THE PAST HUNDRED
YEARS
In the previous section we used the basic results of the application to stochastic dominance theory to investment decisions and inequality measurement, to propose an ordinal as well as a cardinal classification of level of health of a country at different times or of different countries at a given point of time. The measurement of health was limited to that of the mortality level. Let us now present the results of an empirical investigation we made by using the life tables established by Preston et ~2. [S], for England and Wales, from 1861 to 1964. at intervals usually equal to 10 years. In Table 1 we present the ordering obtained by using respectively FSD and SSD. As we see the classification obtained is more precise in the sense that more countries can be classified when using SSD which puts stronger restrictions on the utility function In Table 2 we present data for the same years on infant mortality and life expectancy. It appears that
1665
here also one notices decades of regression (1871, 1891) but this does not prevent us from ordering the years and this ordering is, in fact, quite similar when one uses the criterion of infant mortality or that of life expectancy. Whether one uses the latter criteria or those of FSD or SSD, it is clear that the mortality level is much lower nowadays than it was a century ago, a fact which could explain. according to Becker [9], why people today study more, do more on-thejob training or migrate more easily. Such conclusions would relate the ordinal measure of health we propose to the profitability of investing in human capital. However, if one does not wish to tie the measure of the state of health of a country to the possibility of making profitable investments, one can still derive some interesting conclusions from the application of SD theory. Rather than taking a somehow narrow view of the health of a country, we may assume that what is important to a country is not the income people earn, but the number of years they live. This is the basis of the cardinal measures I(x) and xro of the health of a country, that we proposed. In Table 3 we present estimations of I(x) in England and Wales for every decade of the past hundred years, for different values of E, the elasticity of the marginal utility of the length of life. During the whole period there has been a clear improvement in the sense that there are today less differences among people in the number of years they live [l(x) is much higher today than it was a hundred years ago]. But the quantification of this improvement is very different for different values of E. The decrease in I(x) varies from more than 907, for E = 0.50 [l(x) decreasing them from 0.067 to 0.0073 to 40% for E = 1.75 [l(x) decreasing from 0.81 to 0.491. In Table 4 we compare the evolution of the distribution of deaths by age among men and women. This comparison of the value of I(x) for men and women is based on the assumption that it is possible to separate the social welfare derived Tram the number of years lived by men from the social welfare derived from the number of years lived by women. The data of Table 4 indicate that already in 1861, there was less inequality among women, and this difference still exists in the 1950’s and 1960’s. This is also the case for the life expectancy (see Table 4 also) so that from both points of view, that of the average number of years lived and that of the differences between individuals in the
Table 4. Male vs female mortality in England and Wales in the past century
Y’ear
Life expectancy
1861 1871 1881 1891 1901 1911 19’1 1931 1940 1951 1960 1964
40.5 39.2 44.3 41.9 45.3 49.4 55.9 58.2 59.4 65.9 68.2 68.6
Males Inequality index I(x) For E = 1.25 For E = 1 0.65 0.65 0.58 0.62 0.61 0.56 0.43 0.36 0.31 0.18 0.14 0.13
0.10 0.10 0.09 0.10 0.10 0.09 0.06 0.05 0.04 0.03 0.02 0.02
Life expectancy 43.1 42.5 47.5 45.7 49.4 53.4 59.9 62.4 63.9 71.0 74.1 74.8
Females Inequality index I(\-) Fore = 1.25 For E = I 0.60 0.60 0.53 0.57 0.56 0.51 0.39 0.31 0.26 0.15 0.12 0.1 1
0.09 0.09 0.08 0.09 0.09 0.08 0.05 0.04 0.04 0.02 0.02 0.02
JACQUES SILBER
1666
number of years lived, the situation of women is today better than it was a hundred years ago. Although we have analyzed only data concerning the level of mortality in .a given country, we could have made a similar analysis, had we had detailed data, on the number of sick days per year, such data being classified by the length of the period of sickness (classification in deciles or even finer classification). Unfortunately, there are very few reliable data of such a type and, in most countries, a dichotomic life-death distinction is still the most reliable one. IV. SUMMARY
AND CONCLUSIONS
We have tried to indicate in this paper how techniques which were originally devised for the study of other fields of economics like finance or income inequality, could be applied to the field of health economics provided health is seen as influencing the profitability of investment in other forms of human capital or as being the main determinant of the social welfare of a country. Given the shortage in detailed reliable data on the number of sick days per year in various categories of the population, we have proposed, as a first attempt of using the measures proposed, to apply the stochastic dominance criteria and Atkinson’s measure of inequality to life tables describing the level and structure of mortality in England and Wales since 1861. Our analysis has indicated that the inequality
before death is less important today than past. this improvement being parallel to in the average number of years lived (life It appears also that both improvements for women than for men during the sidered.
it was in the the increase expectancy). were greater period con-
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