On the Measurement of Inequality in the Quality of Life in Israel
by
Joseph Deutsch
Jacques Silber
and Nira Yacouel
Department of Economics
Bar-Ilan University
52900 Ramat-Gan
Israel
Email: [email protected]
May 2001
To be presented at the conference on
Justice and Poverty: Examining Sen’s Capability Approach
Cambridge, 5, 6 and 7 June 2001
The Von Hügel Institute, St Edmund's College, University of Cambridge
Key words: commodities - corrected ordinary least squares - distance function efficiency - functioning - Israel - Malmquist index - quality of life resources - standard of living - stochastic frontier
I. Introduction: Making a Distinction Between the Standard of Living and the
Quality of Life:
Most of the international comparisons of the standard of living are based on indicators
such as the per capita Gross Domestic Product or Disposable Income (see for example
the annual reports of the World Bank). Sometimes use is made of data on household
expenditures or consumption rather than on income, either because this type of data
may be more commonly available than income data or because consumption is
assumed to be a better approximation of the permanent income of the household (see
for example, I.L.O., 1992).
Attempts have also been made to measure the quality of life and not only the
efficiency in producing goods and services. Nordhaus and Tobin (1972) for example
suggested to include leisure when measuring welfare while Usher (1973) proposed to
take into account changes in life expectancy when measuring economic growth. This
emphasis on life expectancy appears also in Hicks and Streeten’s (1979) survey of the
literature dealing with “basic needs”.
Other authors (e.g. Chenery et al., 1974) have stressed that distributional
considerations should not be ignored in computing development indices. Sen (1976)
has indeed insisted on the fact that one cannot say what total national income is, at any
point of time, until one knows its distribution.
The strongest departure from the traditional approach which identifies development
levels with standards of living may indeed be found in several of Sen’s works, in
particular in a study where he explains the difference between Commodities and
Capabilities (Sen, 1985). Following earlier studies recommending to look at
commodities in terms of their characteristics (Lancaster, 1966) and at the household as
a producing unit (Becker, 1976), Sen (1985) suggested to make a distinction between
commodities, characteristics, functionings and capabilities. The characteristics refer
to the various desirable properties of the commodities. Food for example is used “to
satisfy hunger, yield nutrition, to give eating pleasure and to provide support for social
meetings” (Sen, 1985). But knowing the characteristics of the goods does not allow us
to tell what the individual will be able to do with these properties. A person suffering
from a parasitic disease may suffer from undernourishment even though he
theoretically has “enough food”. This is why Sen proposed to look at the
“functioning” of individuals, that is at what the person succeeds in doing with the
goods and characteristics he has command over. The “capabilities” finally refer to the
various combinations of functionings an individual can achieve.
The desire to translate Sen’s ideas into applied research is probably at the origin of
the concept of “Human Development Index” which is computed each year for many
countries and is published annually in the Human Development Report ( e.g.
U.N.D.P., 1997). This index takes into account not only the real per capita G.D.P. of a
country but also its life expectancy at birth and the value of an index measuring its
educational level and the data indeed show that there are important differences
between the classification of countries on the basis of the G.D.P. per capita and
according to the level of the Human Development Index.
Although the computation of the latter is not a straightforward task, there have been
recently more sophisticated attempts to give an empirical content to Sen’s distinction
between commodities and functionings. Lovell et al. (1994) have for example applied
the concept of distance function, quite widely used in production theory, to Australian
household data on the standards of living and compared inequality in resources
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(standards of living) with that in functionings (quality of life) and in the ability to
transform resources into functionings. This very original approach was later adopted
also by Delhausse (1996) who analyzed data on the standard of living and the quality
of life of French households and individuals.
The purpose of the present study is to apply Lovell et al.’s (1994) methodology to data
collected in a survey on the allocation of time of individuals in Israel. Particular
emphasis will however be given to the differences which exist between individuals of
different (ethnic) origin in the standards of living, the quality of life and the efficiency
in transforming resources into functionings. The paper is organized as follows. The
next section defines the concept of distance function as it has appeared in the literature
on duality in production and consumption. Section III then shows how distance
functions may also be used to estimate standards of living, quality of life and the
efficiency in transforming resources into functionings. Section IV summarizes the
econometric techniques which have to be applied while Section V describes the data
base. Section VI shows how different measures of inequality and poverty may be
derived when they are based on the standard of living or on the quality of life indices.
Section VII then tries to better understand these differences and looks at the
determinants of the standard of living, of the quality of life and of the efficiency with
which individuals transform their standard of living into quality of life. Concluding
comments are given in Section VIII.
II. Duality Theory and the Concept of Distance Function
A) The Distance Function in Consumption Theory:
Let q be a vector of quantities and p the corresponding price vector. One way of
defining the consumer’s problem is to assume that she wants to maximize utility u for
a given outlay o = p.q . It is well known that the solution to this maximization
problem yields the Marshallian demand functions q =g (p,o). By substituting these
solutions into the original problem we then derive the indirect utility function ψ (p,o),
the maximum utility which may be reached at prices p with an outlay o.
An alternative approach is to assume that the consumer wants to minimize the
expenditures required to reach utility u. This cost minimization problem yields the
Hicksian or compensated demand functions q = h(p,u) , which, when substituted into
the original problem, gives the cost function o = c(p,u) which is defined as the
minimum cost of attaining u at prices p.
It may also be proven that by deriving the cost function with respect to prices we
obtain the Hicksian demand functions while by using Roy’s identity one may derive
the Marshallian demand functions from the indirect utility function. This duality
between a direct and indirect representation of preferences explains why, for reasons
of convenience, one may sometimes prefer to use the cost rather than the indirect
utility function. We will now show (see, Deaton, 1979, and Deaton and Muellbauer,
1980, for more details) that it is also often helpful to have an inverse representation of
the direct utility function which is called the distance function.
Let q represent an arbitrary quantity vector and u an arbitrary utility indifference
curve. It is not assumed that u = v(q) . The distance function d(u,q), defined on u and
q, represents the amount by which q must be divided in order to bring it on to the
indifference curve, so that v[q/d(u,q)] = u. Geometrically in Figure 1, d(u,q) is the
3
ratio OB/OA. Note that if q happens to be on u, B and A coincide so that u = v(q) if
and only if d(u,q) =1. There is thus an analogy with the indirect representation of
preferences since u = ψ (p,o) if and only if c(p,u) = o.
This link between the distance and the cost functions may be expressed also in the
following way. Let q’ be equal to OA in Figure 1 and let λ be equal to the ratio
OB/OA. We can then write that q’ = q/λ so that
d(u,q) c (p,u) = λ c (p,u) ≤ (λq’) p = q p
In other words q’, which is obtained by a proportional change λ in the quantities
defined by q, does not necessarily yield the cheapest cost c(p,u) of reaching utility u at
prices p. But there exists at least one vector price p for which d(u,q) = q.p/ c(p,u).
We may therefore write that
d(u,q) = Minp p.q such that c(u,p) =1
as well as
c(p,u) = Minq p.q such that d(u,q) = 1.
The distance and cost functions are clearly dual to one another: just as the cost
function seeks out the optimal quantities given u and p, the distance function finds the
prices that will lead the consumer to reach utility u by acquiring a vector of quantities
proportional to q.
When deriving the distance function d (u,q) with respect to prices we obtain functions
ai(q,u) which are dual to the Hicksian demand functions hi (p,u) and are compensated
inverse demand functions, that is we may write:
δd (u,q) / δqi = ai (q,u) = pi /o.
For any quantity vector q and any utility level u, these compensated inverse demand
functions give the prices as a function of o that will cause u to be reached at a point
proportional to q (see Figure 2). These inverse demands may be given a simple
intuitive interpretation: they represent the amounts of money an individual on u,
consuming quantities proportional to those defined by the vector q, will be willing to
pay for one more unit of each of the goods. It should be stressed that such inverse
demand functions are convenient to use when prices do not exist because they then
define shadow prices associated with u and the quantity proportions defined by the
vector q. It is therefore not surprising that the concept of distance function is often
used when prices are not available and such an application has turned out to be very
useful in production theory (see, Coelli, Prasada Rao and Battese, 1998, for a detailed
introduction to the topic).
4
B) The Distance Function in Production Theory:
1) Input Distance Functions:
Let L(y) represent the input set of all input vectors x which can produce the output
vector y. That is
L(y) = {x: x can produce y}.
The input distance function di (x,y) involves then ( in a way similar to that in which
this concept is used in consumption theory) the scaling of the input vector and will be
defined as
di (x,y) = Max {ρ : (x/ρ) ∈ L(y) }
It may be proven that
(1) the input distance function is increasing in x and decreasing in y;
(2) it is linearly homogeneous in x;
(3) if x belongs to the input set of y (i.e. x∈L(y) ) then di (x,y)≥1;
(4) the input distance function is equal to unity if x belongs to the “frontier” of the
input set (the isoquant of y).
A representation of the input distance function is given in Figure 3.
2) Output Distance Functions:
Distance functions may also be used in the context of a multi-input, multi-output
production technology. Let P(x) represent the set of all output vectors y which can be
produced using the input vector x. That is
P(x) = { y : x can produce y }
The output distance function do(x,y) is then defined as
do(x,y) = Min {δ: (y/δ) ∈ P(x) }.
The following properties of the output distance function may be proven:
(1) do(x,y) is increasing in y and decreasing in x;
(2) do(x,y) is linearly homogeneous in y;
(3) if y belongs to the production possibility set of x (i.e. y∈P(x) ), then do(x,y)≤1;
(4) the output distance function is equal to unity if y belongs to the “frontier” of the
production possibility set (to the production possibility curve of x).
An illustration of this concept is given in Figure 4.
5
3) Productivity Indices and the Distance Function:
Another interesting application of the concept of distance function concerns
productivity indices.
a) The Malmquist Output Index:
Let yt, ys, xt and xs represent respectively the quantities of output y produced by a firm
and the inputs x it uses at times t and s.
Using the concept of output distance function defined previously, the Malmquist
output index, based on technology at time t, is defined, for an arbitrarily selected input
vector x, as:
Qot (ys, yt, x) = dot (x, yt ) / dot (x, ys)
A similar Malmquist output index may be defined using the technology available at
time s. Naturally we can define alternative indices using different levels of x. It should
be stressed that the Malmquist output index will be independent of the technology
involved if and only if the technology exhibits Hicks output neutrality. Similarly it
will be independent of the input level x if and only if the technology is output
homothetic.
If we consider output quantity indices based on technology in periods s and t, together
with the inputs used in these periods, we have then two possible measures of output
change, Qos (yt, ys, xs ) and Qot (yt, ys, xt ).
It can be proven that if the distance functions for periods s and t are both represented
by translog functions with identical second-order parameters, then a geometric
average of the Malmquist output indices Qot (ys, yt, xt ) and Qos (ys, yt, xs ) is
equivalent to the Tornquist output quantity index (this index is a weighted geometric
average of the ratios of the quantity at periods t and s, the weights being the simple
arithmetic averages of the value shares in periods s and t). Note also that the
Malmquist index is the only one which satisfies the homogeneity property according
to which if yt = λ ys, then Qot (ys, yt, x) = Qot (ys, λys, x) = λ.
b) The Malmquist Input Index:
Using the concept of input distance function, the Malmquist input quantity index
Qit(xs,xt,y) will be defined as
Qit(xs,xt,y) = dit (xt, y) / dis (xs, y)
Naturally we can define in a similar way an input quantity index Qis(xs,xt, y) based on
period-s technology. It can also be proven that when the distance functions are of the
translog form and have in both periods identical second-order parameters and if the
assumptions of allocative and technical efficiency1 hold, then the geometric average of
these two indices is equal to the Tornquist input quantity index (this index is a
1
Technical efficiency refers to the fact that the firm operates on the production frontier while
allocative efficiency refers to the selection of a mix of inputs which produces a given quantity of output
at minimum cost.
6
weighted geometric average of the ratios of the inputs at periods t and s, the weights
being the simple arithmetic averages of the input cost-shares in periods s and t).
c) The Malmquist Productivity Index:
- output-oriented indices: They focus on the maximal level of outputs that could be
produced using a given input vector and a given production technology relative to the
observed level of outputs. Their computation is based on the use of output distance
functions and the period-s Malmquist productivity index mos (ys,yt,xs,xt) is then
defined as
mos (ys,yt,xs,xt) = dos (yt, xt) / dos(ys, xs)
If we assume that the firm is technically efficient in both periods, dos (ys, xs ) = 1 so
that
mos (ys,yt,xs,xt) = dos (yt, xt)
One can define in a similar way an output-oriented Malmquist productivity index
based on period-t technology.
- input-oriented indices: It can be shown similarly that the input-oriented Malmquist
productivity index mis (ys, yt, xs, xt) is equal to dis(yt,xt) if the firm is technically
efficient in both periods.
III) Estimation Procedures:
Let us take as a simple illustration the case of a Cobb-Douglas production function.
Let ln yi be the logarithm of the output of firm i (i=1 to N) and x i a (k+1) row vector,
whose first element is equal to one and the others are the logarithms of the k inputs
used by the firm. We may then write that
ln (yi) = xi β - ui
i = 1 to N.
where β is a (k+1) column vector of parameters to be estimated and ui a non-negative
random variable, representing the technical inefficiency in production of firm i.
The ratio of the observed output of firm i to its potential output will then give a
measure of its technical efficiency TEi so that
TEi = yi /exp (xi β) = exp (xi β - ui) / exp (xi β ) = exp (-ui)
One of the methods allowing the estimation of this output-oriented Farrell measure of
technical efficiency TEi (see, Farrell, 1957) is to use an algorithm proposed by
Richmond (1974) which has become known as corrected ordinary least squares
(COLS). This method starts by using ordinary least squares to derive the (unbiased)
estimators of the slope parameters. Then in a second stage the (negatively biased)
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OLS estimator of the intercept parameter β0 is adjusted up by the value of the greatest
negative residual so that the new residuals have all become non-negative. Naturally
the mean of the observations does not lie any more on the estimated function: the
latter has become in fact an upward bound to the observations.
One of the main criticisms of the COLS method is that it ignores the possible
influence of measurement errors and other sources of noise. All the deviations from
the frontier have been assumed to be a consequence of technical inefficiency. Aigner,
Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) independently
suggested an alternative approach called the stochastic production frontier method in
which an additional random error vi is added to the non-negative random variable ui.
We therefore write
ln (yi) = xi β + vi - ui
The random error vi is supposed to take into account factors such as the weather, the
luck, etc...and it is assumed that the vi‘s are i.i.d. normal random variables with mean
zero and constant variance σv2. These vi’s are also assumed to be independent of the
ui’s, the latter being taken generally to be i.i.d. exponential or half-normal random
variables. In the latter case where the ui’s are assumed to be i.i.d truncations (at zero)
of a normal variable N(0,σ), Battese and Corra (1977) suggested to proceed as
follows. Calling σs2 the sum σ2 + σv2 , they defined the parameter γ as γ=σ2/σs2 (so
that γ has a value between zero and one) and showed that the log-likelihood function
could be expressed as
ln(L) = -(N/2) ln(π/2) - (N/2) ln(σs2) + ∑i=1 to N [1-Φ(zi)]-(1/(2σs2))∑i=1 to N (ln yi-xiβ)2
where zi =((ln yi - xi β)/σs) √(γ/(1-γ)) and Φ(.) is the distribution function of the
standard normal random variable.
The Maximum Likelihood estimates of β, σs2 and γ are obtained by finding the
maximum of the log-likelihood function defined previously where this function is
estimated for various values of γ between zero and one. More details on this
estimation procedure is available in programs such as FRONTIER (Coelli, 1992) or
LIMDEP (Green, 1992). The same methods (COLS and Maximum Likelihood) may
naturally be also applied when estimating distance functions.
IV) Applications of the Distance Function to the Analysis of the Standards of
Living and the Quality of Life:
As indicated in the introduction what we call resources will be used to estimate the
standard of living while functionings will be the basis for the derivation of the quality
of life.
A) Estimating the Standard of Living Index:
Let xs and xt be two different resources vectors of length N (the number of resources)
and u some functioning vector. As indicated earlier the idea behind the Malmquist
index is to provide a reference to judge the relative magnitudes of the two resource
8
vectors. This reference is the isoquant L(u) and, as we have seen, if xs is radially
farther from the isoquant L(u) than xt , it corresponds to a higher standard of living.
There is however a difficulty because the Malmquist index depends generally on u.
One could use an approximation of this index such as computing a Tornquist index,
but such an index requires price vectors as well as behavioral assumptions. Since we
do not have prices for resources we have to adopt an alternative strategy. The idea is
to get rid of u by treating all individuals equally and assume that each individual has
the same level of functionings: one unit for each functioning. Let e represent such a
vector of functionings (a M-dimensional vector of ones, where M is the number of
functionings). The reference set becomes therefore L(e) and individuals with resource
vectors x∈L(e) share the lowest standard of living, with an index value of unity.
Individuals with large resources vectors will then have higher standards of living, with
index values above unity.
To estimate the distance function we have to remember that di(e,x)≥1 and that
di(e,λx)=λdi(e,x). Let λ=(1/xN) and define a (N-1) dimensional vector z as
z={zj}={xj/xN} with j=1 to N-1. We may then write that di(e,z)=(1/xN) di(e,x) and
since di(e,x)≥1, we may conclude that (1/xN)≤di(e,z). This implies that we may also
write that
(1/xN) = di(e,z) exp(ε) with ε≤0.
By assuming that di(e,z) has a translog functional form, we derive that
ln(1/xN) = a0 +∑j=1 to N-1 aj ln zj + (1/2) ∑j=1 to N-1 ∑k=1 to N-1 ajk ln zj ln zk + ε
Estimates of the coefficients aj and ajk may be obtained using COLS (corrected least
squares) or Maximum likelihood methods (see section III above) while the input
distance function di(e,zr) for each individual r is provided by the transformation
di (e,xr) = exp { (maximum positive residual ) - (residual for individual r) }.
This distance will by definition be greater than or equal to one (since its logarithm will
be positive) and will hence indicate by how much an individual’s resources must be
scaled back in order to reach the resource frontier. This procedure guarantees that all
resources vectors lie on or above the resource frontier L(e). The standard of living for
individual r will then be obtained by dividing di (e,xr ) by the minimum observed
distance value di, but that latter is by definition equal to 1.
B) Estimating the Quality of Life Index:
It is possible in a similar way to derive quality of life indexes. This time we use an
output distance function do(x,u) defined as
do(x,u) = Min {θ: (u/θ) ∈ P(x) }
where P(x) is the set of all functioning vectors which can be realized with the resource
vector x. The output quantity index Q (x, ur, us ) = do (x, ur ) / do (x, us ), where ur and
us are two functionings vectors and x is a resource vector, may then be considered as a
9
quality of life index. It is clear that the further inside the isoquant P(x) a functioning
vector is, the more it must be radially expanded in order to meet the standard and the
lower the corresponding quality of life.
Here also the problem is to choose a reference vector, in our case a resource vector x.
We will this time define a N-dimensional vector e of ones, that is, we will assume that
each individual is endowed with one unit of each resource. This implies that we define
a reference set P(e) which bounds from above the observed functionings of the various
individuals. All the individuals are compared to this reference set. If an individual has
a vector of functionings which put him on the frontier P(e), this implies that he has the
maximal level of quality of life and hence an output index of unity. Individuals with
smaller functionings will have a lower quality of life and hence index values below
unity.
To estimate the output distance functions we proceed as in the input distance case. We
assume a translog functional form and write
ln (1/uM ) = bo + ∑i=1 to M-1 bi ln vi + (1/2) ∑i=1 to M-1 ∑h=1 to M-1 bih ln vi ln vh + ε
where vi = (ui /uM ), i = 1 to M-1. Here again we may use COLS (corrected least
squares) or maximum likelihood methods to obtain estimates of the coefficients bi and
bih. The modified residuals which are then derived will provide output distance
functions for each individual by means of the transformation
do (e,ur ) = exp { (maximum negative residual) - (residual for individual r) }
This distance will by definition be smaller than one (since its logarithm will be
negative or at most equal to zero) so that all individual functionings vectors will lie on
or beneath the functioning frontier corresponding to P(e).
The output distance function do (e,ur ) gives therefore the maximum amount by which
individual functionings vectors must be radially scaled up in order to reach the
functioning frontier. By dividing all the output distance functions obtained by the
maximum distance observed (for which the distance is unity) , we obtain a quality of
life index Q(e, ur ,us ).
C) The Concept of Transformation Efficiency Index:
If (xr, ur ) and (xs, us ) are two different resource and functioning vectors, the
Malmquist productivity index M(xr,xs,ur,us), as indicated previously, will be defined
as
M(xr,xs,ur,us) = do (xr, ur ) / do (xs, us ).
Note that the reference set P(xt) will this time be defined as
P(xt ) = { (ut/θ ): do(xt,(ut/θ)) = 1 } for t=1,....T where T is the number of individuals.
All the individuals will therefore be compared to the relevant reference set and for
those who are able to convert relatively small resources into relatively large
functionings the distance function will be unity while less efficient individuals will
have a smaller score. The same technique used previously in estimating distance
functions for the standard of living and the quality of life will be applied here. Note
10
however that this time both resource and functioning data will be used. The translog
output distance function will then be expressed as:
ln (1/uM ) = ao +∑j=1 to N aj ln xj + (1/2) ∑j=1 to N ∑k=1 to N ajk ln xj ln xk +∑i=1 to M-1 bi ln vi
+(1/2) ∑i=1 to M-1 ∑h=1 to M-1 bih ln vi ln vh + ∑i=1 to M-1 ∑j=1 to N cij ln vi ln xj + ε
Here again we will use COLS (corrected least squares) or maximum likelihood
methods to obtain estimates of the various coefficients. The modified residuals which
are then derived will provide output distance functions for each individual by means
of the transformation
do (e,ur ) = exp { (maximum negative residual) - (residual for individual r) }
This distance will by definition be smaller than one (since its logarithm will be
negative or at most equal to zero) so that all individual resource and functionings
vectors will lie on or beneath the frontier P(x).
These output distance functions will measure the efficiency with which individuals
convert their resources into functionings. Since the maximum observed output
distance function is unity by construction, the distance do (xr, ur ) (when divided by
this maximum output distance) will also be equal to the Malmquist Productivity Index
M(xr,xs,ur,us).
V) The Data:
The data sources are a time-survey conducted in 1992-1993 by Israel’s Central Bureau
of Statistics among 5000 households. Since in many cases the value of some of the
variables were not available for some households, we ended up with only 1005
observations. The survey includes three parts: a questionnaire, an exact diary of the
activities and basic information on the individuals surveyed.
In the questionnaire the individual was asked to give information on his activities such
as the number of times in which a given activity took place or the time allocated to
such an activity ( e.g. trips abroad, “listening to radio”, etc...). There were also
subjective questions where the individual was asked whether he was happy with his
life, his income, etc.... Finally there were also questions on his wealth, in particular on
the durables which were in his position (apartment, car, etc...). In the diary the
individual had to report how much time he devoted on a given day to various kinds of
activities defined by the survey. The individual data finally give information on the
gender, the age, the country of origin, etc... of the individual as well as on her income
sources.
These data allowed us to build the following vectors of resources and functionings.
A) The elements composing the vector of resources:
- Is the apartment/ house owned or rented by the individual?
- Number of rooms per individual in the apartment/house.
- Number of cars per individual in the household.
- Number of TV sets per individual.
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- Quintile of the standardized gross family income to which the individual belongs?2
B) The elements composing the vector of well-being:
- Is the individual satisfied with her health3?
- What does the individual think of the amount of free time which is available to
him?
- Is the individual satisfied with her income?
- Is the individual happy with his life?
- Is the individual satisfied with his work?
C) Other Individual Characteristics:
- Gender
- Age (in years)
- Ethnic Origin4 and Period of Immigration
- Marital status
- Educational Level (in years of schooling)
- Is the individual the head of the household?
- Area of residence5
VI) The Results of the Empirical Investigation: Summary Measures
A) The Link Between the Various Indicators:
Table 1 gives the coefficients of correlation between the Standard of Living Index, the
Quality of Life Index and the “Transformation Efficiency” Index. It appears that there
is a small but significant positive correlation between the Standard of Living index
and the Quality of Life Index. The correlation between the Standard of Living Index
and the Transformation Efficiency Index is however not significantly different from
zero. Finally there is a very high positive relationship between the Quality of Life
Index and the Transformation Efficiency Index.
2
This variable was constructed in several steps. First it had to take into account various components of
the family income such as labor income, family allowances, old age or other allowances received from
the Social Security System, pensions and other income sources. Second this total gross family income
was divided by some equivalence scale reflecting the number of household members and its
composition. Finally given the unreliability of some of the income data, those in charge of the survey
decided to convert the standardized income data into income distribution quintiles, the variable which
was finally taken into account.
3
For each of the variables considered as determining the level of well-being, the individual could give
four answers: 1=Not satisfied at all, 2=Not so satisfied, 3=Satisfied , 4=Very satisfied .
4
Four “ethnic” origins were considered: 1) the individual was born in Israel 2) the individual was born
in Asia or Africa 3) the individual was born in Europe or America (as well as Australia and South
Africa) but arrived before 1971 in Israel 4) the individual was born in Europe or America but
immigrated to Israel after 1971.
5
Three areas of residence were distinguished: rural areas, cities at the exception of Haifa, Jerusalem
and Tel-Aviv, and these three big cities.
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TABLE 1: Coefficient of Correlation Between the Three Indices
Standard of
Living
Standard of Living
1.000
Quality of Life
0.086*
Efficiency Transformation
0.007
* indicates a 1% degree of significance.
Quality of Life
Efficiency
Transformation
1.000
1.000
0.962*
B) Measures of Inequality and Poverty Based on the Distribution of the Three
Indicators:
Table 2 gives the value of some location measures such as the mean and the median as
well as the value of the Gini Index for each of the three distributions considered. It
also gives, in each of the three cases, the value of the poverty line where the latter is
successively defined as being equal to 50%, 60% and 70% of the median value of the
corresponding distribution.
TABLE 2: Inequality in the Distribution of the Three Indices
Number of
observations
Minimal value of
index
Maximal value of
index
Median value of
index
Mean value of
index
Gini Index of
inequality
Value of Poverty
Line (defined as
being equal to 50%
of median value of
index).
Value of Poverty
line (defined as
being equal to 60%
of median value of
index)
Value of Poverty
Standard of
Living
1125
Quality of Life
1125
Efficiency
Transformation
1125
1
0.233
0.237
3.443
1
1
2.256
0.694
0.679
2.203
0.683
0.674
0.1
0.082
0.082
0.622
0.178
0.178
5.511
1.067
0.622
9.689
2.489
2.667
13
Line defined as
being equal to 70%
of median value of
index)
It appears that there is more inequality between the individuals in the Standard of
Living than in the Quality of Life or in the Value of the Index of “Transformation
Efficiency”. These results seem to indicate that individuals with a relatively low level
of Standard of Living may still be efficient enough in transforming their resources into
“quality of life”.
In Tables 3 and 4 we computed some indicators of poverty, first with respect to the
Standard of living, second with respect to the Quality of Life.
1) Defining the “Poor” on the basis of the Standard of Living Index:
To define the “poor” we have used an approach which is commonly adopted in the
poverty literature. There a “poor” is often defined as an individual earning less than
half the median income. Similarly here we define as “poor” those individuals whose
Standard of Living is smaller than some percentage of the median of the distribution
of Standards of Living. We have chosen two alternatives: the poverty level is equal to
60% or to 70% of the median standard of living. Table 3 gives the value of some
indicators of poverty based on the two poverty levels which have been chosen. It
appears that in the first case (poverty level equal to 60% of the median standard of
living) 5.5% of the individuals are poor while this percentage reaches 9.7% when the
second poverty level is adopted (70% of the median value). One may also note the low
value of the Gini index among the poor, whether it concerns the inequality of the
standard of living or that of the transformation efficiency index.
TABLE 3: Distribution of the Standard of Living Among the Poor
Percentage in
population
Mean value of
index among
poor
Minimal value
of index
among the
poor
Maximal value
of index
among the
Poor (defined as those for which
the index is smaller than 60% of
its median value)
Standard of
Efficiency
Living
Transformation
5.511
5.511
Poor (defined as those for which
the index is smaller than 70% of
its median value)
Standard of
Efficiency
Living
Transformation
9.689
9.689
1.226
0.681
1.319
0.668
1
0.371
1
0.298
1.342
0.906
1.569
0.906
14
poor
Value of the
Gini index
among the
poor
0.037
0.082
0.056
0.096
2) Defining the “Poor” on the basis of the Quality of Life Index:
Here an individual will be considered as “poor” if the value of his quality of life index
is smaller than 60% or 70% of the median of the distribution of the quality of life
indices. Table 4 gives the value of some indicators of poverty which are based on the
use of the quality of life index.
TABLE 4: Distribution of Quality of Life Among Poor
Percentage in
population
Mean value of
index among
the poor
Minimal value
of index
among the
poor
Maximal value
of the index
among the
poor
Value of the
Gini index
among the
poor
Poor (defined as those for which
the index is smaller than 60% of
its median value)
Quality of
Efficiency
Life
Transformation
1.067
1.067
Poor (defined as those for which
the index is smaller than 70% of
its median value)
Standard of
Efficiency
Living
Transformation
2.489
2.489
0.368
0.38
0.42
0.423
0.233
0.237
0.233
0.237
0.415
0.427
0.483
0.492
0.072
0.072
0.072
0.068
It appears that only 1.1% of the population should be considered as “poor” on the
basis of their quality of life index, when the poverty level is defined as being equal to
60% of the median quality of life. When a 70% threshold level is chosen, the
percentage of poor remains low (2.5%). It is interesting to note that although the
percentage of poor is smaller when it is based on the distribution of the quality of life
index, one finds a higher degree of inequality among the poor when the latter index is
used than when one bases the computations on the distribution of the standard of
living. This should not be surprising as the percentage of poor and the Gini index
15
among the poor measure different things. It is well known that the percentage of poor
does not say anything about how poor the poor are while the Gini index among the
poor does not give any indication on how common poverty is in the population.
Finally it should be of interest to know whether those individuals who are considered
as poor (whether by the 60% or the 70% of the median criterion) from the point of
view of the quality of their life are also classified as poor when one looks at their
standard of living. Table 5 summarizes the degree of overlapping between the two
poverty concepts. It appears that none of the 12 individuals who are poor (given a
60% of median criterion) according to the quality of their life are poor according to
Table 5: Degree of Overlapping between those who are poor according to the
quality of their life and those who are poor according to their standard of living
(Number of individuals in each case)
Not
poor
according to
the quality of
life (60% of
median
criterion)
Not
poor 1051
according to
the standard of
living (60% of
median
criterion)
Poor according
62
to the standard
of Living (60%
of
median
criterion)
Total Number 1113
of Cases
Not
Poor
according to
the quality of
life (70% of
median
criterion)
Not
poor 1006
according to
the standard of
living (70% of
median
criterion)
Poor according 107
to the standard
of living (70%
of
median
Poor according Total Number
to the quality of Cases
of life (60% of
median
criterion)
12
1063
0
62
12
1125
Poor according Total Number
to the quality of Cases
of life (70% of
median
criterion)
10
1016
2
109
16
criterion)
Total Number 1113
of Cases
12
1125
their standard of living. When a 70% of the median criterion is used, then out of 12
individuals who are poor according to the quality of their life (note that this number
was the same when a 60% of the median criterion was used), 2 are also poor
according to their standard of living. It is therefore clear that the standard of living and
the quality of life measure different things. To better understand the origin of these
differences we have run three regressions where the dependent variables are
respectively the indices of standard of living, of quality of life and of “transformation
efficiency” while the exogenous variables are the variables called “Other individual
characteristics” in section V. The following section summarizes the main results of
such an investigation.
VII) The Empirical Investigation: The Determinants of the Standard of Living,
of the Quality of Life and of the Efficiency Transformation Index
In Tables 6 to 8 we present results of regressions where the dependent variables are
successively the index of standard of living, that measuring the quality of life and the
transformation efficiency measure. Concerning the standard of living it appears (see,
Table 6) that the gender of the individual does not affect the standard of living. Being
head of the household has also no effect on the standard of living. Age however has a
significant effect in so far as the standard of living increases with age, though at a
decreasing rate. One may also observe that individuals who are single have a higher
standard of living than those who are married, divorced or widow(er)s. Education has
a positive effect and it appears also that individuals who came from Europe and
immigrated before 1971 as well as individuals born in Israel have a higher standard of
living than individuals born in Asia or Africa. Moreover all these three categories
have a significantly higher standard of living than individuals who immigrated from
Europe after 1971. Finally the standard of living of individuals living in Tel-Aviv,
Haifa or Jerusalem is higher than that of those living in other cities, the latter itself
being higher than that of those living in rural areas.
If we now look at Table 7 where the dependent variable is the index measuring the
quality of life, we observe that the quality of life decreases with age but at a
decreasing rate. Note that the minimal value of the index is reached at the age 53 so
that for people older than 53 the quality of life rises again. Single individuals seem to
have a higher quality of life than individuals with another marital status. Being the
head of the household as well as the gender of the individual have no significant
impact. Education, on the contrary, has a positive effect on the quality of life, as it had
one on the standard of living. The coefficients of the variables measuring the area of
residence are not significant so that there apparently is no difference between the
quality of life in cities and in rural areas. Finally one may observe that individuals
who were born in Europe or America and immigrated after 1971 have a lower quality
of life than the other categories. Note however that here the highest quality of life is
observed among individuals born in Israel rather than among those born in Europe or
America but who immigrated before 1971.
In table 8 finally the dependent variable is the transformation efficiency index whose
definition was given earlier. It appears that neither the gender nor the marital status
17
has a significant effect on this index. This transformation efficiency however
decreases with age, though at a decreasing rate, and the minimal value is reached at
age 57, so that beyond this age the ability to transform resources (the standard of
living) into quality of life increases again with age. It appears also that neither the
educational level nor the area of residence has an impact on the transformation
efficiency index but the country of origin has again a significant impact. Individuals
born in Europe or America and who immigrated after 1971 are the least able to
transform resources into quality of life while people born in Israel have the highest
capacity of transforming the standard of living into quality of life.
VIII) Concluding Comments
In this paper an attempt has been made to measure separately the standard of living,
the quality of life and the efficiency of transforming standard of living into quality of
life among Israeli individuals in 1992-1993. The approach is based on the theoretical
distinction made by Sen between resources and functionings and on the use of the
concept of distance function which originated in production theory. The methodology
had been originally proposed by Lovell et al. (1994) who applied it to Australian data
while Delhausse (1996) replicated it with French data. The estimation procedure
utilizes either the stochastic frontier approach or what is known as corrected ordinary
least squares. The main findings of this research may be summarized as follows.
- A small but significant positive link was found between the standard of living and
the quality of life but mostly a strong positive relationship appears to exist between
the quality of life and the transformation efficiency index.
- There seems to be less inequality in the quality of life than in the standard of living.
Similarly the percentage of poor, whether one defines it as being equal to 60% or 70%
of the median of the relevant distribution, is smaller when one bases the computation
on the standard of living than when one derives it from the quality of life. These
results seem to indicate that individuals with a lower standard of living may be
efficient enough in transforming their resources into quality of life to finally reach a
level of functioning (quality of life) which may not be so much different from that of
individuals having a high standard of living.
- We observed also that those individuals who are poor according to their quality of
life are not necessarily poor according to their standard of living. In fact when the
identification of the poor was based on a 60% of the median criterion, none of the
individuals who were poor according to their quality of life were poor according to
their standard of living. There was a small degree of overlapping when the
identification of the poor was based on a 70% of the median criterion.
- Concerning to the determinants of the standard of living, the quality of life and the
ability of transforming resources into functionings (efficiency transformation index)
we first observe that new immigrants have a lower standard of living, a lower quality
of life and are less able to transform resources into functionings. The gender has never
a significant effect while age has a positive effect on the standard of living but first a
negative than a positive effect on the quality of life and on the transformation
efficiency index. Having a higher level of education or being single has a positive
effect on the standard of living and the quality of life but does not affect the ability of
transforming resources into functionings. Finally living in one of the three big cities
18
(Tel-Aviv, Jerusalem or Haifa) has a significant (and positive) effect only on the
standard of living.
19
References
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Yardstick,” World Development 7(1979): 567-580.
International Labor Office, Statistics Sources and Methods, Volume 6, Household
Income and Expenditure Surveys, Geneva, 1992.
20
Lancaster, K. J., “A New Approach to Consumer Theory,” Journal of Political
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Functionings: A New View of Inequality in Australia,” in Models and
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Columbia University press, New York, 1972.
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21
Table 6: Regression Results with the Standard of Living as the Dependent
Variable
Exogenous
Variables
Mean of
Variable
Standard
Deviation of
Variable
0.4105
0.4964
13.1577
1155.0319
0.3677
0.4238
0.1674
3.5660
Coefficients
t value of
coefficient
Intercept
2.2231
1.0094
Male
0.5592
-0.0468
Age
39.7641
0.0231
Square of Age
1754.3174
-0.0001
Single
0.1611
0.2291
Married
0.7651
0.0097
Widower
0.0288
0.1298
Years of
13.0616
0.0225
education
Head of
0.5462
0.4978
-0.0389
household
Born in Israel
0.5064
0.4999
0.2925
Born in Asia
0.1970
0.3977
0.2125
or Africa
Born in
0.1442
0.3513
0.2918
Europe or
America but
immigrated
before 1971
Lives in a
0.2248
0.4175
-0.0694
city*
Lives in a
0.0716
0.2578
-0.0960
rural area
Adusted R2= 0.18606 , Standard Error = 0.37056, N= 1005
7.029
-1.299
4.002
-1.917
3.222
0.163
1.434
6.161
-1.020
8.221
4.939
6.343
-2.392
-2.068
* This category does not include the cities of Haifa, Jerusalem and Tel-Aviv.
22
Table 7: Regression Results with the Quality of Life as Dependent Variable
Exogenous
Variables
Mean of
Variable
Standard
Deviation of
Variable
0.1002
0.4964
13.1577
1155.0319
0.3677
0.4238
0.1674
3.5660
Coefficients
t value of
coefficient
Intercept
0.6827
0.67791
Male
0.5592
0.00110
Age
39.7641
-0.00316
Square of Age
1754.3174
0.00003
Single
0.1611
0.03289
Married
0.7651
0.01848
Widower
0.0288
0.00714
Years of
13.0616
0.00191
education
Head of
0.5462
0.4978
-0.00716
household
Born in Israel
0.5064
0.4999
0.04143
Born in Asia
0.1970
0.3977
0.03021
or Africa
Born in
0.1442
0.3513
0.03556
Europe or
America but
immigrated
before 1971
Lives in a
0.2248
0.4175
0.00674
city*
Lives in a
0.0716
0.2578
-0.00011
rural area
Adusted R2= 0.04332 , Standard Error = 0.09810, N= 1005
17.832
0.115
-2.068
1.911
1.747
1.178
0.298
1.973
-0.708
4.399
2.652
2.920
0.877
-0.008
* This category does not include the cities of Haifa, Jerusalem and Tel-Aviv.
23
Table 8: Regression Results with the Efficiency Transformation Index as
Dependent Variable
Adusted R2= 0.04096, Standard Error = 0.09671, N= 1005
Exogenous
Mean of
Standard
Coefficients
Variables
Variable
Deviation of
Variable
Intercept
0.6730
0.0985
0.72096
Male
0.5592
0.4964
-0.00224
Age
39.7641
13.1577
-0.00456
Square of Age
1754.3174
1155.0319
0.00004
Single
0.1611
0.3677
0.01640
Married
0.7651
0.4238
0.00954
Widower
0.0288
0.1674
0.00095
Years of
13.0616
3.5660
0.00115
education
Head of
0.5462
0.4978
0.00052
household
Born in Israel
0.5064
0.4999
0.03518
Born in Asia
0.1970
0.3977
0.02110
or Africa
Born in
0.1442
0.3513
0.02410
Europe or
America but
immigrated
before 1971
Lives in a
0.2248
0.4175
0.00996
city*
Lives in a
0.0716
0.2578
-0.00280
rural area
Adusted R2= 0.04096, Standard Error = 0.09671, N= 1005
t value of
coefficient
19.236
-0.238
-3.026
2.855
0.883
0.617
0.040
1.206
0.052
3.789
1.879
2.007
1.315
-0.231
* This category does not include the cities of Haifa, Jerusalem and Tel-Aviv.
24
Figure 1
q2
B
q
A
u
O
q1
25
Figure 2
q2
q
1/a2(u,q)
u
O
1/a1(u,q)
q1
26
Figure 3
x2
x2A
A
L(y)
B
C
Isoq-L(y)
O
x1A
x1
27
Figure 4
y2
B
y2A
A
C
PPC-P(x)
P(x)
O
y1A
y1
28