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The long-range population projections of the United Nations issued in 2003 span three centuries and are elaborated for all countries of the world according to the present-day political map. This note discusses the merits and limitations of this ambitious enterprise. The numerical implications of various contrasting assumptions concerning fertility, in combination with single hypothetical future schedules of mortality and international migration, provide a valuable frame of reference for contemplating possible long-range demographic trajectories. The dominant suggestion of these projections of a surprise-free convergence to a stationary or slowly declining population is, however, questionable: with respect to global numbers, relative magnitudes of the constituting units of the global total, and the time pattern of change the demographic future is likely to be far less orderly.