More generally, as Willis 1986 observed, ‘‘the term ‘earnings function’ has come to mean any regression of individual wage rates or earnings on a vector of personal,
market, and environmental variables thought to influence the wage’’. Departing from the approach of Gibrat 1931, Rutherford 1955, Mandelbrot
1960, 1963, who considered chance as a dominant variable determining the levels of earnings, whereas individual choice was not, Mincer 1958, 1970 specified the
first model of earned income derived from individual investment behavior, i.e. individual choice. He took ‘‘the length of training as the basic source of heterogene-
ity of labor incomes’’ Mincer, 1970. This training raises productivity and post- pones the entrance age into the labor market. The specification of Mincer’s earning
function is derived from four highly simplifying assumptions, where the first and the fourth are only mathematical devices to rationalize an obvious explanatory
variable, i.e. years of schooling, which suggests instead a direct specification based on factual observations.
The assumptions of Mincer 1970 are, A.1: ‘‘In a competitive equilibrium, the distribution of earnings is such that the
present values of future earnings discounted at the market rate of interest are equalized al the time training begins’’.
A.2: ‘‘The model is formulated in terms of training periods which are completed before earnings begin’’.
A.3: ‘‘No further investments in human capital are undertaken by individuals after completion of their schooling’’.
A.4: ‘‘The flow of their earnings is constant throughout their working lives’’. Becker 1975 extended Mincer’s approach by including postschool investments
in HC, which can be disaggregated to deal with different types of investment, such as, training, health, and migration. After a sequence of simplified assumptions,
Becker specified earned income as a function of years of schooling and postschool investments in HC, i.e.
log E
t
= log E
+ rs + rP
t
, 3
where, in period t, E stands for earnings, s for years of schooling, and P for net postschool investment, whereas E
stands for raw earnings, i.e. earnings without schooling and postschool investment in HC; r, r and E
are parameters to be estimated.
3. Human capital: methods of estimation
Traditionally, two methods of HC estimation were advanced, i the prospective, and ii the retrospective methods. In the 1980s, a third method was proposed as a
proxy or an indicator of the stock of HC, i.e. iii a measure of the average education stock, enrollment data, or years of schooling of the working age
population. In Section 4, we will present a new method of estimation of the personal HC as a latent variable combined with the prospective method of
estimation. We also introduce the average HC by age of the economic units under inquiry.
3
.
1
. The prospecti6e approach This was the first method used to estimate personal and national HC. Petty
1690 was the most prominent founder of the Political Arithmetic school of economics and a precursor to modern applied econometrics. He was the first author
to apply the prospective method to estimate the HC of a nation with the purpose of assessing the loss sustained by a plague, by the slaughter of men in war, and by
migration. His method purported to offer also a sound base for taxation and to evaluate the power of a nation.
Petty estimated the HC of England as the difference between his estimation of national income £42 million and property income rent of land 8 millions and
profit 8 millions capitalized in perpetuity at a 5 interest rate, arriving at a total HC estimation of £520 million; or a per capita estimation of £80. Although Petty’s
approach was a very crude one, it had the merit of raising the issue, giving an answer, and making an economic and social interpretation of the result obtained.
A rigorous scientific approach, that applies actuarial mathematics to estimate individual human capital, was developed by Farr 1853. He provided the basic
approach and the scientific standard to estimate the gross and the net economic value of a human being to offer it as a basis for an equitable taxation of individual
physical and human capital stocks. Farr 1853 stated that ‘‘the characteristic of life property in wages, …, is that it is inherent in man, and is the value of his services
— of the direct product of his skill and industry, …, [which] is not the less on that account property’’.
Observing the salary and the maintenance cost of agricultural laborers by age, and adopting a 5 discount rate and a life table, Farr estimated £349 to be the
average gross value of an agricultural laborer, £199 to be the average maintenance cost, hence, he estimated £150 to be the average net HC value.
Wittstein 1867 applied Farr’s prospective method to estimate the HC of an individual at several ages. His results are distorted by the adoption of the unaccept-
able postulate that at birth, the flow of incomes of an individual throughout his life and the expenses of a given individual’s maintenance are equal. This assumption
contradicts human history which testimonies that, in average, individuals add to the national well-being of societies.
Marshall 1922 adopted Wittstein’s controversial postulate and stated that, ‘‘many writers assume, implicitly at least, that the net production of an average
individual and the consumption during the whole of his life are equal, or in other words, that he would neither add to nor take from the national well-being of a
country, …. On this assumption, the above two plans of estimating his value would be convertible; and then, of course, we should make our calculations by the latter
and easier the consumption method’’.
Unlike Farr who estimated individual HC, Nicholson 1891 adopted Petty’s macroeconomic approach combined with the retrospective cost of production
method developed by Engel 1883 to estimate HC. He added the value of what he called ‘domesticated humanity’ sic, to arrive at an estimation of the living capital
of Great Britain.
A French actuarian, Barriol 1910 used the prospective method to estimate the social value of an individual. By social value, he meant the amount that an
individual restores to society out of his earnings. Barriol’s social value would be equivalent to Farr’s present net value of an individual at birth.
de Foville 1905 estimated HC in France by capitalizing labor income net of consumption expenditures. Fisher 1927 applied Farr’s prospective method to
estimate average U.S. human capital and then, by applying a mortality table, he advanced an estimation of the cost of preventable illness.
Following Farr’s seminal contribution, Dublin and Lotka 1930 published an influential research monograph on the money value of a human being. Their model
is now discussed. Let Vx be the net value of a person of age x; 6
x
= 1 + i
− x
the present value of a unit of money due x years later, where i is the discount rate; pa, x = la + x
lx the probability at age a of living to age a + x; lx the population of age x; yx the annual earnings of a person of age x; Ex the annual rate of employment
at age x, hence, Ux = 1 − Ex is the annual rate of unemployment at age x; and cx is the annual cost of living of a person at age x. For the sake of notational
simplification, we work with age x, instead of x + 12 as the representative age in a calendar year.
The net value of a human being net HC at age a is the present actuarial value of a flow of net annual expected earnings, i.e.
Va =
x = a
6
x − a
[yxEx − cx]pa, x 4
Hence, at birth, V0 =
x = 0
6
x
[yxEx − cx]p0, x, 5
i.e. Barriol’s social value of an individual. It follows from Eq. 4 that the net cost at age a of rearing a person from birth
to age a is, Ca =
a − 1 x = 0
1 + i
a − x
[cx − yxEx] px, a
6 The denominator in Eq. 6 means that Ca includes the per capita net cost for
the surviving population at age a of those that died at age x B a. It follows from Eqs. 4 – 6 that,
Va = 1 + i
a
p0, a
x = a
6
x
[yxEx − cx]p0, x =
V0 1 + i
a
p0, a +
Ca. Hence,
Ca = Va − V0 1 + i
a
p0, a 7
The gross HC at age a is obtained from Eq. 4 after making cx = 0 for all x, i.e.
Gross HCa =
x = a
6
x − a
yxExpa, x 8
In the second half of the 20th century, little research was done using the prospective method. Among them, we mention the comprehensive estimates of
Jorgenson and Fraumeni 1989. These authors proposed a new system of national accounts for the U.S. economy that included market and non-market economic
activities with the purpose of assessing the role of capital formation in U.S. economic growth. Jorgenson and Fraumeni 1989 defined full labor compensation
as the sum of market and non-market labor compensation after taxes. They assumed p. 233 that expected incomes in future time periods are equal to the
incomes of individuals of the same sex and education, but with the age that the individual will have in the future time period, adjusted for increases in real income.
The authors estimated the human and non-human capital for the US from 1949 to 1984. They used hourly labor compensation annually for individuals classified by
the two sexes, 61 age groups, and 18 education groups for a total of 2196 groups.
Macklem 1997 estimated quarterly per capita HC for Canada, from 1963 to the second quarter of 1994. He computed aggregate HC as the expected present value
of aggregate labor income net of government expenditures based on an estimated bivariate vector autoregressive VAR model for the real interest rate and the
growth rate of labor income net of government expenditures.
3
.
2
. The retrospecti6e approach Although A. Smith implicitly proposed the cost of production ‘‘a man educated
at the expense of much labor and time’’ as a main determinant of differential wages, Engel 1883 was the first to advance and apply the retrospective method of
HC estimation. He was not attracted to the prospective method because of the weight he gave to outstanding outliers such as Goethe, Newton, and Benjamin
Franklin. He argued that the HC of these extreme cases could not be estimated for lack of knowledge about their future earnings, instead, he said it was possible to
estimate their rearing costs to their parents.
Studying the budget of Prussian working families, Engel adopted very crude assumptions to arrive at the estimation of the cost of production. He considered
three lower, middle, and upper classes, assumed a cost c
i
i = 1, 2, 3 at birth of the ith class, increasing it annually in an arithmetic progression until the age of 25.
At 26, he considered that a human being was fully produced. For each year of age, Engel increased the cost c
i
at birth by the constant amount c
i
q
i
, hence, the annual cost of rearing a person of age x B 26, belonging to the ith class, becomes c
i
+ xc
i
q
i
. Adding the historical cost from birth up to the age x B 26, he obtained
c
i
x = c
i
1 + x + q
i
xx + 1 2
n
, i = 1, 2, 3,
9
as the cost of production of a human being up to the age x B 26. Engel assumed q
i
= q = 0.10 constant and c
i
equal to 100, 200, and 300 marks for the lower, middle, and upper German social classes, respectively.
Besides the simplicity of Engel’s assumptions, his approach should not be taken as an estimation of individual HC. It is only a historical cost estimation that
neglects to capitalize the imputed cost of past years and ignores the imputation of social costs such as education, health services, sanitation, recreation, and the social
cost of mortality and emigration.
Until the third decade of the 20th century, Engel’s approach inspired much research which focused on the estimation of monetary losses of preventable illness
and death, emigration and war. Among the authors that dealt with these issues were Pareto 1897, Beneluce 1904, Pareto 1905, Sensini 1908, Ros Jimeno
1931, Pietra 1931, Ferrari 1932, Mortara 1934. Gini 1931, 1954 made a cogent analysis of the components of economic cost of a human being.
More recently, Machlup 1962, Nordhaus and Tobin 1972, Kendrick 1976, Eisner 1978 employed the cost of production approach to the estimation of stocks
and flows of investment in HC, opening the way to the construction of HC time series with the application of the perpetual inventory method.
3
.
3
. The educational stock approach Unlike the prospective and the retrospective methods that deal with HC estima-
tion, this approach considers the educational attainment or school enrollment by countries or regions as proxies for HC. Hence, it circumvents the estimation of HC.
Barro 1991, Mankiw et al. 1992 used measures of school enrollment; Romer 1989, Azariadis and Drazen 1990 used adult literacy rates; Psacharopoulos and
Arriagada 1986 adopted as a proxy the average years of schooling embodied in the labor force; Lau et al. 1991 considered as the educational capital stock, the
number of person-school years of the working age population; Nehru et al. 1995 estimated the educational stock of the working age population for 85 countries
from 1960 to 1987. They defined it as the sum of primary, secondary and post-secondary education stock of the population between the ages 15 and 64. The
series are built from enrollment data using the perpetual inventory method, adjusted for mortality.
Mulligan and Sala-i-Martin 1997 considered estimates based on the number of individuals in an economy with s years of schooling weighted by an efficiency
parameter that the authors made equal to the wage rate of individuals with s years of schooling divided by the wage rate of individuals with zero years of schooling.
After reviewing the approaches presented in the literature to estimate the educational stock of the working age population, Laroche and Me´rette 1999
estimated an index number of HC for Canada, with base 100 in 1976. Working with the quinquennial Canadian Censuses from 1971 to 1996 and with the annual
inflows of high school and postsecondary graduates, these authors estimated an index number based on the years of schooling and the years of working experience
of the Canadian working age population 15 – 64 weighted by an efficiency parame- ter defined as the proportion of wage income of workers with s years of schooling
and x years of experience in the total wage bill of the economy.
4. Proposal of a new approach to estimate the average human capital of a population and its distribution by size
The new approach presented in this paper to estimate HC further develops the method introduced by Dagum 1994, Dagum and Vittadini 1996. It estimates
personal such as households, families, member of the labor force, and working age population HC, its size distribution, the average level of HC by age, and the
average level of HC of the population. From this approach, we arrive at a specific monetary value of HC and not just at a proxy index number for HC.
The estimation of personal HC, its distribution, the average HC by age, and the average HC level of the population of economic units are obtained from sample
surveys of income and wealth data as explained below in points 1 – 6. 1. From the information available in a sample survey, we choose what we retain as
the most relevant indicators that determine the HC of each economic units. Unfortunately, the available sample surveys do not provide socioeconomic
information of the parents of the household head and spouse, nor measure of intelligence, ability and other indicators of genetic endowment of the household
head and spouse. From a selection of p indicators, we specify the following HC linear equation,
z = Lx
1
, x
2
, …, x
p
, 10
where z stands for the standardized zero mean and unit variance HC latent variable, and x
1
, x
2
, …, x
p
are p standardized indicators. 2. Once Eq. 10 is estimated, to pass from zi in Eq. 10 to hi in an
accounting monetary value, where i stands for the ith economic unit, we apply the following transformation:
hi = exp z
i
. 11
The average value of hi is A6h =
i = 1 n
hi fi
i = 1 n
fi 12
where n is the sample size and fi is the weight attached to the ith sample observation, because these observations are not purely random.
3. To estimate average personal HC, we proceed as follows, 3.1. We order the sample observations by age of the economic units age of the
head when the economic units are households or families.
3.2. For each age x we obtain the total earnings and the size of the population they represent.
3.3. The total earnings by age is equal to the sum of the products of the earnings of each economic unit of age x times the number of economic
units it represents in the population, i.e. its weight. Dividing this total amount by the total weight by age, we obtain the average earnings by age.
3.4. To eliminate large random fluctuations, we smooth the average earnings and the total weights by age, applying a seven-term weighted moving
average 3 × 5 ma. Hence, our smoothed average earning yx and its corresponding weight fx are our representative cross-section data for the
estimation of HC. The levels of yx reveal mainly the ability, drive, determination, dynamism, choice investment in education, on the job
training, postschool investment, health, etc., home, and social environ- ment of the average economic units of age x.
3.5. In the absence of temporal technological changes and without increases in HC productivity, the representative average earnings of the economic units
of age x, t years later, is given by the average earnings yx + t of the economic units of age x + t. Hence, under these simplified assumptions, the
cross-section and life-cycle average earnings are equal. Thus, given a discount rate i and the mortality table of a population, the HC of the
average economic unit of age x is,
hx =
70 − x t = 0
yx + tpx, x + t1 + i
− t
13 for x = 20, 21, …, 70, i.e. for a working population age 20 – 70.
It follows from Eq. 13 that the weighted average of the population HC is,
A6HCh =
x = 20 70
hxfx
x = 20 70
fx 14
3.6. In real life, economic processes incorporate technological changes, higher educational levels, hence, the productivity of HC increases through time
yielding a process of economic growth. For these reasons, the cross-section average HC hx will not be equal to the life cycle time series realization
of average HC at age x.
Assuming a HC productivity increase at the annual rate r, it follows from this assumption and Eqs. 13 and 14, that average HC at age x is
hx, r =
70 − x t = 0
yx + tpx, x + t1 + r
t
1 + i
− t
15 and the corresponding average HC of the population is
A6HCh =
x = 0 70
hx, rfx
x = 0 70
fx 16
4. To arrive at the current monetary value of the HC estimation of the n sample observations we obtain the ratio between the average HC given by Eq. 14 and
the average of the transformation Eq. 11 given by Eq. 12, and multiply it by hi as given by Eq. 11. Hence, the HC of the ith sample observation is,
HCi = hi A6HCh
A6h ,
i = 1, 2, …, n. 17
which gives the vector of human capital in the corresponding national monetary unit. It represents the empirical HC corresponding to the sample survey object
of research. 5. To the vector of empirical HC obtained from Eq. 17 we fit the three-parameter
Dagum 1977, 1996 model Fh = 1 + lh
− d − b
, h \ 0,
b, l \ 0, d \
1, 18
or the four-parameter Dagum model Fh = a + 1 − a1 + lh
− d − b
= [1 + lh − h
− d
]
− b
, h \ h
] 0,
a 0,
b, l \ 0, d \
1, 19
to obtain a parametric representation of the level of personal HC and an estimation of the degree of inequality in the distribution of HC, as measured by
the Gini ratio. 6. Table 1 presents an illustration of the proposed new method of HC estimation.
Step A in Table 1 deals with the development presented in 1 and 2. It is concerned with, i the specification and estimation of Eq. 10 for each
economic unit in a sample survey as a standardized latent variable; ii applying Eq. 11, the latent variable is transformed into an accounting monetary unit,
i.e. hi ; and iii applying Eq. 12, the average accounting monetary value of HC is estimated.
Step B deals with the prospective actuarial mathematics method of HC estimation. It is applied as follows a as explained in points 31 – 4, the flow
of earned income by age, yx, is obtained; b applying Eq. 13, we obtain the monetary estimation of the average HC by age, hx, of the economic units
considered; c applying Eq. 14 to hx, we obtain the weighted average of the population HC. This weighted average is a cross-section estimation of the
average HC because it did not incorporate the expected time path productivity increase of each economic unit. Consequently, the average HC by age follows a
stationary process with a linear representation. Therefore, for large samples, both averages in time and frequency domain converge to the same value.
To incorporate the dynamic process of economic growth, technological changes and HC accumulation, we have to assume a given pattern of productiv-
ity increase by age to obtain, applying Eq. 15, the life cycle estimation of HC by age and, applying Eq. 16, the corresponding HC population average.
The combination of Steps A and B allows the monetary estimation of each economic unit HC. This synthesis is explained in 4 and presented in Table 1.
Point 5 specifies the three- and four-parameter Dagum probability distribu- tion function as a model of the size distribution of HC to fit the economic unit
HC data obtained by application of Eq. 17.
Table 1 Illustration of a new proposed approach to estimate the level and the distribution of human capital of
the members of a sample survey
5. Methods of HC estimation: an evaluation
