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B Mathematical Derivations This appendix explains the basis for the mathematical results presented in Chap- ter 4. The form in which the equations are presented follows that of Rogers and colleagues (e.g., Rogers, 1995), who have put emphasis on their links to stable population theory. Alternative approaches can also be found in the literature; for example, some authors develop their equations in terms of net rather than gross migration rates, and others define rates of migration in terms of destination rather than origin populations. The fixed-rates model described in the first section of Chapter 4 can be written in matrix form as follows: [ UtUt_1 1 _ ~ ridmoor mu 1 ~ Ut-1 1 (B.1) RtRt-1 ~ L mn,r Ormr,~ ~ L Rt-1 ~ with all terms defined as in the text. This is the most convenient form for present purposes. If analytic solutions are desired, the equations can be re-expressed as Ut ~ ~ 1+~nmoor Rt ~ = L mr'n ~ ~ Ut-1 1 moor 1 + Ilrmr,~ ~ L Rt-1 ~ and the stable solution is one in which the urban and rural populations grow at the same rate r, such that in equilibrium, Rt ~ =(l+r) [Rt_l ~ As in conventional stable population models, r can be derived from the eigenval- ues of the projection matrix. Rogers (1995: 15-16) outlines the method for the general case; a survey of solution techniques can be found in Simon and Blume (1994~. These techniques are needed in models with age structure, but in the simple model at hand, r is easily found (Ledent,1980~. 484

MATHEMATICAL DERIVATIONS 485 Returning to the form of the model shown in equation (B.1), we obtain expressions for urban and rural growth rates and the share of total urban growth attributable to migration from the rural sector. These are, respectively, UtUt- 1 rut - 1 = r7~mn,r + TT · mr,~' Rt-1 Ut_l RtRt-l Ut-1 = Ilrmr,~ + . m t1 R~ ""n,r t1 and (B.2) (B.3) MSt = (a m ~ Ut_1 + mr,~ Rt_ ~ U'_ 1 no, me, ~ ~ Rt-1 mr'n , (B.4) Equations (B.2) and (B.3) can be solved for the equilibrium urban/rural population balance, · Ut b _ llm taboo Rt provided that this limit exists and is greater than zero (see Schoen and Kim, 1993). If such a limit exists, then asymptotically the urban and rural growth rates both equal r, the stable population growth rate. To find b, one equates the right-hand sides of equations (B.2) and (B.3), and the value of b is obtained as one of the roots of a quadratic equation. With the solution for b determined, equation (B.2) establishes the long-term rate of urban growth. The long-term share of migration is similarly found by inserting b into equation (B.4). The other measures we discuss can be derived from these equations. The level of urbanization, denoted by Put in the main text, is expressed as p _ Ut ~ t in the terms employed here. The rate of national population growth is rim ~ Pt-1 ~~ + P Art (B.5) which is a weighted average of the urban and rural rates of natural increase. The rate of urbanization, which can be expressed as the difference between the urban growth rate and that of the national population, is given by U U _ P P m + Rt _ + Rt m (B 6) ~ ~ ~ ~ ' Pt-1 ~ ~ Ut-1 '

486 The difference between the urban and rural growth rate is URGDt_ ~ = (~nmarryFormr'n) + U CITIES TRANSFORMED mr'n Ut_i Rt_i This relationship plays an important role in United Nations projections. · mar. (B.7)