Alternative Quota Formulas - Considerations
Statistical Appendix

Alternative Quota Formulas - Considerations

Abstract

Alternative Quota Formulas - Considerations

I. Origins of PPP Data in the World Economic Outlook

1. The World Economic Outlook (WEO) presents annual calculations of individual country GDP valued at purchasing power parity (PPP) for 176 countries.1 Here GDP is converted from national currency units to “nternational dollars,” not by market exchange rates, but by PPP conversion rates based on the estimated purchasing power of countries’ currencies. The applied PPPs are obtained from a number of intermediate sources, and the source data are updated every few years. The current series are based on data provided by the OECD and the World Bank in 1999.

2. There is a long history of work aimed at constructing internationally comparable measures of purchasing power centering on the International Comparisons Program (ICP) involving the United Nations Statistical Office and the University of Pennsylvania.2 The core of this effort is a systematic survey of price levels in individual countries, which is used to construct measures of absolute PPP. This work has gradually become more comprehensive, but country coverage is still limited and data are gathered only for infrequent "benchmark" years. To overcome the lack of coverage, data for remaining countries are estimated with regression techniques, and missing years are filled in by extrapolation.

3. As the ICP has evolved, the program has become increasingly decentralized, with data collection carried out by a number of participants each responsible for a given group of countries, and prominent parts are played by, in particular, the World Bank and the OECD. The procedure thus involves a multi-step exercise where at first countries of a particular group, typically a region such as the EU, are compared. These groups are subsequently compared to each other.3 Since 1970, price surveys have been carried out at approximately five-year intervals and the total number of surveyed countries has increased to about 100.

4. The latest round of ICP price surveys for which final data are available took place in 1996. The OECD, based on surveys taking place in 1994–96 and carried out by a number of different agencies, provides PPP results from this round for OECD members, Israel, Slovak Republic, Slovenia, and CIS countries—52 countries in all. The World Bank also coordinates data collection efforts carried out by a number of different agencies and publishes PPP-based GDP series for a large number of countries in its World Development Indicators (WDI) from which the PPP rate may be backed out by comparing to values for GDP in national currency. Data for a significant number of countries in the WDI, however, are based on estimates rather than actual price surveys. Of the 108 countries for which the latest WEO relied on WDI data for PPP-based GDP, values for 55 countries are based on ICP price surveys.4 Of the remaining countries, the value for China is based on an independently performed price comparison with U.S.A. dating from 1986 and the values for another 52 countries (mostly small developing countries, but also India) are based on estimates.

5. For WEO purposes, calculation of PPP-based GDP has since 1999 relied on the 1996 values of the (at that time) most recently available data series from the OECD and, for countries not covered there, WDI (see Table 1).5 For 16 countries not represented by these sources, numbers were estimated based on cross-section regressions relating PPP-based GDP to GDP at market exchange rates, trade openness, and regional dummies. The complete set of 1996 numbers for PPP-based GDP are each year extended forward in time by applying the growth rate of the product of the IMF country desk-provided real GDP and the level of the U.S. GDP deflator. This procedure implicitly assumes that relative prices remain unchanged.

Table 1.

Origin of PPP Conversion Factors Used in the Latest World Economic Outlook

article image
article image
article image
article image
article image

OECD refers to data on OECD members and countries in Eastern Europe and the CIS provided by the Organisation for Economic Cooperation and Development. WDI refers to data from the World Development Indicator database provided by the World Bank. IMF refers to data by Fund staff estimated for countries not present in the OECD or WDI data.

Blanks indicate that PPP conversion factors are estimated rather than based on actual price surveys.

The value used is for 1995 rather than for the 1996 base year.

The WDI-based PPP conversion factor is based on a price comparison with the U.S.A. performed independently of the ICP framework.

Based on the market exchange rate instead of the official rate.

6. An important limitation of the available ICP data is that price surveys should make comparisons of goods, which are of the same quantity and quality across all countries.6 Since goods often differ between countries, differences along these dimensions may well be mistaken for price differences. In addition, a number of indexing problems mean that there is no clear-cut way of aggregating from the level of an individual country to the group level, and on to the world level.7 This implies that results are not invariant to how the ICP delegates data collection and that results may change depending on which countries are included in the price surveys. Consequently, PPP estimates for a given country from different sources are not readily comparable. Finally, although the estimation techniques have been shown to have satisfactory statistical properties,8 the fact that a large number of primarily small developing countries are not included in the price surveys raises obvious questions about the validity of the results for these countries.

II. The Effect of Weights of Variables on Calculated Quota Shares: Analytical Results

7. Whether the calculated quota share of a country will go up or down as a result of a change in the weight of a particular variable depends on relationships involving a host of factors, including the distribution of variables, the quota formula under consideration, and the initial set of weights of variables. Evaluating the mathematical derivatives of quota shares with respect to the different weights can illuminate the interrelationships. In particular, these derivatives, which can be established using analytical methods, may be used to express the criteria that determine the direction of change in calculated quota shares due to a change in weights.

8. The analytical criteria that are established here to determine if a country will benefit from a change in the weights of variables hold up well in practice. The criteria hold with certainty when considering very small changes in weights but are not necessarily valid when considering larger changes. For the magnitude of changes considered in this paper, however, the analytical results are generally robust. Even when considering changes in weights that change the combined quota share of country groups by several percentage points, the predicted direction of changes in quota shares is typically correct for all but a handful of the 183 member countries.

9. The two types of formulae considered in the paper can be expressed as follows:

  • Linear formula: qi=jajxi,j

  • Multiplicative formula: qi=jxi,jaj

Here qi is the notional calculated quota with the subscript indicating the relevant country; xi,j is the value of a variable with the first subscript indicating the country and the second subscript indicating the type of variable; and aj is the weight on a variable with the subscript indicating the relevant variable.9 The calculated quota share of a country, si, is then given by

si=qijql.

10. The effect on the calculated quota share when weights of variables are changed will depend on the manner in which they are changed. The following methods may be distinguished:

  • Increase one weight while holding all others constant (Method A).

  • Increase all weights by the same proportion (Method B).

  • Increase one weight and reduce another by an equal amount to keep the sum of weights constant (Method C).

11. This implies the criteria shown in the table below. Considering first the linear formula, note that the calculated quota share, qi, is simply the (arithmetic) weighted average of the variables within a country. Consequently, the table shows that an increase in a single weight (method A) has a positive impact on the quota shares in those countries where the relevant variable is large relative to the average size of variables in that country. Changing all weights in equal proportion (method B), in contrast, has no effect on quota shares since this implies the same proportional increase in the calculated quotas of all countries. When weights are changed pair-wise so as to keep the sum of weights unchanged (method C), the impact on quota shares will be positive if the variable for which the weight increases is larger than the variable for which the weight decreases.

Table 2.

Criteria for Rise in Calculated Quota Share When Weights are Increased

article image

Increase in aj.

Proportional increase in all weights.

Increase in aj and corresponding decrease in af.

12. With the multiplicative formula the impact on calculated quota shares depends not only on the relative magnitude of variables within a country but also on the relative magnitude of variables across countries. In the case of an increase in a single weight (method A), the criterion for an increase in the quota share is that the relevant variable is greater than the (geometric) average of that variable across all countries where variables are weighted by countries’ calculated quota shares. In the case where all weights are changed in equal proportion (method B), the criterion is that the calculated quota of a country is greater than the (geometric) average of calculated quotas across all countries. When weights are changed pair-wise (method C), the criterion is that the ratio of the two relevant variables is greater than the geometric average of that ratio across all countries.

13. A significant difference between the linear and the multiplicative formula is how they respond to changes in the sum of weights of variables. This is clearly reflected in the case where all weights are changed in equal proportion, which with the multiplicative formula benefits countries with large quotas but has no effect with the linear formula. When the sum of weights increases in the multiplicative formula the distribution of quotas becomes more expanded as large variables increase their relative leverage. With the linear formula, in contrast, an increase in the sum of weights has the same relative impact on all variables and hence there is no impact on quota shares.

14. The importance of the sum of weights in the multiplicative formula implies that a sufficiently small country will be negatively affected by an increase in any weight. Intuitively, for a small country the effect of an increase in the sum of weights, which benefits large countries at the expense of small countries, is likely to dominate the effect of a greater weight on even the variable that is most favorable to that country. This effect is not present with the linear formula or if the sum of weights is held constant. In these cases there is always a weight that will have a positive impact on the quota share of any given country when it is increased.

15. The analytical criteria listed in the table above can be quantified with the data used in this paper. The table below shows the critical values for the formulas used in Tables 5 and 8 in the main text. The critical values with method A are broadly similar to the variable averages of the advanced countries (see Tables 1 and 3 in the main text). This implies that an increase in a weight of a variable would raise the calculated quota share of only the few countries that are above the average for the advanced economies, and that the majority of countries would be negatively affected by an increase in any one weight. The same result holds with method B. With method C the largest critical value relates to the situation where GDP and variability are changed together reflecting that these two variables are on average the largest and the smallest of the four. In all cases these critical values hold up well also when allowing countries to select the formula that gives them the higher quota share as done in Table 8.

Table 8.

Calculated Quota Shares: Two Formulas

(In percent)

article image
article image
article image
article image
article image
article image
Source: Staff estimates.

Based on actual quotas except for the nine countries that have not yet consented to their quota increases, for which 11th Review proposed quotas are used.

Computed as traditionally specified, except that current receipts and payments have not been adjusted for official transfers, reexports, and international banking interest.

See Table 8 in the main text for the weights of variables.

The highest of the two shares is selected for each member. Selected shares are proportionally reduced so that they sum to 100 percent.

Table 5.

Calculated Quota Shares: GDP with Largest Weight

(In percent)

article image
article image
article image
article image
article image
article image
Source: Staff estimates.

Based on actual quotas except for the nine countries that have not yet consented to their quota increases, for which 11th Review proposed quotas are used.

Computed as traditionally specified, except that current receipts and payments have not been adjusted for official transfers, reexports, and international banking interest.

See Table 5 in the main text for the weights of variables.