Burden Sharing in Support of the United Nations
IV. Some Estimates of Progressive Formulas

We have made several set of calculations using the following assessment rules:

1. By national shares of world GDP.

2. Proportional to a normalized size of per capita GDP.

2.1 Proportional to an unweighted average of 1 and 2.

3. Proportional to an unweighted average of normalized GDP share and normalized per capita GDP.

4. The maximum contribution by any single country must not exceed 25% of the total and the poorest countries shall contribute only at the minimum rate of 0.001% of their per capita income.

5. In addition to structuring the maximum-minimum points, as in 4, the rate for the median income countries is fixed at 0.01%.

6. Same as in 5, except that the assessment rate for the median income countries is raised to 0.02%.

In the study of economic aspects of Peacekeeping our calculations for burden-sharing for a hypothetical UN standing army were simply made according to the first four principles. Although data were available for many countries, nearly all, the burden assignments for Peacekeeping were made only for 43 relatively prosperous countries. By restricting the contributions in this way, some attention was paid to ability to pay.

In this paper, we have altogether experimented with seven different models or allocation structures. The detailed results are summarized, with brief comments, in Appendices A and B. The three additional new structures (4-6) were devised by imposing some common sense rules and by using some practical criteria. We have also covered more countries, and used two sets of data in this paper. The first set, taken from PENN TABLES, is based on national GDP values converted into 1985 US$ by using PPP conversion factors. This set includes 176 countries. A prefix notation FN denotes the use of these data. The second set is based on GNP values converted into 1993 US$ by using market exchange rates (MER). There are 146 (or 147) countries included in this set. The calculations based on these data are identified by a prefix notation TX.

Rule 1. Except for the truncation point, excluding poor countries, the first method essentially follows equation (1). The shares in the truncated listing were modified so that the total sum targeted would be realized, if all assessed countries paid their allocated share.

Above the truncation point (income minus the minimum value) it is a flat assessment rate; so it is not entirely progressive, but the shares are calculated so that they add to 1.0 for all the countries included. This means that the summed product of each country's share and the total sum to be raised will automatically provide the required total. The average per capita burden across countries turns out to be a little over half a penny (0.66, to be precise) out of $100. And, the per country contribution to the total amount is around $8.5 million.

However, it was noticed that under this simple rule large but much less prosperous countries, such as India and China are called upon to bear an unduly large proportion of the total targeted amount. Yet much richer countries (by per capita income level) but small in size, like Sweden, Norway and New Zealand, capable of making large contributions, have much smaller shares.

Rule 2. The values of per capita GDP for each country are divided by the sum of all per capita values; so ratios add to 1.0, and these are multiplied into the total sum of funding to be raised. This calculation greatly increases the amounts to be raised from small, prosperous countries. It also narrows the variation across countries.

One may visualize the Rule 2 scenario as one in which each country is inhabited by one person, and the total world population is simply equal to the total number of countries. These (hypothetical) people are now made to share the total financing of the United Nations in accordance with their respective share of the total world income.

Rule 2.1. The next formula is a compromise between the first two principles. The simple unweighted average of Rules 1 and 2 lies, for each country, between the values obtained separately under each, as expected. But it also moderates the size of the assessment of small prosperous countries and gives a small amount of relief to small developing countries (also in comparison with Rule 3).

Rule 3. This approach also forms a compromise between the relative importance of total income shares and the shares of the total of per capita values. Before averaging the two concepts, they are each normalized, i.e. expressed as ratios to their respective standard deviations (across countries). The normalized values are then averaged without weights to get another rule of distribution of burden sharing, the values for each country lie between the values of Rules 1 and 2. But, on the average, they are relatively closer to the values estimated by Rule 2.

Alternative progressive approaches (Rules 4-6)

Rule 4. The linear logarithmic (exponential) function and also some quadratic functions have been evaluated. They show some interesting properties of an alternative approach to assessment. The preceding approaches are automatically structured so that the individual assessments sum to the total target values to be raised. In the logarithmic or polynomial formulas, coefficients must be selected so that a target sum will be raised.

There is no theoretical, optimal value for gradient and curvature of a tax function. There are arbitrary rules about how much a person or a nation can be taxed. We simply impose some common sense rules such that average and marginal tax rates shall each be less than unity. Specifically, in order to locate a function in the 2-dimensional area associating contribution per capita with income per capita, we have chosen two practical criteria.

The United States contribution should amount to 25% of the total. (ceiling)

The poorest countries should pay only 0.001% of their per capita income. (floor)

The basis for the first of these two criteria is simply that, as noted above, there have been a certain amount of editorial commentaries that the US could reasonably be expected to bear 25 percent of the total tax burden. And, for the second, the General Assembly document of 1995 remarks that "most countries with a per capita income below the world average of $3,200 contribute between 0.001 percent and 0.006 percent of their income to the United Nations regular budget." We interpreted this to mean 0.001 percent of the per capita income of the poorest countries should be the per capita contribution.

The simple linear logarithmic assessment function (equation 3) has just two parameters; therefore these two points in the space of (Ti/Ni, Yi/Ni) provide two equations for estimating the two unknown parameters. In other words, we make the assessment function pass through these two points.

In the case of the US, 25 percent of $1.5 billion is $375 million, and with a population of 257.9 million people, this amounts to a US per capita contribution of $1.45405. The per capita GNP of the US in 1993 (Human Development Report, 1996) is $24,740. It may be noted that in this first calculation based on progressive rules we are using the GNP concept because the Ad Hoc Working Group expressed a preference for such a concept. Also, we are using market exchange rates (MER) to convert all values to $US. This too is the Working Group's choice.

In the case of the poorest countries, we considered the per capita GNP in 1993 of the "Least Developed Countries' given in the Human Development Report, 1996, at $210. Correspondingly, the point on the assessment function for the poorest countries is set at values, Ti/Ni = 0.0021 and Yi/Ni = 210.

To make the curve,

(Ti/Ni) = A(Yi/Ni)a ,

pass through the two points for the USA and for the poorest countries, we simply find estimates of A and a from

ln (1.45405) = ln A + a ln (24740)

ln (0.0021) = ln A + a ln (210)

The estimated values are

a = 1.3714

A = 137 * 10-8

Next, the question to be asked is whether assessments, country-by-country, evaluated according to Rule 4,

(TX04) Ti/Ni = 137 * 10-8 (Yi/Ni)1.3714 ,

will, in fact, yield the value of $1.5 billion. This question does not have to be asked if the assessment is simply based on a country's share of world GNP. The total obtained from the numerical function above comes to $1.217 billion for 1993. The mean per capita assessment is 27 cents, the mean contribution per country is $8.3 million, and the standard deviation of contribution across countries is close to $40 million.

A graphical picture of this function is provided in Figure 5. The estimated values for some particular countries from a random selection of noteworthy cases, according to Rule 4 (TX04 evaluation) are:

$ million $(million)
Argentina 0.2733.8 9.13
Canada 1.0828.8 31.1
Chile 0.0913.8 1.24
China 0.00671196.4 8.02
Costa Rica 0.053.3 0.16
Ethiopia 0.000851.9 0.04
France 1.2757.5 73.23
Germany 1.3680.9 109.82
India 0.0034901.5 3.08
Ivory Coast 0.009513.3 0.13
Japan 2.02124.5 251.58
Malaysia 0.0919.2 1.73
Mexico 0.190 9
Mozambique 0.000715.1 0.01
Nepal 0.001820.8 0.04
Nicaragua 0.00414.1 0.02
Russia 0.0572147.8 8.45
Sierra Leone 0.00134.3 0.01
South Korea 0.2944.1 12.79
USA 1.45257.9 374.35
Viet Nam 0.001671.3 0.11

A more meaningful summary array, mainly according to classification in the Human Development Report, 1996 is:

Industrial Countries $1,103.29 million
Very Large Developing Countries

and Some Transition Countries

Less prosperous transition


Prosperous developing


Medium prosperous developing


Poor developing countries 2.61
Very poor developing


Total $1,217.85 million

Figure 5

Figure 6

Next we shall look at a combination of two variations on this process. The previous calculation was based directly on data from the Human Development Report, 1996. We shall now use data directly from the PENN Tables, for 1993. As noted before, the two data sets are not identical in coverage. More importantly, in using the data from the PENN Tables we shall shift from GNP at market exchange rates to (a) GDP (b) at PPP conversion rates. Furthermore, the GDP data reported in the PENN Tables are for 1993, but at 1985 PPP valuations. We have therefore scaled these data to 1993 valuations, to conform to the previous set of calculations, by a simple ratio transformation based on

US price index for 1993, GDP = 102.6

US price index for 1985, GDP = 78.6

Base value = 100 for 1992.

We have also ranked the PENN Tables data according to the size of per capita GDP (at PPP conversion factors) from the poorest, Zaire, to the most prosperous, USA. These two cases make our ceiling and floor values on the assessment function. That is, we have used

ln (1.45441) = ln A + a ln (24453) USA

ln (0.0039) = ln A + a ln (391) Zaire

to get

a = 1.4312

A = 76.238 * 10-8

The estimated function,

(FN04) Ti/Ni = 76.238 * 10-8 (Yi/Ni)1.4312 ,

is graphed in Figure 6. The total amount that would have been collected for this case is $1.277 million, which is slightly larger than the assessment according to GNP per capita, at market exchange rates. It should be noted, however, that the sample is slightly different. Zaire, for example, has no estimate for GNP or GDP in the Human Development Report, 1996, but is listed in the PENN Tables. As mentioned earlier, the PENN Tables provide data for 175 countries and one residual group. The Human Development Report, 1996, lists 174 countries but has no GNP estimate for 18 countries.

Rule 5. The calculations with simple 2-parameter log linear assessment formulas (TX04, FN04) illustrate the nature of an approach that can achieve progressivity, and it can be seen from our numerical plots of graphs (Figs. 5-6) that there is a moderate degree of curvature in the proposed functions. There are many ways to introduce more curvature, and therefore more progressivity.

One way would be to estimate a 3-parameter (A, µ, a) log linear function of the form

Ti/Ni = A (Yi/Ni - µ) a

where µ is a minimum income. Taxes would be exempt for countries with per capita incomes of µ or less. That is,

Ti/Ni = 0, if Yi/Ni £ µ .

This type of function would be similar to that implicit in Figure 2. We have already seen, for example under Rule 4, that the very poorest countries with per capita GNP values less than $800 (1993), making up the 23 lowest scores on the human development index, contribute only a minuscule amount of $0.42 million to the computed total assessment and have only symbolic participation according to the numerical formulas cited above (TX04). The value of parameter µ can be set equal to a per capita income level around $800 or so, and these countries could be exempt from making any contribution to the total assessment. Methodologically, the remaining two parameters, A and a, would be estimated as in TX04 or FN04 before, but by shifting the minimum point on the per capita income axis.

Another way of changing the curvature (progressivity) of the function is to estimate a 3-parameter function by forcing it to pass through three points of the payment schedule, an intermediate point, say at a median income per capita, as well as at the maximum (high) and minimum (low) points used earlier. The median point occurs at per capita GNP of about $1000 for 1993. Market exchange rates are used for conversion to $US in this estimate. For PPP conversion rates, the median is at the Ecuador value of $3622 for 1993. The Ad Hoc Working Group estimates that countries at the median per capita income ($500 in their case) contribute $.01 per capita. Their median value is low, but for indicative purposes, we have made some estimates, using GNP per capita at market exchange rates and

Ti/Ni = 0.01 Yi/Ni = 1000 median

This point, combined with

Ti/Ni = 1.45405; Yi/Ni = 24,740 USA

Ti/Ni = 0.0021; Yi/Ni = 210 minimum

yield the following 3-parameter assessment function.


This assessment formula has a fine progressive appearance, shown in Fig. 7, but it generates only $857 million in 1993 prices for income values at PPP prices. The curvature produced by PPP evaluated data is depicted in the top panel (FN05) of Fig. 7, and by MER data in its bottom panel (TX05).

Rule 6. If we were to force the assessment function to yield Ti/Ni = 0.02 instead of 0.01 at the median income, and combine it with the same maximum and minimum points as in Rule 5, the assessment function is estimated as


In this case the estimated total yield for PPP income values in 1993 prices comes to $1.257 billion which is much closer to the necessary amount. See Fig. 8 for parallel plots of PPP evaluated data (FN06) in the top panel and by MER data (TX06) in the bottom panel.

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This file last modified 11/24/2008