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Frank T. Denton and Byron G. Spencer

and a subsequently maintained low level. The Children of the boom are in Middle Age at
t = 0, and a generation later they will be Seniors. The population is aging.
Alpha is closed to trade but open to immigration — indeed, there is an infinite supply of
potential immigrants available, and thus the possibility of using immigration as a tool to
offset what is going on in the domestic population. (Note that we are talking about immi-
grants as permanent additions to the population, not temporary “guest workers”.) The
government can set the immigration quota — the number of immigrants to be admitted in
each generation — and it can set the immigrant age distribution. What follows in this pa-
per is a model and assessment of the longer-run implications of those choices and related
considerations.

2.2 The Model
The dynamics of the population and income generation are simple. Let the column vector
n stand for the population by age and sex: the first five rows are female age groups
(youngest to oldest), the second five are male age groups. The progress of the population
from generation t to generation t+1 can be represented as
n t+ 1 = Qn + m t+ 1 (1)
t
where m is a vector of immigrants (with age-sex elements corresponding to those of n, all
nonnegative) and Q is a 10×10 Leslie matrix (Leslie, 1945, 1948); its nonzero elements are
determined by age-sex-specific survival rates, the fertility rate, and the male/female birth
ratio. If there were no immigration, and all rates were constant, n t 1 + = Qn would hold
t
exactly for all t. (The matrix is defined more precisely in the Appendix.) There is no emi-
gration, only immigration.
The vector m can be separated into two components, one representing the total number
of immigrants, the scalar M, the other their proportionate age-sex distribution, the vector
A, where A is the set of all possible age-sex distributions. We refer to M as the immi-
α
gration quota. The quota is set as a proportion q of what the total population would be in
any given generation without immigration. The actual total population in generation t+1 is
un ′ t 1 + , where is a column vector of ones, and the total population as it would be if there
)
′
′
were no immigration is u Qn . The immigration quota is then M t 1 + = ( q u Qn . Making
t
t
the substitutions, Equation (1) can be rewritten as
′
n t 1 + = Qn + M α t 1 + Qn + ( q u Qn t )α = (2)
t
t
Thus q and are the policy choices for the government.
The employed labour force — or simply labour force — is determined by the popula-
tion vector n and a vector of constant participation rates r, shared by both immigrants and
the domestic population: thus L = ' rn .
Output Z (in real terms) is generated by a constant-returns-to-scale Cobb-Douglas pro-
duction function, with inputs L for labour and K for capital: in log form,
lnZ = t µθ+ t β+ lnK + t (1 β− )lnL (3)
t
where θ is the intergenerational rate of neutral technical progress, or equivalently, total
factor productivity. Investment I is supported by a constant saving rate γ : thus
I = S γ= Z . The stock of capital is subject to a rectangular or “one horse shay” deprecia-
tion function (Hulten and Wykoff, 1981). A unit of stock is undepreciated for one genera-
tion, and is then terminated; hence K = = γ Z , a convenient simplification for our pur-
I
poses. Note that since a generation is 20 years, the rectangular depreciation function pro-

International Journal of Population Studies | 2015, Volume 1, Issue 1 77