City structure and interdependence

REGIONAL SCIENCE ASSOCIATION; PAPERS, X, ZURICH CONGRESS, 1962

CITY STRUCTURE AND INTERDEPENDENCE

by Benjamin Ward ~

1. INTRODUCTION

Every now and then in the history of science a surpr is ing regular i ty has emerged f rom empirical work, surprising in the sense that there is no saris- fory explanation for it in the exist ing body of theory. Some of these regulari t ies have been finally explained only a f te r a substantial revision of theory has been undertaken. For example the accumulat ing evidence that p lane ta ry orbits were elliptical ra ther than circular played an important role in the over- throw of Ptolemaic as t ronomy. I t is the possibilities such regulari t ies offer for expanding and deepening theory which explains their special a t t ract ion to in- vest igators . However , not all such regulari t ies make good on their bri l l iant prospects. One r emember s the sun spot theory of the business cycle, which began as an improbable correlation, was developed into a theory of c l imatic influence, and collapsed as a resul t of careful examinat ion of the data.

The so-called "city-size l aw" of regional economics is still, a f te r fo r ty years of discussion, in this class of intr iguing but as yet unrefu ted and unexplained phenomena. I t s ta tes that the relat ive sizes of a nat ion 's cities fall into a regular pa t te rn which can be described by an exponential equation. T h i s suggests that if one knew the present size pa t te rn and the ra te of growth of an urban population one would be able to predict accurately the population of the nat ion 's cities at some t ime in the future. This is surprising, even un- likely, given the large number of ra ther variable factors which we believe influence the growth of cities. Most especially, one wonders how this can be compatible wi th the various sorts of locational advantages and d i s advan tages which each of the world 's cities possesses. Are these mixtures , often t r ea t ed as unique to each location, characterizable in such a way that they can be in- corporated into so apparent ly rigid a f ramework? Can a nat ion 's cities rea l ly be so s imilar ly consti tuted that, whether the country be poor and crowded or emp ty and rich, pa t te rns of urban interdependence are the same? And w h a t of the role of institutions? Must we conclude that when it comes to influencing ci ty size these are mere epiphenomena, colorful but minor variat ions ort some grand determinis t ic theme? I t is I think because it suggests such questions as these that the city-size law has continued to arouse curiosity over a period of four decades.

Before taking a look at the present s ta te of the discussion, it may be worthwhile to raise one more of these broad questions for which ci ty size

* The author is associated with the Economic Research Center, Athens, Greece, and the Department of Economics, University of California (Berkeley), U.S.A.

208 REGIONAL SCIENCE ASSOCIATION: PAPERS X, EUROPEAN CONGRESS, 1962

patterns are relevant. At the present time two other empirical size distributions are under discussion in economics. Recent interest in the distribution of personal income has focussed, especially in the work of Champernowne ~, around the problem of tying the Pareto law of income distribution more spe- cifically to the economic theory of distribution. The general approach is to interpret the " law" of income distribution as the stationary state of a stochastic process, an approach which Champernowne pioneered. The second of these size distributions is that of firms, where the stochastic process approach is also being adopted 2. Is it possible that these two distributions and the city size distribution are all manifestations of the same causal mechanism? At least they have the same form; they all, assertedly, satisfy Pareto 's law ~. In ex- amining the evidence and constructing a conceptual framework for dealing with ci ty size distributions, this question may also obtrude.

2. SOME EMPIRICAL EVIDENCE

Probably the most extensive survey of international data on city size is tha t made by G. R. Allen in 1954'. He tested goodness of fit for 44 countries, using census data from the period around 1950. His test was to calculate the average relative error of estimate of the number of cities in each size class. He found that this average error exceeded 7 per cent for 17 of these countries and 10 per cent in nine of them. However, most of these censuses defined city size in terms of legal boundaries. Of the 21 countries for which conurbation was the basis for size determination, at least in the larger cities, the error exceeded 7 per cent in six cases and 10 per cent in only two cases. It was also true that for seven of the ten worst fits, the census definition of cities was in terms of legal boundaries. It seems then that while the fit is far from perfect, there is a strong international tendency for city size distributions to fit roughly the Pareto formula, and that the fit is improved by using conurbation as a definition.

Allen also found however that at the upper end the fit was often poor. Following Singer, he called the fit at the top good if the population predicted by the Pareto formula lay between the actual values for the two largest cities. However, by limiting himself to data based on conurbation and eliminating a number of countries for which he felt a good fit could not be expected (for

1 D.G. Champernowne, "A Model of Income Distribution," Economic Jou'~nal, vol. 63, June 1953, pp. 318-351. See also: B. Mandelbrot, "The Paretc.L~vy law and the Distri- bution of Income," International Economic Review, vol. 2, May 1960, pp. 79-106, esp. pp. 97- 99.

2 H. A. Simon and C. P. Bonini, "The Size Distribution of Business Firms," American Economic Review, vol. 48, Sept. 1958, pp. 600-15. Irma Adelman applied Markov chain analysis to the size distribution of United States steel firms in an article in: "A Stochastic Analysis of the Size Distribution of Firms," Journal of the American Statistical Associa- tion, vol. 53, Dec. 1958, pp. 893-904.

3 H. Simon in "On a Class of Skew Distribution Functions," ch. 9 in Models of Man, Wiley, New York, 1957, pp. 145-64, (originally published in: Biometrika, vol. 42, Dec. 1955), suggests that the same probability scheme defines the process of generation of a- number of similar distributions. What we are saying is something different: that the same ~=ausal factor is at work on some of these distributions.

4 G.R. Allen, "The 'Courbe des Populations'; a further analysis," Bulletin of the Oxford lnstitute of Statistics, vol. 16, May-June 1954, pp. 179-89.

WARD: CITY STRUCTURE AND INTERDEPENDENCE 209

example, because they had had a short history under present boundaries, or were made up pr imar i ly of islands, or possessed a nonhomogeneous population) he found that eleven of the twelve remaining countries satisfied this upper tail fit criterion.

At the lower end Allen found good fits down to cities of about 2000 population. The point at which the fit began to deteriorate varied somewhat . Allen in te rpre ted this as resul t ing f rom the vary ing role and significance of vil lages f rom one country to another. In countries where nucleated se t t lements of f a rmers are re la t ively rare, as in the United States, the relat ive f requency of the smaller ci ty units mus t be far less than in a country where the vil lage form of se t t lement is typical, as in Greece.

Now one may raise some objections to Allen 's tes t procedure. I t is weak in the sense tha t it measures accuracy in predict ing the number of cities in each class in terval (chosen pr imar i ly by reference to the U.N. classification of cities) ra ther than the sizes of individual cities in a ranking by size. Even an accura te fit by this criterion, though it would be intriguing, would be of lit t le use as such for a planner s. Also in tes t ing the fit for the largest cities Allen 's tes t leaves out some quite good fits; especially those in which the largest ci ty is only sl ightly below the Pareto line; and one is a little suspicious of the large number of ad hoc exclusions. Even so it seems evident that wi th respec t to the data used by Allen much is explained and much is unexplained by apply- ing the Pareto formula.

Finally, A l l en es t imated the slopes of the l inear-logari thmic lines of the Pare to formula. He found some var ia t ion in the slope, but tha t most countries lay in the range 0.9 to 1.1, and that in many countries the slope appeared to be quite s table be tween censuses. For a few nineteenth century censuses, aside f rom the United States, coefficients well above uni ty were observed, but wi th a tendency to decline toward uni ty as the century progressed.

Hoyt has also looked at data relevant to the Pareto law, wi th part icular reference to the effects of redefinition of larger cities to cover metropol i tan areas 6. His conclusions are negative. However he is considering only a special case of the Pareto formula, one used by Zipf in 1949, which fits a rectangular hyperbola to ci ty populations ranked by size. This is the special case of the formula for which the exponent (which is also the slope of the linear loga- r i thmic line) has the value --1, a value which was observed often by Allen, but by no means universally. Hoyt also plotted the hyperbolas through the -value for the largest city, so that if this city lay far off the line of " b e s t " l inear fit for the whole set of data, the deviations f rom Hoyt ' s line seemed -very large. Hoyt also considers the fact that reversals of rank order occur fair ly often between censuses to be a refuta t ion of the validity of the law; that is, he is tes t ing a determinis t ic ra ther than a statist ical hypothesis.

Tab le I shows the nature of the fit for the twenty-five largest metropoli tan areas in each of two countries, France and the United Kingdom, for which relat ively poor resul ts were obtained using Hoyt ' s data and Allen 's tes t of fit.

5 One would like to see a test of performance of the Pareto equation against alternative distributions such as the lognormal. Unfortunately this has not yet been attempted.

6 H. Hoyt, "World urbanization," Technical Bulletin no. 43 of the Urban Land Institute, Washington, D.C., 1962.


TABLE I: Fit of City Size Data to Pareto Formula for 25 Largest Metropolitan Areas

France 1954 United Kingdom 1956

1. Average error 11% 10% 2. Average error (Paris, London, Manchester excluded) 8% 7% 3. Largest error (Paris excluded) 30% 40% 4. Rho (slope) -0 .9 -1 .0

Source: Calculated from data in H. Hoyt, "World urbanization," Technical Bulletin no. 43 of Urban Land Institute, Washington, D.C., 1962, pp. 39, 42.

In both cases the errors of estimate for the largest city or two are very high, being at least 30 per cent in each case. Paris in paticular was very far from the Pareto line. However, by omitting Paris and changing the exponent somewhat more satisfactory results were obtained; in fact, for the largest cities these fits are not bad, comparatively speaking. For two countries of the nine for which data was processed by Hoyt (India and the United S ta t e s ) the fit using his procedure was quite good. In the remaining five countries, the situation was roughly comparable to that presented in the table, though in the case of the Soviet Union a linear fit of rank to population would work very well if the two largest cities were excluded.

Hoyt ' s data refers to the upper portion of the distribution. It was here that the largest errors occurred in the older data, some of which dealt with legal boundaries, some with conurbations. These errors are not reduced by taking the latest data in which cities are defined in terms of metropolitan areas. And again the slope of the linear-logarithmic line at times deviates from unity. But it is in the middle range of city sizes that the best fit occurs, and Hoyt is not dealing with this section of the distribution. Also he does not use enough cities to get a good estimate of the slope or to test the overall fit.

3. SIMON'S MODEL

It is only during the postwar period that serious attempts have been made to explain these distributions. By explaining I mean the derivation of the distribution from reasonable assumptions, whether these be taken from some theory of human behavior or merely satisfy some subjective criterion of reasonableness. Clearly, standing by itself, the Pareto formula does neither of these things. Until the results are rationalized in this way one is bound to be a bit sceptical as to the prospects for putt ing the distribution to work; ill other words, developing a theory from which the distribution can be derived is in itself a form of hypothesis testing.

The most interesting work along this lille is that of Simon 7. Simon argues

"On a Class of Skew Distribution Functions," op. cir. George F. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, Reading, Mass., 1949, has also theorized on this subject. However his theory has not, so far as I can determine, been worked out to the point at which one can derive the empirical results as a consequence of the theory. There is a connection between this class of distributions and maximization however; see B. Mandelbrot, "Paretian distributions and income maximization," Quarterly Journat of Economies, vol. 76, Feb. 1962, pp. 57-85.


that there are a number of behavioral phenomena, such as income distribution, word frequencies in prose works, the number of contributions of authors to scientific journals and the size of cities, which can all be explained in the same way. He is not arguing of course that all these distributions are caused by the same factors; ra ther he is saying that, wha tever the causal factors, the pro- cesses by which they operate to create effects have many of the same properties. These common propert ies may be described as an urn scheme, a pro- babilistic method of selecting some events from among a set of possible events.

Simon's s t a tement of the assumptions needed and his defence of them are wor th quoting in full:

One assumption is satisfied " i f the growth of population were due solely to the net excess of bir ths over deaths, and if this net growth were proportional to present population. This assumption is certainly satisfied at least roughly. Moreover, it need not hold for each city, but only for the aggregate of cities in each population band. Finally, the equation would still be satisfied if there were net migrat ion to or f rom cities of part icular regions, provided the net addition or loss of population of individual cities within any region was proportional to ci ty size. T h a t is, even if all California cities were growing, and all New England cities declining, the equation would hold provided the percentage growth or decline in each area were uncorrelated with ci ty size . . . . The constant a would then be in terpre ted as the fraction of the total population growth in cities above the min imum size that is accounted for by the new cities that reach that size. ''8

The first assumption mentioned above is as follows: the probabil i ty that the next additional m e m b e r of the urban population will come to inhabit a ci ty of population i is proportional to the total population of cities of this size. If there were no migrat ion so that all cities g rew only f rom the difference between bir ths and deaths among their inhabitants, this assumption would hold, provided bir th and death ra tes were the same in all cities. Migrat ion would also be possible provided the net effect were to keep the probabil i ty of an increase proportional to if, where f is the number of cities of population i. In discus- sing regional variat ions in growth rates, Simon is re in terpre t ing the model so tha t i represents the mean population of a group of cities in a class interval. I t is consistent wi th the model that the probabil i ty of increase of each of these cities varies widely, so long as the group of cities satisfies the assumption. The constant, a, depends on the exponent of the city-size law and will be zero when that exponent is - 1 , corresponding in the model to no fur ther growth of the urban population by means of the creation of new cities.

What is the relation between these assumptions and the city-size law, which may be expressed as:

(1) N=~io where N is the number of cities having a population of at least i and fl and p are constants with p < --17 The answer is that, for sufficiently large values of i equation (1) describes an asymptot ic s teady s ta te distribution generated by random variables having the propert ies specified in the assumptions. In other words if the assumptions hold, the formula describes accurately the distribution

Simon, loc. cir.


of values of the random variables, the city sizes in our case. Of course, being a distribution we never expect the actual values of the var iables to be exact- ly the values predicted f rom the formula. In fact, for any g iven amount of error in est imation we can calculate a posit ive probabil i ty of an es t imat ing error of this magni tude occurring. But if actual convergence to the s teady s tate is rapid enough and this mechanism dominates the selection of values for ci ty sizes, the formula may give quite good results.

This seems to me a fascinating explanation of the city-size law; however, it is unsat isfactory for several reasons. In the first place there is the problem of fit in the upper tail; this is bad enough in most cases to be useless for pre- dictive purposes. It might be added that this is a problem also in the appli- cation of the model both to income distributions and to scientific journal contributions. The contrast be tween the ve ry good fits in the middle range and the bad ones at the upper end suggests that there is some sys temat ic error in the theory.

Another problem has to do with the l imited scope of the theory. The Pareto formula describes the steady s ta te distribution only for values of p less than - 1 . Several examples have already been mentioned however of distributions whose l inear-logarithmic slope is grea ter than --1. In part icular it appears that data in t e rms of metropoli tan areas are of somewhat smaller slope than was the older data. Of course this may only apply in the area in which the fit is bad anyway; but there are a number of countries where slopes less than one have persisted throughout the length of the l inear-logarithmic distr ibution. Another stochastic process may describe this situation; the Simon model in its present form cannot.

Also the f ramework is ra ther rigid wi th respect to declines in city-size. A decline can only occur if compensated by growth of another ci ty in the same class interval (more precisely, this s ta tement applies to the probal-ilities of growth or decline). The re is no a priori reason to suppose that this is t rue. It is s imply a device to make decline possible in a s imple stochastic growth process.

Finally there is the question of the reasonableness of the basic assumptions. The use of a min imum cutoff point seems appropriate , since there is certainly some, if perhaps vaguely defined, min imum population below which a c i t y . . , wha tever the d e f i n i t i o n . . , ceases to be a city. The assumption of the " law of proport ionate effect," as it is somet imes called, is a bit harder to swallow, at least in its present form. The problem is that much city growth is not accounted for by natural growth of the population, but instead by migration, ei ther f rom rural areas, or f rom other cities, or f rom abroad. To explain migrat ion in these t e rms requires something else than the bio-economics of human reproduction.

These difficulties lead us to seek another explanation for city-size distributions. T h e two general propositions on which we will t r y to build are: 1) that there is enough regular i ty in the city-size distributions to make some explanation seem feasible; and 2) that Simon has led the way as far as the determination of a s t ra tegy of approach to the problem is concerned.


4. A CONCEPTUAL FRAMEWORK It seems to be well established now that the main reason for migrat ion is

employment opportunity. Recent studies, such as those of Hoover and Thomas 9, have indicated that this applies both to short and to long distance movements . Except ions can always be found- -mos t obviously of course in the case of cata- clysms such as war, but also for certain groups, such as older people choosing a place of r e t i r emen t - -bu t the rule seems to apply with sufficient s t rength so that a t tent ion should be paid to it, at least as a first approximation.

One might well say then that all changes in ci ty sizes are the result of migrat ion, since a person who grows up in a ci ty and remains there on reach- ing working age may be thought of as having chosen the city which offered him the best opportunity, of having migra ted a distance of zero. This does not mean that everyone i s continually searching for be t te r opportunit ies away from home. Surely the vast major i ty of the act ive population are sufficiently content with their exist ing circumstances that they are not engaged at all in a quest for a be t te r position. And many of those who are looking are not looking beyond their immediate location. Employment opportunit ies which will a t t rac t migrants are not, re la t ively speaking, large at any one point in t ime. Over the long run it is sufficient that there be some information available as to the nature and location of opportunities, and that a marginal pool of potential migran ts examine this information.

If employment opportunit ies are the dominant cause of rural-urban or inter- urban migrat ion, it seems reasonable to look to the nature of opportuni t ies in t ry ing to explain pa t te rns of ci ty size. In doing so it is useful to generalized somewhat the concept of an opportunity. An opportuni ty is an event which consists in the possibil i ty of a long run expansion in the level of an economic act ivi ty at some location. Without t ry ing to be very precise, an oppor tuni ty m a y be said to exist if, g iven technology and reasonable es t imates of demand, it is profitable, to car ry out the expansion of ac t iv i ty wha tever the cri terion used by ent repreneurs . If the s ta te is the en t repreneur some social welfare function may be the basis of the profit calculation; if en t repreneurs are pr iva te citizens the basis is marke t calculation 1~ A negat ive opportuni ty , calling for permanent reduction in economic act ivi ty, is also possible.

An opportunity, when it is realized, manifes ts itself in several ways. I t typical ly requires capital and genera tes employment during the period of con- s t ruct ion of permanent facilities. I t genera tes employment and income (or out- put) over the long run a f te r it is completed. From the point of view of long run changes in ci ty size, employment in the construct ion sector may be important; cer tainly the s teady employment generated by the continuing operat ion of an ac t iv i ty is important .

What do we know about the occurrence of opportunities? In general termsj . it can be said that opportunit ies may occur ei ther because of an expansion of the marke t or because of technological change. One might think of a country as containing a given stock of ci ty sites. Not all of them contain cities at any

o E. Hoover, "People and Jobs," Supplement I to: Robert M. Lichtenberg, One Tenth o f a Nation, Harvard University Press, Cambridge, Mass. 1960, pp. 201-26; and B. Thomas, Migration and Economic Growth, Cambridge Press, Cambridge, 1954.

10 External effects are discussed below.


one point in t ime (with changing technology the number of ci ty si tes may also be changing) but all of them have the proper ty that there is some positive probabil i ty of urban- type opportunities occurring there. Expansion of the marke t m a y occur in a way which permits realization only within a particular city; thus growth of the retai l grocery marke t in a town will rare ly be satisfied by put t ing up a grocery store in a different town. On the other hand, some marke t expansions are national or regional in scope, embracing a number of cities. The act ivi ty which will sat isfy this demand may often locate in only one of them because of indivisibilities in the activity. Some cities are more l ikely to receive the ac t iv i ty than others, while outside the area there is no chance to realize the opportunity.

In the case of technological change too there will be a set of cities which m a y potential ly contain the new activi ty. If the ac t iv i ty is s t rongly resource- based, the act ivi ty will have to sett le on one of the ci ty si tes near the resource; in other cases demand considerations may dominate. Again, some cities for economic and resource reasons will be more likely recipients than others.

Some activit ies may be interdependent. The location of one process in a part icular city may increase the chances that another activity, linked either on the supply side or the demand side to the given one, will also locate in that city. Or the use of specialized services may make it ve ry likely that an increase in the level of an act ivi ty will occur in a city where the act ivi ty is already carried out. A technological innovation which leads to raising the level of o n e act ivi ty may lead to a lowering in the level of subst i tute activit ies and rais ing of the level of complementary activities.

An opportuni ty has been defined as a discrete event occurring at some one part icular location. This does not mean however that each new opportuni ty when realized entails the creation of a new economic unit. An already exist- ing firm may set up a new plant or mere ly expand the capaci ty (or the long run level of operation) of an exist ing plant. Both would const i tute opportuni- t ies under this definition.

Of course an opportuni ty may occur but not be realized. One possible reason for this is that it is not observed. Entrepreneurs , whether individuals, large corporations or even s tates may not search with respect to this part icular act ivi ty. The process may be so new that regularized search has not been in- itiated, or that information has not crossed over the l ine between, say, scien- t ist and promoter. Or ent repreneurs may for some reason be somewhat un- enterpris ing, for example in a s tagnant society.

Even though observed, an opportuni ty may not be taken up. I t has been suggested that innovations occur in bunches, both with regard to long swings of economic act ivi ty and in the shorter movements of a period of several years. If this is t rue an opportuni ty may occur in the sense that a long run increase in act ivi ty is profitable but not taken up because in the short run capital is too dear. In Schumpeterean t e rms the growing understanding of the opportuni ty by ent repreneurs is another important factor influencing the probabilities. Such an opportuni ty will eventual ly be taken up, but not as soon as it occurs. If entrepreneurial at t i tudes toward risks vary, ei ther in t ime or space, yet another factor influencing the picking up of opportunit ies is isolated. Some societies, the high-saving ones, devote more of their resources to realizing


opportunit ies than do others. A pers is tent ly low-saving society may find over the long run that opportunit ies which occurred and were not taken up disap- pear. The overall inves tment rat io of a country makes its impact within this f r amework on the probabil i t ies of realizing opportunities, once they have occurred, and inves tment insti tutions may of course have a differential effect on the various kinds of opportunities.

Internat ional variat ions in insti tutions also affect the realization probabilities. Markets which are badly organized, or not capable of picking up external economies, or en t repreneurs who do not cooperate in their realization provide examples of the possibilities.

Obviously opportunit ies are of different kinds. In the absence of technical change, an expansion of say the text i le marke t by a given amount should lead to generat ion of roughly the same amount of additional employment regard- less of the number of opportunit ies that have already occurred. But the same marke t expansion in the steel industry or in retai l t rade will have quite different effects. When fundamental technical change occurs it may change the employment effects in one or a number of activit ies quite substantial ly.

A complete theory of the dis tr ibut ive effects of opportuni ty-generat ion would be a complicated one. But speaking broadly the main factors can easily be isolated: for example by following through the effects of a technological change. We s ta r t with a given size distr ibution of cities, both in t e rms of act ive population and of marke t size (which may embrace a number of cities, depending on the activity). A technical change creates opportunit ies in several types of activit ies. For the resource-based industries these are realized in the most likely among the feasible cities; for other activit ies the new opportunit ies may be more scat tered. When realized they genera te employment , a higher level of economic ac t iv i ty and grea ter marke t demand. Depending on the ac- t iv i ty the increase in demand which resul ts may be national or regional in scope or may apply wholly within a single city. The expanded marke t genera tes new opportunities, both in the originating act ivi t ies and in others. In each case the realization of an opportuni ty induces migrat ion and expands the market , thus paving the way for new opportunit ies and new migrations. In this way the dis tr ibut ive mechanism responds to the exogenous changes corresponding to long run economic growth (and decline) and technological change.

We can now res ta te the Simon model within this conceptual f ramework. Though there are several possible ways to do so, the following seems to be a reasonable interpretat ion: opportunit ies may, at least for some purposes, be aggregated. For a given period we may think in te rms of the probabi l i ty of some given aggregat ive opportuni ty occurring, the opportuni ty producing general employment ra ther than employment in specific activities. No distinction need be made be tween technological and market-expansion opportunities, pos - sibly because the former are dominated in the long run by the latter. Tha t is, a f te r some initial expansion a technological opportuni ty leads to fur ther employment generat ion only because the marke t is expanding, so that over the long run the initial impetus may be ignored. The size of the population of a ci ty is a measure of its m a rke t size so that market-expansion opportunit ies de- pend for their occurrence on the number of inhabitants a ci ty possesses. The opportunit ies characterized in this way then occur with a relat ive f requency


(probability) which is proportionate to the size of the market. There is some minimum city size below which the probability of occurrence of opportunities is much smaller, partly because there is less effective search by both entrepreneurs and prospective migrants, and partly because the basic natural and technological material for opportunity creation does not exist in any quantity. These assumptions are sufficient to generate the Paretian city size distribution 11.

Of the special assumptions needed to generate the city size distribution on th i s interpretation, two stand out as being especially restrictive: 1) the aggre- gation of opportunities, which leaves no room for taking into account variations in technological developments or in demand-mix among cities and over time; and 2) the omission of the unique properties of some cities which give them special advantages over their rivals and, presumably, should be reflected in relative growth patterns. The conceptual scheme outlined above is broad enough to be capable of taking these factors into account and we will turn now to a brief discussion of some of the ways in which a more powerful model may be developed.

5. SOME FURTHER COMMENTS

Disaggregation of Opportunities. How do sectoral employment distributions compare with city-size distributions? The following comments are based on recent Yugoslav and Greek census data 1~. In general, Yugoslavia appears to fit the city-size law for population fairly well, though there appears to be a small steepening of the slope of the linear-logarithmic line for cities of size less than about 15,000. For Greece the linear-logarithmic fit is not so good and applies only to cities in the range 5,000-100,0001~. The two largest cities lie well above the Pareto line, a situation which also occurs in a number of other relatively small countries possessing one or more "p r imate" cities. Though the data available for these two countries is somewhat limited they are at least representative, in their city-size distributions, of two internationally rather common situations.

It is not surprising to find that the distribution of the active population has roughly the same shape as that for the population as a whole, including the bend in the Yugoslav case and the primate cities in the Greek case. Also there is a very high rank correlation among cities ranked by population size and active population respectively. There is however no noticeable improvement in the fit. From the point of view of the present framework active popu-

1~ In either Simon's form or as a strict growth process, depending on the choice of class intervals.

15 Unfortunately from the point of view of the theory under discussion the choice of countries was purely accidental, a matter of access to the data. For Yugoslavia sectoral employment data was available for the year 1951 for the 315 cities having a total population exceeding 2000. For Greece population data was available for five censuses between 1920 and 1961 for cities of population above 10,000 (in 1951) with extensions to 5,000 for 1951 and 1961. Sectoral employment data were available only for one recent year (either 1951 or 1958) and generally covered only the eight largest cities. For further description and analysis of the Greek data see Benjamin Ward, Problems of Greek Regional Develop- merit, Center of Economic Research, Athens, 1962, oh. 4 (charts 4.1 to 4.9 depict some of the relationships discussed in sections of the present paper).

18 See below, p. 218, for an alternative statement.


lation is to be preferred as a measure of ci ty size; the fact that it does not fit be t te r than population as a whole at least indicates that the use of population was not shielding a be t te r fit for aggregat ive data.

A breakdown of the Yugoslav data into employment distributions by five sectors for 315 cities gives the following results: administrat ion, t rade and agricul ture follow the linear logari thmic line very closely, though wi th a sharp tailing off f rom the line at employment levels respect ively of 250, 200 and 300, The re are no persis tent bends or sharp breaks in the plot though the fit is, as wi th population, much be t te r in the middle and lower range than at the top. Average re la t ive errors of es t imate before the tailoff are in the range of 6 to 8 per cent for these three distributions, and if the top twen ty cities are omitted, the fit can be improved by one or two percentage points. By this measure the fit is a clear improvement over both population and act ive population where average errors are above 10 per cent 14.

Greek sectoral data were only available for the eight most important cities (except for data on public servants) . All one can conclude is that trade, both wholesale and retail, and public servants are dis t r ibuted in roughly the same way as population, with the two largest cities lying well above the line and the remaining six making a good linear fit. Public servants tail off at about 400 and the average relat ive error, omit t ing Athens and Salonica, is around 9 per cent. Th is too is an improvemen t on the population fit.

Construct ion also appeared to be linear in the case of Yugoslavia, but wi th a ra ther large error of es t imate . I t should be noted tha t the rank correlation of cities ranked for each of the sectoral employment categeries, together with the same cities ranked by size of act ive population, var ies widely, wi th both adminis t ra t ion and t rade being fairly closely correlated with active population (but not so closely as was population), industry somewhat less so, and agricul ture having a much smaller correlation.

Yugoslav industry does not fit the Pareto line closely. A fit giving about the same average relat ive error as those for t rade and adminis t ra t ion would require three line segments , with the slope s teeper as one moves toward the smaller cities. T h e reason for this may be that the Pareto distribution does not apply to industry, or that the special c i rcumstances of industrial h is tory in Yugoslavia dominate the "un ive r sa l " forces. Another possibili ty is that many industries do fit the Pareto distribution but that industry as a whole is too heterogeneous to be aggregated. The re is wide variat ion f rom one industry to another in the min imum efficient plant size. The probabil i ty of an opportuni ty occurring to produce at levels below the min imum size will be much reduced as a result, though some such opportunit ies may occur when it is s imply a ma t t e r of expansion of exist ing plant. The consequence of this var ia t ion in the cutoff points among branches of industry, assuming the Pareto formula applied above that point, would be to produce a distribution like that actually found.

It was possible to get employment distr ibutions for Greek industry by two- digit industrial classification, but only for the eight largest cities. Th is does

14 These relative errors shou!d be distinguished from Alien's. They refer to the average percentage deviation of each city from the Pareto line.


not permit a resolution of the question as to whether disaggregation will re- veal a series of Pareto industrial distributions. Of the nineteen industries tested, twelve either followed a linear fit or were linear except for the two largest cities (in every case Athens and Salonica) which lay substantially above the line. The remaining six did not have sufficient employment to cause any noticeable variation in the aggregative manufacturing distribution. Possibly the first tailoff point had not been reached after only eight cities.

In summary, it appears that the Pareto distribution applies well to sectoral breakdowns, possibly better than to more aggregative distributions. The industrial distribution probably is the cause of the slight bend in the Yugoslav population distribution, though its influence on the distribution of administra- tive, commercial and agricultural opportunities appears insignificant. Where a primate city or cities exist, the sectoral distributions tend to follow the primacy pattern.

As for the slopes of these lines, Yugoslav administration, trade and con- struction show - 1 while agriculture is approximately --0.9 and industry varies from - 0 . 9 to - 4 . A linear fit to population and active population gives a slope close t o - -1 . The interpretation of unit slope in Simon's model is that no new employment is occurring in cities not previously represented but that all growth is occurring in "old" cities, that is, in cities which in the previous period also lay above the tailoff point. Tha t industry should not fit this picture for postwar Yugoslavia is not surprising: there have been a large number of new industrial cities created. As noted before, there is no interpretation of a slope greater than --1 in Simon's model.

For the Greek data to be made consistent with this interpretation of the slope of the Pareto line, the fit must be made between Athens and Salonica at one end and cities of population below 25,000 at the other. A linear fit to these cities gives a slope which yields a rate of growth of new city populations consistent with actual results over the last three intercensal periods (since 1928 this slope has remained constant). The interpretation is that Athens and Salonica are absorbing, in their capacity as dominant cities, population which in other countries would "normal ly" migrate to the cities in the middle range. Because of their special position as both a central city and a resource-based manufactur ing center importing raw materials, one notes a higher probability of occurrence of opportunities in Athens, and, to a lesser extent, in Salonica, and a lower probability of occurrence of opportunities in middle range cities. The small number of cities for which sectoral data was available precluded a determination as to whether all sectors participated in this central absorption ~s.

When a sectoral distribution has the same slope as that of active population and when there is a high rank correlation between the respective lists of cities it would seem reasonable to interpret this as meaning that it is the size of the c i ty 's active population which is determining the probabilities of the

15 Another possible interpretation stems from the fact that one can argue that Athens and Salonica are not above the Pareto line but that middle range cities are beIow it. Primacy does not increase the probabilities of occurrence of opportunities in the primate cities but simply reduces the probabilities for middle-sized cities. Should international stability in the parameters of sectoral distributions be demonstrable, this hypothesis might be tested.


sectoral distribution. 16 In Yugoslavia this would be t rue for adminis t ra t ion and trade, that is, for service activit ies where the marke t for the ac t iv i ty is wholly local. (This is also t rue of construct ion but it does not have as high a rank correlation, nor does it show as good a fit to the Pareto line.) It is obviously not the case in industry and agriculture, though it may be for branches of the former. Agricul ture in part icular has a ve ry low rank correlation with act ive population, suggest ing that some ent irely different factor is at work. Also the mechanism generat ing the s ta t ionary s ta te is probably different because of the slope. In industry especially, deviations f rom the slope of the act ive population distribution suggest that either the marke t area is not coextensive with the cities in which p l an t s are located or that technological considerations ra ther than changes in marke t size are de termining the growth of the industry. This la t ter si tuation is likely to last for no more than a few years for any given fundamental innovation, as for example during the rise of a new industry. It could easily persis t over one or two intercensal periods, however, and in the case of new industries, dominate the censal employment distribution for that branch.

City Typologies. Several classifications of cities have been made based on the relat ive levels of various economic act ivi t ies 17. These classifications have li t t le or no relation to exist ing economic theory and it is not at present wholly clear to what use they m a y be put, though it would obviously be of the grea tes t interest to have a theory as to how significant urban differences come about. At any rate, a disaggregated city-size model wilt generate these various " t y p e s " of cities in a predictable way. For example, if the sectoral distr ibutions were mutual ly independent the relat ive f requency with which a given type of ci ty (that is, a ci ty with a specified amount of employment in each sector) will occur is s imply the product of the probabil i t ies for each individual distribution. Classes of cities and their expected f requency of occurrence could then be est imated, possibly wi th considerable accuracy.

From the point of view of our conceptual f ramework another distinction could be made: tha t between cities in which the unique propert ies of the ci ty significantly affect its s t ruc ture and cities for which this is not the case. Such special-case cities cannot a lways be isolated s imply by looking at the distributions. However in cases where substantial deviations f rom expected joint distr ibutions of sectoral employment occur, or in cases where the expected number of rank inversions in a given size class do not occur, it is l ikely tha t these special factors are at work. I t may also be that the deviations f rom the Pare to line at the ex t reme upper end of the distribution may be explained ill this way. Tha t is, a large fract ion of the largest cities in a country acquire their size because of special features of thei r environment , thereby inducing deviations f rom the standard opportunity-occurrence distribution. On the other hand, it appears that there is a large class of cities in the middle and smaller

10 It is not clear whether total or active population is more satisfactory as a measure of market size. Probably neither is satisfactory, though the latter is frequently an increasing function of the former.

17 For example, C. Harris, "A Functional Classification of Cities in the United States," Geographical Review, vol. 33, 1943, pp. 86-99; and P. Gillen, The Distribution of Occupa- tions as a City Yardstick, Kings Crown Press, New York, 1951.


range which are sufficiently similar in their re levant propert ies that a re la t ively simple opportuni ty distribution dominates in determining their size. At the momen t this is merely speculation.

The relation between s t ruc ture and interdependence among cities is perhaps worth a comment at this point. Even though the sectoral distr ibutions are as- sumed to be generated by mutual ly independent random variables, this does not mean that cities grow wholly independently of one another. Cities are intercon- nected in the model by the relat ive probabili t ies of occurrence of opportunit ies (it will f requently be t rue that an increase in these probabili t ies for one city is at the expense of other cities), and by marke t l inkages in the case of ac- t ivi t ies whose marke t area comprises more than one city.

On the Definition of "City" and "Opportunity". The city has presented a fairly serious problem of definition, especially when it comes to deciding who is in it and who is not. Within the present context a clear and reasonably s imply criterion can be used for deciding what is and what is not par t of a conurbated area. A city is a collection of interdependent places of work and residences. In determining a c i ty ' s boundaries one could s tar t wi th the ac- t ivi t ies located, say, in the densest section of the ci ty and add to this step by step other areas that are interdependent with the initial and gradually expanding a rea- - in te rdependent in the sense that ei ther the work place is in the central area and the residence in the area under consideration, or vice versa. Some reasonable cutoff point for minimal interdependence could provide a s tandard basis for the definition of conurbation. Such a cri terion ties the ci ty to the search for opportunit ies and also roughly fixes the size of the interdependent marke t area. This definition seems intui t ively reasonable, is not far removed from current (ideal) conceptions, and is related to a theoretical framework.

I t may seem that ty ing city-size distr ibutions to an abs t rac t conception of an economic (or social) opportuni ty is a ra ther indirect way of get t ing at the problem. Opportunit ies as defined here would be difficult to observe and especially to measure. The i r evanescent quali ty smacks of a re turn to neoclas- sical abstract ion ra ther than a serious interest in the real world. However , I think such a cri t icism would be misplaced. In the first place most of the more important economic concepts are of this nature. National income seems to be a "ha rd" practical concept only because we have general ly accepted stat is t ical procedures for es t imat ing it. The closer one looks at the conceptual basis the more it seems divorced f rom observable reali ty. Opportunit ies seem vague and unworldly pr imari ly because there is not a generation of stat ist ical work behind their est imation as there is wi th income concepts. Secondly, every oppor tuni ty which is realized must have been observed by someone, though a dif- ficulty remains in the likelihood of making mistakes, of perceiving opportuni- t ies where none exist. Planning bureaus in many countries are explici t ly concerned with the discovery of opportunities, though they have not so far tr ied to systematize their investigations in a way which would be useful for the problem at hand. It is not inconceivable that some sort of es t imate of the frequency and place of occurrence of opportunities could be made using data of this kind.

However, it is not really necessary to do this. The function of opportuni-


t ies within the class of models present ly envisioned is as an in tervening var i - able. The manKestat ions of opportunities, the emtJloyment generated, the effect on marke t size, etc., c a n be observed direct ly and this data used to calculate size distributions. The direct observat ion of opportunit ies can be avoided provided certain relat ions between opportunit ies and their manifestat ions remain stable.

6. CONCLUSION

No a t t empt has been made here to offer a specific model of the phenomena under consideration. It has seemed desirable first to survey the available international data, par t icular ly that showing intercensal transit ions, in order to see what sort of s tabi l i ty over t ime the various distr ibutions may exhibit and to consider some fur ther disaggregation. At the moment the most pro- mising models would seem to be some forms of nonhomogeneous first order Markov chains in which transit ion probabili t ies are themselves functions of of the previous state; S imon 's model is an example. Act ive populations would be generated as the sums of the sectoral random variables. Whether cases of urban p r imacy can be explained within this f ramework or whether they will require a more generalized t ime-dependence assumpt ion remains to be seen.

Finally, a comment on possible extensions of the approach. Opportunit ies lie at the heart of all economics concerned with long run change. The present conceptual f ramework provides a general basis for discussion of the distr ibutive implications (in the economic sense) of long run change. The size distribution of income in the upper range seems obviously related to the nature and frequency of occurrence of opportunit ies in capitalist societies. I t may be tha t e i ther a direct or indirect es t imate of the propert ies of opportunit ies will yield a basis for es t imat ing the resul t ing income distribution. The same may be t rue of the size distribution of firms. In other words, the analysis of opportunit ies m a y open the door to a unified explanation of a considerable body of long run dis t r ibut ive phenomena as well as having some conditional predic t ive power.

City structure and interdependence

Documents

Transcript of City structure and interdependence