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OCR for page 484

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:
[
Ut—Ut_1 1 _ ~ rid—moor mu 1 ~ Ut-1 1 (B.1)
Rt—Rt-1 ~ L mn,r Or—mr,~ ~ 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+~n—moor
Rt ~ = L
mr'n ~ ~ Ut-1 1
moor 1 + Ilr—mr,~ ~ 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

OCR for page 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,
Ut—Ut- 1
rut - 1 = r7~—mn,r + TT · mr,~'
Rt-1
Ut_l
Rt—Rt-l Ut-1
= Ilr—mr,~ + . m
t—1
R~ ""n,r
t—1
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 '

OCR for page 484

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