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D United Nations Estimates and Projections The urban/rural growth rate difference, or URGD, is the centerpiece of the method developed by the United Nations Population Division to estimate and project ur- ban populations. In its initial formulation, however, the United Nations method did not exploit the full implications of equation (B.7) in Appendix B. Here we first discuss the extrapolation method as it was presented by the United Nations (1974: 29), and then show how the refinements later introduced by the United Nations ( 1980: 10-1 1 ) exploited the equation. With data at two points in time, say, t1 and t, the urban/rural growth rate difference can be approximated by URGDt_i = inin t ~ . Rt Rt_i Taking the growth rate difference to be constant at URGD = d and expressing the urban/rural balance in t as a function of a base year balance, we may write _ _ Uo edt Rt Ro To convert the urban/rural balance at time t into an expression for the level of urbanization, we make use of the relation Ut/Rt l+Ut/Rt' and obtain, after substitution, p (UO/RO)edt ant 1 + (Uo /Ro j edit 495 Jo -

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496 CITIES TRANSFORMED With SO _ in (Uo/Ro), the level of urbanization can be written in a standard logit form, em+ ~ 1 + e~o+d t (D.1) This is the essence of the so-called "United Nations method" for estimating the level of urbanization from data on a country's urban and rural populations at two points in time. ~ In the format of World Urbanization Prospects, estimates (and projections) are published for 5-year intervals beginning in 1950. To produce these estimates, the United Nations uses equation (D.1) to interpolate between censuses and to pro- vide reasonable guesses of urban proportions between 1950 and the first national census available after that date (United Nations, 1998b: 33~. The weak point of the United Nations method emerges when it is employed to generate forward projections of urbanization levels. The method assumes con- stancy in the URGD, and this is an untenable assumption in medium- and long- term projections. The simple analytic model of Appendix B shows that, with other things held constant, the URGD should decline with urbanization. Refer- ring to equation (B.7), we see that as the urban/rural balance Ut_~/Rt_i rises, a reduction in the growth rate difference must result, except in the unlikely case in which both mr,~' and m~`,r are zero. Noting this, the United Nations (1980: 10-11) sought and found in the empirical record supporting evidence for URGD decline. Since 1980, therefore, the United Nations has incorporated in its projections a function that ensures a progressive decline in growth rate differences over the course of each country's projection. The United Nations (1998b: 33-4) gives a concise summary of the function, which is based on a simple regression model estimated on data from all countries, whether developing or developed, with more than 2 million inhabitants in 1995 (a total of 113 countries). The function has the effect of slowing the projected rate at which countries with high levels of ur- banization approach the upper limit of unity, but it also induces somewhat faster projected urbanization for countries with low levels of urbanization (United Na- tions, 1980: 11~. Note also that whereas the United Nations estimates of urban population are based only on country-specific census data, the projections of ur- ban population use both country-specific and cross-country data, with the latter including developed as well as developing countries. The United Nations method for estimating and projecting the population of individual cities is, in broad outline, little more than an application of the URGD method to different data. Rather than working with the urban/rural balance Ut/Rt, however, the method considers a city's population in relation to the national total less the population of the city in question.2 Let i denote the city whose population iNote that the maximum urbanization level implied by the equation is unity. 2The approach is a revised version of a method set forth earlier by the United Nations (1974), in which each city's population was examined in relation to the total urban population of the country,

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UNITED NATIONS ESTIMATES AND PROJECTIONS 497 is being estimated and projected, and let Ui,~ be its population in year t. The estimation exercise for city i begins with ui't AtUi,t (D.2) and the population of city i is then estimated from the differences in growth rates of numerator and denominator found in the most recently available intercensal period. The method is applied independently, city by city within a country, to all those cities whose populations are to be estimated (United Nations 1980: 44-47, 1998b: 35). As is the case when urban totals are projected, the projections of city popu- lation must rely upon additional research judgments. The United Nations (1980) introduced a method for keeping the projected growth rates of large cities within reasonable bounds. In this study, the United Nations noted a U-shaped relation- ship between city size and city growth rates, with the highest growth rates evident in small (under 500,000 population) and very large (over 4 million population) cities. At the time of the study, however, there were relatively few very large cities in developing countries, and the United Nations concluded that the relation- ship between city size and growth rates was predominantly negative. City size was found to be only a weak predictor of city growth, however, with the raw cor- relation between city growth rates and the log of city size being only0.083. The United Nations (1998b: 36) revisited the issue, again using both developed and developing countries, and confirmed what had been seen earlier: a weakly negative relationship between city size and growth.3 Although mindful of the thin empirical justification, the United Nations has judged it sensible to embed a negative relationship between city size and growth in its city population projections. As explained by the United Nations (1998b), the method relies upon the results of a cross-country regression, estimated using the sample of 113 developed and developing countries described above (which con- tains some 1,982 cities). The dependent variable of the regression is the difference in growth rates between the city and the country's total urban population (rather than the total population used in the denominator of equation (D.2)); the explana- tory variable is the log of initial city size. This regression produces a predicted value for the city-urban growth rate difference given the size of the city. rather than the national total. The growth rate of city i is of course linked to the growth rate of the country's urban population the linkage is obvious in countries where the urban sector is dominated by a few large cities. Hence in the approach favored by the United Nations (1974: 46-47), the city projections were to be made iteratively, beginning with the country's largest city and the projected urban total, and at each subsequent step removing from the denominator of the equation (an expression analogous to equation (D.2)) those cities whose populations had already been projected. Tin view of this low bivariate correlation, it is surprising that in the multivariate regressions reported by Preston (1979: Table 2) and Brockerhoff (1999: Table 4), where one might have expected a weak city size effect to disappear altogether, city size remains negatively and significantly associated with the rate of city growth.

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498 CITIES TRANSFORMED Consider a projection of the population of city i in year t, which relies on the size projected for that city in year t1. The projection for t relies in large part on the URGD for city i in the most recent intercensal period, but the projected value is modified to reflect the regression equation described above, which is mul- tiplied by a year-specific weight. The sequence of weights is chosen so as to force the projected city-urban growth difference toward the relationship implied by the regression equation. The United Nations (1980: 45-47) gives a lucid account of this procedure. We would not want to leave the impression that these city estimates and pro- jections are little more than a mechanical extrapolation of assumptions. United Nations researchers have long recognized the possibility of errors and have at- tempted to take them into account. The United Nations (1974: 45, 48) warned of the possibility of considerable error if the city projection method were to be applied mechanically and urged that the results of the exercise be carefully scruti- nized. Recently, the United Nations (1998b: 36) conceded that, even with a damp- ening function in place, when the city population projections are aggregated to the country level, they can show a tendency to grow more rapidly than the country's total projected urban population. This is not, in itself, evidence of an inconsis- tency between the two types of projections recall that small urban areas do not usually have their populations estimated and projected. But care must obviously be taken to ensure that the total of the projected city populations does not exceed the total projected for all urban areas. When the aggregate of the city-specific projections happens to grow more rapidly than the projected urban total (the United Nations, 1998b: 36, suggests that this situation is common), yet another adjustment is applied to bring the pro- jected rates of city growth more in line with that of the urban total. Each city's growth rate is reduced by the same amount in such a way that, over the full course of a projection, the projected city total can never exceed the projected urban to- tal. Finally, additional adjustments are made in cases of negative estimated or projected city growth rates; in the final adjusted estimates and projections, such negative growth rates are generally replaced by zero growth rates (or so it appears, to judge from the United Nations, 1998b: 36~. The result of all this is a complicated algorithm whose success in estimating and projecting city populations is far from being assured. Although the United Nations has been careful to describe the main features of the methods it applies, it has not fully developed the demographic justification for its projection adjust- ments, nor has it presented a clear rationale for commingling developed- and developing-country population data in these projections. The consequence is that the United Nations' city population projections are linked across countries and over time in ways that are difficult to defend. Criticisms of this nature can also be leveled at the projections of total urban populations, although where these projec- tions are concerned, the adjustments are fewer and somewhat better justified.