Estimates and projection of the population of large cities and their use in urban development planning

OS'S?

Distr.

/ LIMITED

E/CN.14/POP/96

UNITED NATIONS 19 June 1973

ECONOMIC AND SOCIAL COUNCIL original: English

ECONOMIC COMMISSION FOR AFRICA Seminar on Techniques of Evaluation

■L of Basic Demographic Data

Accra, Ghana, 16-28 July 1973

ESTIMATES AND PROJECTIONS OF THE POPULATION OF LARGE CITIES

AND THEIR USE IN URBAN DEVELOPMENT PLANNING**"

+ This document was issued for the United Nations Seminar on Evaluation and Utilization of Population Census Data in Latin America, held in Santiago, Chile, from 30 November to 18 December 1959 and the United Nations Seminar on Evaluation and Utilization of Population Census data in.Asia and the

Far East, Bombay, India, from 20 June to 8 July I960. It is intended to use

this document also as a basis for discussion at the Seminar on Techniques

of Evaluation of Basic Demographic Data, Accra, Ghana,from 16 to 28 July 1973-

£73-1442

E/CN.U/F0P/9S

Estimates and projections of the population of large cities

and their use in urban development 'planning ■

'■;■■■■"■• ■ " -' -'■-"-' :''-"':- ■' ,i; ' INTRODUCTION ■ ■

Surprisingly few people" seem to appreciate the extraordinary Value of censuses. They are costly, and uninformed people' can even consider them as

nothing more than show pieces. ■ It, is therefore incumbent.on.-those who appreciate what the censuses' of l°60 can mean-to their countries to do everything pos.sible to ensure that their nations will derive ,the largest-possible benefit from .them, A good Census illuminates almost every aspect of a nation's development and makes, possible-undertakings which, without it, would be of dubious value. One of these aspects is the estimation of future population. This is always a haaardous

venture under the 'best.circumstances, but the.margins of error become so wide as to make them of little practical value if there is no sound starting point - like the marker in the ground from which a land surveyor starts his measurements..

This paper is particularly concerned with estimates and forecasts of the population of large cities.

To save money and to have advance warning of possible dangers must rank . among the principal purposes of forecasting. But if there has been no census for many years, or if the last one was po,or, the nation or city necessarily

"adapts itself to a situation in which there can be little effective planning.

It is-many times more costly no'Vtb have a1 census than to have one.

Most municipal activities require planning. For example, public health must constantly be guarded in a densely settled area. While water, sewags disposal, 'food inspection, hospitalization and many other things are being currently

looked after, active preparations must simultaneously be under way to meet the demands which will make themselves felt in'the future. Personnel must be hired and trained and equipment provided on time - not too much ahead of time, because that wastes public'money, and certainly not too late, because the* endangers

the public health. How large will be those future demands ? Ldwell J- Reed

of the Johns Hopkins-University noted that: "There is almost no contact between the ideas of these specialized plans and the estimates that have been

made of the future population." (Reference 23, page 5.). ...

Public health is only one area for planning; others come quickly to'mind.

Will highways be adequate ? Are certain areas destined to» become overcrowded slums which will be breeding places of crime,,-discontent and disease'? What

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is the situation with respect to schools, police and fire departments', etc.?

To prepare for contingencies, it is imperative to have an accepted working plan as a guide and one which, as Professor Reed pointed out, is adapted to the size of the population of the future.

It may be helpful to consider briefly, before launching into this subject, what can and what cannot be expected in attempting to forecast city popula

tions. ■ ' . ■-•■;'

li First, let us once for all accept the bare truth that we cannot forecast

the future. ' (>-''■*■ '" :

.."It is.the failure of human history to repeat-itself, the appearance of

■ the new and the unexpected that renders the search for good methods of forecasting hopeless. However much we improve our tools to take care of all that happened in the past, something will sooner or later crop up for which we are unprepared . ■. . New and more complex techniques which may be invented are, I think, just as liable as past techniques to be fairly often upset by the unpredictability of history. They will just as often - and that means rather frequently - give results which are ' very wide of the mark and less accurate than crude guessing."

(Reference 9)» ' v .

This statement is far from constituting a plea to< dispense with forecasts - quite the contrary. Of course they will continue to be wrong, but, its author says, they will also continue to meet "the need to take decisions. This

seems to be the meaning of the demand so often faced by, the statistician

*Give me a forecast, any figure is better than-none*." (page 313).

2. We can in fact know a good, deal about the future, and our knowledge of it comes ent'irely from the past. (Appreciation of this fact shows how

valuable is the census record as a tool for seeing into the future.) The most elaborate statistical models - in fact all serious attempts to look into the future - are based entirely on past trends. This knowledge of the past may

•take different forms, such as regression coefficients, correlation coefficients, age distributions, measured trends, averages, etc., yet all of these measures have this same common denominator and to admit of an exception is to postulate

the supernatural. -

3w "The success of a population projection as a piece of analysis is not

■ ■ ' i- '

measured by the percentage difference between the projected and the actual

populations." (Hajnal, loc.cit.t page 314)- Accuracy of prediction is no

Page 3

sine qua non of forecasts. There are other factors' which have equally important places in the scale ;of values: a superb set of forecast numbers ' given too late for their intended use, or so late that basic conditions ■ have changed, is no better than the worst forecast. A projection bya metho dology so complex that the forecaster for .sheer lack of time is himself ignorant of the implications is also a wasted effort: if those who'must use The rorecast cannot be toli its neaning, where, indeed, is its value ?- .Again to quote from Hajnal: "Forecasts, should flow from the analysis of the past.

Anyone who has not bothered with analysis should not forecast. The labour spent in doing elaborate projections on a variety of assumptions by a ready- made technique v/ould often be much better employed in a study of the past.

Out of such study may. occasionally come important insights about unexpected possibilities in the future." (loc.cit.', page 321).

. A population projection is not a wager in which the contestants await the outcome to settle the score, because the score is' settled shortly after the projection is made and usually many years prior to its outcome. In fact, most forecasts will long have been forgotten by the time of outcome, and their merit will have consisted solely in the understanding they engendered of the popula tion trends at' the time they were made and hence in the wisdom of the decisions based upon them.

There has grown up a literature on the relative accurancy of various

methodologies to which only a few references will be made here. It- is difficult to see what these tests prove. They assuredly do not test those additional attributes of methodologies whichjimst be considered if small area population projections are being taken up for the .first time. (The following references are almost wholly devoted to such tests-r Numbers 28, 29, 37. So also is • Number 24, but it should be read. in,:,, conjunction with Numbers 13, 25, and pages 78 and 79 of Number 48).

There are today a great many methodologies for making projections of the population of smaller areas. A careful-classification has been given by . •

Siegel (28),' although his evaluations (quite properly) are from, the point of view of showing at a world population csndTerence wha-fc is done in the United States, with the statistical apparatus peculiar to it, and they cannot have

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Page A

been offered as a prescription f^^ery county. '

for lamentation, however, that"Wne W the .ore elaborate .ethologies'

«-* be laid aside by-a great ^/countries of the world because

,data are notavailable,

, There ^ ,„ ^^ ^^ ^^^

There ^

over t e siller ones. HaJnal believes that .'si.ple, mpretenticu8 sh ,f '

: t°:; T " "" " ^ ^——-Polt-f : t:;0 T " "" " ^ ^-——-Populate,-fore

casts (loc^., page 3Oy), The ,r..ter.has q t,

heavy deraanas for fiftesri-year as for Wy^' forecasts, '..; / , '

in J\ T r ^»ialrole of

m the estimation .of city, populations: . , . . ,...,;

(a) If there is a- ,cloa.d« grouper study. one which is 3ubjeot ! only to ffains frOm births and losses. fr.om,deaths, with'no .i^aticnto or from

bHl' Pr0Uem " POPUl3tlOn ^i—«-. • Migratioh ten

be erratio m a.ou.t and sti i —«-. Migratioh ten

m a.ou.t and so.eti.es. in direction of .ove.ent, even though the

^'*ti«- of Migrants preserve, considerable stlbility

_). The facts about mi?ration ^ ^ ^ & ^

a matter of official record, mt.rnal Ration is .uoh .ore difficult" to '

lT " I3 SUbJ6Ct ^ id

clT I e ^ -S °f measuring,et inter-

censal Ration between areas of a country, usin, nothme mOre than ^e • ■ ■ distributions in successive censuses (Reference 12), Dut in general the "

me.sure.ent of internal .^ration iB dependent on vital statistics

■accurate record of births to, and deaths of, reEidentS; recorded as of th.

tl-e periods in which they actually occurred. Anyth^ less entaUs the '

risk, of large error in population predecticns founded upon then,- In' •

particular, it .ay be noted that large citiesare usually subject to :heavy •

migration across their borders. , ; ' ■...-■•'•

birth^ +! lnVlt6d ^ the Carem ^^^PPraisal of the .

birth statistics of Latin African countries by *. Lvnn s.ith (Reference 30), as ^ as to the^analyse, by the Secretariat of the United Nations (Hefrence 36 and later.issues of the sa.e publication). Desplte i.prove.ent, S.ith.

notes "the utter lack of dependability of the birth rates as reported". ....

7TZ V ^ ^1-^.W-h has

both the choice of topics and the varying e.phasis placed on"what

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follows. ' Because of these considerations it has seemed wisest to concentrate rather heavily on the use of growth curves, and the ratio method, neither of which makes any demand for vital statistics. With regard to the former,' Hajnal has said: "In some circumstances, the fitting of simple growth curves will be useful. On the whole, however, growth curves are most likely to

find application in under-developed countries'with poor statistics." (loc.cit.,

:page 317).; " ■.

< II. -MATHEMATICAL PROJECTION' ' ■ - -

Attempts to predict the population of large cities and countries by mathematical curves are frequently made. A brief analysis of the methods employed will clarify the reasoning in other types of projections.

A. Malthusian law

Thomas Malthus (I766-I834) contended that- without some external limitation (particularly the supply of food), the growth of a population tends to be

proportional to its size. Wilson and Puffer comment that: "It'is curious to call the exponential law Malthusian when Malthus' Chief thesis was that it could not be followed, but the usage seems established."._ .(Reference.38, page 362).1

If we denote the size of a population at timevt by P(t), with the variable t

1 1

measured in years, and with n the constant of proportionality, then the Malthusian or exponential "law" states.that

The symbol P'(t) represents the annual number of people who would be added

to the population if the rate of growth at t were to stay in effect for a year.

The proportion on the left is the one described in the law. It is like an interest rate, where the increment is divided by the principal, but with the interest compounded continuously instead of once or twice a year. Such a rate is called the "force of interest". There is only one mathematical function which answers to the prescription in (l) and that is:

(2) P(t) = C en\

where "C" is a constant of integration which adjusts the level of the curve to

e/cn.u/pop/96 Page'6

agree .with .that of the population. If t = 0/ then P(0). = C? since e° = 1;

If P(t) in equation (2) is plotted, as a time series, with the ordinates (population) on a logarithmic vertical scale, the result is a rising straight

•line if n .is positive (declining if n is negative). With larger and larger

values of t- this curve reaches such fantastic levels*as to be absurd. Also, if t-he Malthusian were fitted to the populations in several areas (parts of a city, for example), their sum would not be a Malthusian, unless it happened that

n was the same for all of the areas. In spite of its drawbacks, this is one of the most useful estimators of population for short periods: it is almost

universally used toj compare rates of growth over a period of years - ,but with, one minor alteration for convenience alone:; continuous compounding is dropped

in favour,of annual. The conversion of (2) to such' a basis is a simple one:1

(3)' P(t) = C(l + i)\ if i = en - 1 .

The rate i is referred to as an "average annual rate of increase". (For an example of its use in comparisons, see Reference 36, Table 4, last column.

Concerning Malthus and his opinions, see Reference 35 and- its extensive bibliography.).

B." Logistic law or Pearl-Reed curve. ..■■■'

The relentless grovrth of the Malthusian function called for a different

law, and in 1838, P.P. Verhulst (l804-l849)» produced the logistic curve.

(See Reference 35)" Verhulst subtracted from the right-hand side of the equation (l) a quantity which grew with the population:

(4) ■■:■ m - n - a PC*)",-

with "a" a positive constant less than unity. At the very start of growth, the relative gains of the population are the same as the Malthusian since the

population is negligible. As the population grows, however, the subtraction grows larger.' The curve which results from Verhulst's specification in (4)

is the famous logistic:. "

E/CN\ 14/FOP/96

Page 7 ■"•""^ •*. .. ■ .

r

(5) P(,t)- - 1 •■

a + b e. ■ " ■ -_

By taking reciprocals of both sides-of the equation, it can be written:

(5a)' '^ _J__ "„ a-fb e"ntv ■ '"

P(*)" ' . ' . ■ '

The curve is S-shaped, and sets a finite upper limit for population size instead of the infinite'limit of the Malthusian. This upper limit,

or asymptote which the curve approaches is l/a. At P(t) = T/2a - half-way

up - there is a point of inflection; from concave up the curve changes to concave down. The logistic, like the Malthueian, of which it is, a generaliza tion, suffers from the. same lack of "additivity". / ■

The logistic had long been forgotten when, eighty-two years later, in .1920, it was independently discovered by Pearl and Reed,- who then published one

of the most famous of all forecasts of United States population." They determined the three constants by requiring the curve to pass through three decennial

census enumerations: those of 1790 (the nation's first)j I85O and 1910,

the latest then available. They thus ignored ten other available counts. ■ Stating- exactly what they had done, they added: "We attach no particular • '""■' ' significance to the numerical results ..." (Reference 19). For 1950 -

some-forty years after their latest datum and thirty after its publication -

this famous- cur.ve indicated a population which was only 1 l/3 per cent below

the actual census count..-J.S. Davis in a not too complimentary paper on

population forecasts, gave more than a nod of "approval to ,it. (Reference 4-)- It indicates, however a maximum population for the United States-of only 197 ■ million, or roughly seven more years of growth .at present annual increments, and it now appears that the long day of glory of this curve is drawing to a close. Hajnal has stated that "these authors themselves proved that their success was only accidental.' As a consequence "of a 'rapid' fall in birth

rates.... they- issued revised estimates which were lower than the original ones

and turned out to be in worse agreement with the facts." (Reference 9, page 316).

The logistic has been used extensively in forecasts for large cities, but only two well-known examples will be cited: those for New York and Chicago.

(References 15.and 20). For the latter,:however, no logistic could be found to

E/CN.14/POP/96 Page 8

to give credible results. (See Pearl and Reed's comments in Reference 21).

The results for New York were proudly published in 1923 by the committee which had retained the services of- Pearl and Reed. (Wilson and Puffer's comments should also be consulted). With 1920 their last datum, their projection for New York City proper was only 1 per cent above the 1930 census. By 1940, two decades later, the gap had widened to 12 per cent, and by 1950, it"was 23 per cent - but after a lapse of thirty years. 'This • is a remarkably good record. In the interim, the Pearl-Reed forecasts

must liave "been'replaced'several times- ' ' ' '' * ' As descriptive of long-range growth, both in population and economic , . production, the logistic has had few peers. On its theoretical interpretation as the representation of a law of population growth, as well as on its practical

application to the forecasting of populations/the reader is referred especially

to the works of Pearl and Reed (Reference 19), Wilson and Puffer (Reference 38)/

Davies (Reference 2), von. Szelinski (Reference 32) and Feller.(Reference,7).

These works and others, some of which are cited in the list'of references, have shown that the logistic often describes past population growth with great

accuracy, but is by no means an infallible tool, nor necessarily the best tool, for the prediction of population growth. Other-forms of growth curves may' "

give' equally good or better results in given situations, and it is plain that the choice of curves must bs pragmatic, with no law to support them. With no law to contend with, there is a wider latitude of choice among specific types ' than might be supposed. Consider their behaviour at the point of inflection

of four types of curve: ","'"'

■- ' Behaviour at Point of Inflection Ordinate as Inclination of

percentage of tangent as multiple ' ■ ■-

Curve ' . . upper asymptote of upper asymptote Symmetry ■■

Logistic 50.00 • ,2500b . skew symmetric .

Arctangent 50.00 -7329b asymmetric

Curve of ..error . 50.00 . , . .3183 b . . skew symmetric

Gompertz ., .36.79 -,. ■ .3679 b asymmetric

In each case, "b" is that one of the three constants at-our disposal which ' enables us "to fix arbitrarily ... the tangent.at the point of inflection".

Page 9

Apart from this one instance, the other factors are fixed: the. ordinate

at the point of inflection of ali Gompertz curves is 36-79 per cent of the upper asymptote' (lOO/e)-. These are all factors to consider in fitting a

curve to population data and full advantage of such latitude should be taken,

■;■'-■ ■' ' III. COMPONENTS METHOD

-:•'■ The excellence of the results to be obtained by the components method depends to a: large extent on the quality of the statistics of births and deaths.

At the outset let us write P(t) for the number of people in the' population

at date "t". We shall suppose that this variable is measured in years elapsed

from the last census, so that P(o) is the population then enumerated, ;Let us suppose that the census date is always on January 1. We can agree that

B.i D , and M, shall represent, in that order, the births, deaths and net in- migration in the calendar year between dates t and t + 1. The migration can of course.be negative and it may consist Of two streams, the foreign and the internal. The general scheme is as follows:

■ January 1 ' \ _. Population change during year

.. Xeax Population " Births Deaths Migration Total

-° .HO)*' BQ : Do /. . MQ . . ... . Ro

1 .. ' P(l) Bl ' Dl ' Ml ' Rl -

i p(i) ■ • Bi .■" Di •■■ - Mi -- ■

* Assumed to be the census count.

At once we can write down the following expressions which are evident from the table

(1) ' R. = B. - D. + M. - ' .

. i,ii,if

(2) : ' P(i +' 1) = 'P(i) + R(i). '

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■ Now we proceed to the projection. If we could develop rates for births, deaths and migration' in each' projection year, we could project the population.

But we know only'the past, not the future; so with our excellent vital statistics and censuses in the past we develop a table.like that above, but covering the last inter-censal decade. We can fill in the population for only two dates: t = -10, for which we know the January 1 figure, and t = -1, for which we know the December 31 Figure, which is the January 1 figure for t = 0, Both figures are census counts. In the other columns, we can enter only the births:and deaths, which we then add for the decade. We'realize

that we can write equations (l) and'(2) for a decade just as well as for a year.

Starting with the latter we have: ' ' ■ ■-...■,

) = P(0) - P

where S(R. ) means.the sum. Hence we know the net total population gain.

Equation (l) we write as: . . .

S(Mi) = 'SOL) - S(Bi) +

and obtain the net total inter—censal migration to the city.

It is futile to pursue the migration figure further. If there were nothing better, S(M.) .could.be divided by 10; ■ bettor still would be to go

back into more inter-censal periods. With whatever data there were, the object would be to develop, a series of figures of migration in the past.

We could then determine average population and relate the data on births,

deaths and migration to the corresponding averages. The resulting ratios would

be rates per person. . " .

It is generally these rates which are projected, and here it would be a question of projecting them on the basis of past trends, tempered by what

our thoughts were for the future. ...

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IV. RATIO METHOD

The ratio, method is probably one of the most satisfactory -procedures for projecting the populations of cities and other.subdivisions of a country where the country's total population has teen projected by other methods- Simple in concept, it is difficult to apply well. It makes no demand whatever for

■ vital statistics, but the more information a person has at his command about

the areas for which" he is "making predictions,' the better should be1 his fore

casts. 'In view of its potentialities'and the poor reputation it has acquired by inadequate methodology,1 it has'seemed worth while to give it emphasis here.

Ai Basic-concepts and .-rule a ■ , '

s The ratio method does not of itself provide population projections; its

function .is-solely to provide sets of proportions ("ratios") which, when ■

multiplied by a given projection, indicate the population in the constituent parts- of the.area.. The method specifically assumes the availability of a population projection for the over-all,area. By the ratio method, however/

a-national projection can be successively broken down into smaller and smaller subdivisions* An advantage of ,the method is that the given population

projection can be revised repeatedly without necessitating recomputation of.

"the •ratios. ' ■■ '-*■ . ' ■ .

Frequently it is necessary, in making small area forecasts with, the .■ - - ratio method, to start with a projection for a very much larger area. Here. ■■ • the best practice ("long" method) is to proceed in successive stages. As. .

an illustration, the United States Bureau of the Census for its state

projections published in.1952 proceeded in two stages, in the.first of which proportions were derived for the nine\geographic divisions'of the nation.

(Reference 43). In another quite distinct and independent application of the

method, the New England.division, for instance, was subdivided into its six component states, one of which, for example,-is-Maine. This process could

course, have progressed to areas'much smaller'than states. However, the results of these two- stages, the high projection for Maine in I960 .then, consisted of-, the product: , ■ . ■

E/CN.14/PQP/-9-6:

Page \Z ' : ■.

.' Given

Projection • (Stage l) (Stage 2)

Population in Maine = U.S. pop! x -&&• x .

.... ... . "' , .U.S. - ; • -.-.N.-E, .

■ .-■ = ■■' 180,276,000 x-- 0.0597 !x 0.0981 .

' ' = l.O55;OOO.

At each stage, all component parts should be projected, since a.

projection fo* the proportion of one implies a complementary projection for the remainder. Many local area projections seem.to founder through ignoring this safeguard. "By progressing in stages, the parts can be kept to roughly the same order of magnitude (combining smaller parts when-

necessary). For example/ if a projection.for the standard metropolitan

area of Portland, Maine, were an objective one can see the absurdity of ■ balancing its less than 0.08 per cent of the nation's population (in 1950) against the remaining.99.92 per cent.' An absurdly bad blunder for.Portland could have no discernible effect on the projection"for the remainder,

of the country and could go undetected. On the other hand, if the Portland standard metropolitan area were to be related to the state of Maine, the

corresponding proportions .would, be. 13 per cent for the Portland^ area against "-

87 per cent for. the remainder, A bad error for Portland would have consequences

for the remainder of the state which would soon come to light. Securing

reasonable projections for each part, and at +,he same time having the parts-..,, add to the whole is often difficult, but constitutes a valuable.safeguard.

To cope with too many areas in one sxage is false economy. Half a dozen is a reasonable.average. Combination of areas,.again, provides a means of control. The combined areas can later be broken down in an added stage.' It seems reasonable when combining areas to endeavour .to group together those

showing somewhat similar trends. •

B. X^J^H^^^lhe^p^^ortion^s ....•-.. ■

Having planned the stages by which to proceed1, the population data are extracted from all "the past censuses in which they are' available/ Any necessary adjustments are made for significant boundary changes, or for known defects in particular censuses. The1 effects pf some types of defect rray cancel out,

such as s, consistent under-enumeration in all areas.

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For each stage, the proportions are computed and a check is made to ensure that their total is .precisely unity at .-.each census. The .common logarithm of each proportion is then recorded and charted to show clearly what is known of

the past history of each proportion. . . ■ •

These charts usually fell a surprising story - one that would not be at all obvious were population rather than proportions to be plotted. What one is looking for, of courseware trend's - trends which have persisted for considerable periods and which must be taken into account in projection. It is at this

juncture that any information available about the individual area .is helpful.

It*is a safe assumption that trends will be much in evidence for most areas;

If there is any area for which no clear trend is indicated, the proportion for that area should be" projected at the"level indicated in the iast census.

The first objective is to find the rate of growth of each proportion precise ly at the date of the last census. This is not at all' the same thing as the annual average rate of growth between two past dates: it is the rate of growth precisely at.that date, so that the data incorporated in the forecast will be as up-to-date as possible and not an average applicable, to a period many years ago.

"Finding this rate involves fitting a curve to the proportions arid determining,, specifically, the logarithmic derivative from this curve. (Fitting this curve

is usually simple and quick, since it is rarely necessary to compute points

,and-plot them-on a chart. Its sole function is the ascertainment of this rate

'of growth). The rate determined from-the curve is equivalent to the force of

.interest; we shall call it r here.

o

The last census indicated that a certain proportion R(o) of the population iri the entire area covered in some stage was to be found in one of its constituent parts. Treating each such' part independently of the others, the procedure

consists of regarding R(o).as. one might a sum of money put at compound interest, but with certain exceptions: ■ the interest is to be added to the account every instant, and every'instant the rate becomes less (in absolute amount) until it reaches 0 per cent per year some "n" years after the last census. The rates

can be (and frequently are) negative. The rate r is amortized linearly, so that

it becomes less by one n-th of r every year. After n. years, it is assumed that the proportion R(n) remains unchanged - reflecting the fact that there is no basis for forecasting any further changes. In certain cases it may be desirable to alter this assumption,' however.

'..Page 14

In actual application," sbme'period of amortization, * say" n, must be sel'ected. " From'.a study of past census records'of the same or "of a similar country, studies can" shed"light on the values to be assumed for n, Below is a1

tabulation which shows decennially, from 1790 to 1950, the proportion which

the. population in each of the,five boroughs constitutingNew York City bore to the total .population of the City.1 (The area-covered by these'data remained the- same throughout,-..although much .of it was hot incorporated within the City- until the act of consolidation in 1898.). -During these 160 years the" ' ' ■ population grew 160-fold..- Very long-cycles 'can be seen. The' proportion- for '-

Population in Counties Comprising New York City Expressed

" "as Percentages of Total City Population

Census

1790 . 1800 1810

■1820 1830 , 1840 1850 I860 1870 . 1880 1890 ' 1900' 1910 1920 1930 1940 1550

Bronx

-3-6

■2.2

1.91.8 - 1-3-:

1.3 ' 1.2 2.0

■ 2-5 2-7 3.5 5-8 9.0 13.0

18.318.7 I8.4

7-2

6.9 7.49.5

12.2 '23.819-9 28.4

31.4 33.4 33.9 34-3 35-936.9 36,2 34-7

New York

67.1.

76-480.5

81.483.6 80.0

74-0 69.2 63.8 60.9 57.5 53.8 40.648-9 26.9 25.4.

24.8

Queens

■ 8.4 6.2 5-4 3.7 3.7 2.72.8 3.1.

3-0' 3-5

4-5'

6.0

8.3"

15-6 17.4 19.7

Richmond.

7.8-.

5-8 ■ 4.5-

. 4.0

2,92,8 ;■

2.2 ■ 2.2 2.2- 2.0 , . 2.1 2.0 1.8 ' 2,1 2.3 2.3 ..

Total

100.0 - 100.0 100.0

160.0

100.0 100.0 100.0 100*0 ' 100.0 100. 0 100.0

" 100.c

ioo.o

100.0 :.oo.o 100.0

2.4 ^100.0

Basic-data from: Census of Population-: 1950, Vol. II, Characteristics-of the Population, Part 1, U.S. Summary, Table 23, U.S. Government . Printing Office, 1953, Washington, D.C

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Page 15

Kings County. (Brooklyn) rose between 1810 and 1930, a period of 120'years*

That of. Bronx County rose between I85O and 1940, a-period of ninety years.

There has-been a continuous decline for New York County (Manhattan) since its ' peak in 1830; the. unusually .large, decline between 1920 and 1930 reflected -the"

opening of underground railways.enabling large numbers of people to move from New York County to Bronx and Queens Counties. These data are intended to be

suggestive of what,is involved ,in choosing a period of amortization. An ; average for Kings and Bronx Counties gives 105 years as the period of rise. • If these data were representative, and if one were always to use ,a uniform period of amortization, one would use half this figure, since one might be

entering this' cycle at any date. A term of fifty years has been used by the

Bureau of the Census. However, since the period of amortization is a definite , element of control in the forecast, and.since there is no reason whatever

why such period should be uniform, it would be reasonable to make some sort of investigation before adopting this-figure. .

In order to project the.proportions in any stage, the following quantities' are required for each constituent area: •

R(0): The proportion of-total area population.it contained at last censu?;

r^: its initial rate of growth, as described above;

n : the number of years subsequent to the last census irT which it is assumed that r will have been completely amortized.

Such a set of three constants completely determines the proportion of

total population to be contained in a particular constituent area throughout the projection period. Each constituent area in any stage has its own set-of them, and it will be observed that there is no guarantee at this point that at any giyen date in the future the sum of the proportions indicated by the curves for all constituents will be 100 per cent. - Indeed, it would be so'only"by the.

sheerest coincidence, since these projections.have all been arrived at quite independently. It is also to be noted that each curve not only has the same • initial level as that observed in the last census, but that the tangent drawn ' to the curve at the date of that census has precisely the same inclination as the tangent drawn at the same point to the curve which was fit-ted to represent past census observations. . In fact the nominal annual but momentary rate-of

e/CN.H/pop/96

Page 16 '

compound interest growth at the date of/the last census is the same, both in the projected curve and in the curve fitted to past data;, but.,only.;.a;tv:the .-.\- date of the iafffc census. The corresponding rate"of growth in^the projection/, curves for dates' subsequent to the last census, whether the,.growth..is. positive-, or negative, is re'duced linearly, until all change has, ceasegL precisely vn;;j-\v.

years subsequent to the census. (Each part has its'individual. "n'J.,); The- -. -

length of the period "n" is an element of control in the hands of the fore-, .

caster, in the same way that the manner of fitting the curves to past-census—

data'is subject "to some play. . ■ . : ■ ... .

It.- happens that the projected. curves for each proportion are1 parts'Vf >

-the bell-shaped normal curve used in probability - not. by design but by ' '•-•'■"'■

coincidence. With. a. positive rate of growth, the date-of the last' census ' •"'i!-': ^ -would fall, at.some point.to the left-of;the highest point of the normal curve, -

while the point "n" years later, at the end of the period', of amortization,-''' falls precisely at that highest point. ..The tangent to the normal curve 'aii its highest point is of course horizontal* .. When, the^ rate of growth of :a: ' proportion is negative, so that the'proportion constantly grows less;, ...

precisely the same situation holds, except that here we- are dealing wi-thv.fthe -.-'ic.;

reciprocals of the ordinates of, the' normal curve. Since extensive tables of .values-are available for the normal curve,, the task of computing points on

the projected curves.for the proportions is eased. : '

Suppose now-that the required three constants, r(o), f and'n, have been' thus tentatively determined for each constituent,area in a.stage and that future proportions of a total are being determined and the time has come to add them together^ In general, the sum will not.be 100,tper cent, but it ... * is surprising how closely it will come to this goal, with a. difference. of ■ only -^ per cent or less. If this is the order of magnitude of the discrepancy, ' the1 determination of the constants can be considered final. The computed -■ . ■ proportions for" any date can then be divided,.by their actual total, to make '.-..--■.

them add-precisely to "lOO per cent. It is best to start-with one of-the-- -

more distant dates "of th'e projection; the discrepancies..tend ;tp grow larger.- .-.;

the longer the period'of projection. ' . . . .... ... ... . . ,..-.:;;-: -.,,..

x ■-■- --'•■'■■■'.:•

E/CN.l4/P0P/'96 Page 1? ' ,

If one has not been fortunate and the sum deviates by a sizable amount from 100 per cent, one must survey the situation, always remembering that- the next census will show percentages for these.same.areas adding precisely to 100 per cent. 'It seems-poor procedure to cover up a sizable discrepancy by a mechanical adjustment. One has two main controls: the determination of - . x.q and .the period .of. amortization n. There are many ways in which to determine

, rQ and-they will all yield different results. Judgement was needed to descry a trend'and more, judgement to. decide on. .its representation. One can.also

examine the choice of "n", particularly in the case of'large or of rapidly

growing proportions..- There, are. likely to.be..sows stages Mhioh resist every

effort at reconciliation and as a last resort it may be necessary to realign stages to..make the. situation more tractable. It is advisable to plot the projected-proportions ..on the. charts used in. the. analysis of trend; often a diagnosis of the.trouble can be made in this way. The first derivative, or annual change in each proportion, can easily be.-.evaluated- Comparison of these

magnitudes can also help. . '

i

When all adjustments have been.made, the proportions can be multiplied by the given projections of population for the entire area and this part cf

- the work is then complete. . '

It would be a waste of time to carry through, for every single year, the . procedures outlined. It: is. desirable,...however, to go-.through this procedure as of the end of each five-year period subsequent to the census. A nicety which causes, no additional effort is to provide the needed annual or semi annual interpolations, using the Karup^Kihg interpolation' formula^ This is

a "smooth junction" four-point' formula. (See Reference 40, para. 101, p. 137. ■ With modern equipment the TSrmula is more, serviceable in linear compound form -

acting on four ordinates with no. differences - than asWolfenden states it.

(41)). '■

This is not the. place in which to discuss detailed working procedures.

There are many methodologies and short-cuts which cannot be discussed. The

purpose here is to indicate enough to show practicability.

E/CN.14/POP/96 Page 18

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