In a more ambitious spreading of the burden of support, it is conceptually possible to assess individuals, within a country, rather than assess separate nations. If all the income units in the world had their income converted to a common unit of measurement and placed in an array by size. There could be a progressive world assessment for UN support. Collection could implicitly be made at the national level and forwarded to the United Nations.

Recalling the "spirit of 1946", this may have been possible - as a principle of UN support - but in today's world, payment would have to be in terms of national assessment. Some allowance, however, for the degree of inequality of income distribution could be a "conditioning" variable in a progressive assessment function. Instead of (2), we might consider

6

where s_{i} is a measure of
dispersion (or inequality) in the income distribution of country
i.

It may seem to be an overly ambitious data requirement to have a system based on a "world income distribution," but it is not beyond imagination to consider the feasibility of such calculations in the present era, using the advances in sampling, field survey enumeration, and data management.

From an older era of research, there have been a few attempts to construct a world income distribution based on extremely sparse data, by exploiting some of the properties of the lognormal distribution, on the assumption that 2 or 3-parameter versions of that distribution would be appropriate for any individual country.

A parametric specification of equation (2¢) can be obtained by combining (3) with an assumption that each country's income distribution is lognormal.

Let T_{ij} = assessment of the j-th person (income unit)
in country i.

Y_{ij} = income of the j-th person (income unit) in country
i.

N_{i} = population (income unit) in country i.

A single assessment function will determine each income unit's contribution.

(3¢) .

This is the same function that is used for international data, relating per capita assessment to per capita income for each country.

In logarithmic form

(3²) .

Aggregating, for each country across population,

(4) .

And, at the per capita level

(4¢) ,

which may be witten as

(4²) ,

where:

(GMT)_{i} = geometric mean assessment for country i.

(GMY)_{i} = geometric mean income for country i.

The international function calibrated for per capita data (assessment and income) across countries has been interpreted as

(5) ,

where:

(AMT)_{i} = arithmetic mean assessment for country i.

(AMY)_{i} = arithmetic mean income for country i.

For positive values of T_{ij} and Y_{ij}, it is
well known that

GMT £ AMT; GMY £ AMY .

In order to parameterize the relationship between GM and AM, it is instructive to assume that each country's income is distributed according to a lognormal function defined by country-specific mean and variance parameters. This simplifying assumption is used for illustrative purposes, but it carries a degree of realism because the lognormal function is known to be a fair approximation to individual country income distributions.

We then can write

;

And since,

ln T_{ij} = ln A + a ln Y_{ij}
,

var(ln T_{ij}) = a^{2}
var (ln Y_{ij}) = a^{2}s_{i}^{2
}

where s_{i}^{2} is
the variance of the logarithms of the lognormal income distribution.

The relation between (GMT)_{i} and (GMY)_{i} can
thus be transformed in terms of their airthmetic means, and written
as

(5)

Since a > 1, for a progressive assessment
system, s_{i}^{2} has
a positive effect on per capita (arithmetic mean) assessment.
That is, the more unequal the distribution of country's income
the higher will be its rate of assessment. The purpose of this
exposition is simply to show the direction of effect of income
variance (inequality) on assessment. Horizontal equity, in this
formulation, should be interpreted to mean that countries with
the same geometric mean income will bear the same geometric mean
assessment.

Data on income distribution for many industrial and developing
countries have been assembled and listed by quintile distributions
and Gini measures of inequality. This could be the starting point
for introducing distributional measures into the function, as
in equation (2') or (3') and determining its quantitative effects
on assessments.

Yale University Library and Social Science Statistical Laboratory

Revised: June 30, 1997

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