http://cps.sagepub.com/ Comparative Political Studies http://cps.sagepub.com/content/1/1/103.citation The online version of this article can be found at: DOI: 10.1177/001041406800100104 1968 1: 103 Comparative Political Studies Bruce M. Russett System? Is There a Long-Run Trend Toward Concentration in the International Published by: http://www.sagepublications.com can be found at: Comparative Political Studies Additional services and information for http://cps.sagepub.com/cgi/alerts Email Alerts: http://cps.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: What is This? - Apr 1, 1968 Version of Record >> at University of Sydney on March 4, 2013 cps.sagepub.com Downloaded from
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http://cps.sagepub.com/Comparative Political Studies
http://cps.sagepub.com/content/1/1/103.citationThe online version of this article can be found at:
DOI: 10.1177/001041406800100104
1968 1: 103Comparative Political StudiesBruce M. Russett
System?Is There a Long-Run Trend Toward Concentration in the International
Published by:
http://www.sagepublications.com
can be found at:Comparative Political StudiesAdditional services and information for
BRUCE M. RUSSETT is Associate Professor of ~’ot~~;ealScience at Yak University, and Director of the YalePolitical Data Program. He is the author of (amongother books) the World Handbook of Political and SocialIndicators, Trends in World Politics, and InternationalRegions and the International System, as well as nu-merous articles and monographs in comparative and in-t~rr~ia~at politicsand political behavior.
ANY THEORIES OF international politics draw major propositionsfrom the structure of the international system; that is, from thenumber and relative size of the nations that go to make up the system,the degree to which population or other power bases are concentrated ina few states.
NATIONAL SIZE IN HISTORICAL SYSTEMS
We have theories that a bipolar system is or is not more ‘ ~tab~e~than a balance of power system, or, extending the argument, that aworld with very small nations would or would not be less prone to severewars than would a multipolar world with only a few major powers.&dquo;These theories are not at the moment entirely satisfactory for makingreliable statements about the contemporary world. Either they rest uponunrealistic assumptions, or they require many qualfications. Yet some
preliminary efforts to test alternative hypotheses do suggest that over thepast century and a half the structure of the system has been a moder-ately good predictor of the amount and nature of violent conflict in the
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system.* These theories have a long and for the most part honored placein the history of international relations theory~ and contain at least theseeds of important explanations. By analogy, similar theories have sub-stantial explanatory power in economics for the behavior of firms-in anumber of significant ways one can speedy &dquo;the difference it makes&dquo;whether a market is basically duopolis~c (only two firm, or at least twoBmis very much bigger than any others), ohgopohstic (several largeInns), or approximating perfect competition (many small firms, noneof them large enough in isolation to affect the price or quantity of goodssold).
If it is conceded that there is something relevant, for policy as wellas for theory, in the structure of the international system it behooves usto devote some effort to description-to get some sense of the cwrewdistribution of nations by size-and to test some hypotheses about whatthat distribution has in the past, so as to be able to offer- somefurther hypotheses, however speculative, about what that distributionmay be becoming. In this article we shall look at the evidence availableas to trends in the distribution of nations by population size. The datainclude information on the &dquo;birth&dquo; and *’death&dquo; of nations of various
sizes, population growth rates, and static data for the distribution ofnations by size at particular points in time, from which we can test someinferences about random processes that might have generated those <Ns-tributions. To anticipate, we find that the available data are ambiguous,but at least there is no good evidence that the distribution is changing orthat there are forces at work which are notably increasing or WeakeDÚ1gthe degree of concentration.
While population alone is not a good index of power in current inter-national politics, the data for population are very-much better than forindices such as military capability or economic strength that might betheoretically more preferable. Furthermore, the number of people is arelevant measure of size for many purposes, and in the long run it maybe a more important dimension of power than it seems at present. Itscurrent disabilities stem from the power that Western organization andcapital accumulation convey in the industrialized countries, but overmost of history the differentials across the globe were far less than theyare at present. And they may become less again at some time in thefuture. Hence we shall limit our empirical investigation to concentrationof population, while acknowledging that a study of the distribution ofpower more generally would cover other variables also. <
,
.
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The first crude bit of evidence suggests that over the tvV long runthe relative size of the largest nations has not changed greatly. For
virtually all of recorded ~tury, the largest state in the world has beenChina. Currently ~a’~ population of 700 million represents a~p~ai-mately 21% of the people of the world; its estimated 1939 population of452 million was also about 21% of the world’s people th~. On a muchlonger time scale, a census in the second century A.D. turned up ~?.5million Chinese; estimates for world population at the time are roughly210-2SO million. This would put China at about ~~’~-a remarkablyconstant proportion, all things considered.
This picture is not much different if we look at heterogeneous empiresrather than units that more nearly approximate nation-states. The UnitedKingdom and the non-self-governing parts of the British Empire in 1939amounted to possibly 524 million people, or about 24% of the then-current world population, and estimates of the extent of the RomanEmpire contemporary with the census for ancient China put it at around55 million, or perhaps just under 25% of the total. So however we chooseto d~ the appropriate political unit for long-term comparison, it wouldseem that over at least a 2GXkyear period the largest has contained onthe order of o~,e-~ to one-fourth of the entire worlds population a
GROWTH RATES AND ECONOMIES OF SCALE
Pure description of present and past distributions by itself offers littleguidance in suggesting what may occur in the future. F~ately, how-ever, it is possible from the descriptive material on the size distributionof nations to make some inferences about how the distribution got thatway. Essentially the problem is: Do large nations have any particularadvantages of scale, so that big states are likely to grow faster than smallones (and hence become proportionately bigger); or, on the contrary,do big countries face diseconomies of scale, so that they typically growat slower rates than smaller ones? This brings to mind traditional eco-nomic explanations for the relative size of firms, explanations that de-pend upon assumptions about the structure of costs.
One such explanation is that the larger a firm i~, the more efficientit will be, i.e., it will be able to produce at a lower cost per unit of out-put, Obvious examples stem from the contrast of production-line tohand-crafted techmque& of manufacture. If there are efficiencies that canbe achieved only with a very great output relative to the size of the
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market, then small firms ~ have difficulty completing with large onesand the likely result will be an oligopolistic industry. But another ex-planation contends that per unit costs do not decline mdefinitely as thesize of firm increases, that eventually there comes a point where unitcosts begin to rise again and the larger ~n becomes inefficient
Clearly very small arms do suffer real disadvantages, but it seems
unlikely that disecanonfles of scale can be traced simply to productionmethods, or that over a long enough time span to allow for plant andstaff expansion there is, even for a particular industry, any &dquo;optimum&dquo;firm size. It is possible, however, that the size of firms may be limitedby organizational difficulties and problems of management control ofvery large fi=~4 There are a variety of theories about loss of controlin large institutions, especially those that are hierarchically organized.On the other hand, there is evidence that for many industries long-run
costs are stable, once some minimal firm size has been reached. Exceptfor the costs of entry that keep very small firms out entirely, size wouldotherwise be irrelevant to a firm&dquo;s prospect of success. The proportionategrowth rate would thus be a random variable, in the sense of beinguncorrelated with the absolute size of a farm or with its relative ger~.formance in the previous time period. Size would then be determinedby the statistical &dquo;law of proportionate effect,&dquo; sometimes named &dquo;Gibrat’slaw&dquo; for its maker. The assumption that proportionate growth is in-
dependent of size means not that the distribution of 6rms by size willbe normal, but that it will be highly skewed, with a clustering of mostfirms toward the left-hand (small-size) side of the scale and a sharplydecreasing number as one looks farther to the right. If Gibrat’s lawholds, the resulting distribution will be a lognormal one. That is, if forthe numbers representing the firms’ size we were to substitute their loga-rithms, th~t~ the distribution of the logarithms would be norman (ben-shap~6d).
A normal curve is generated when a large number of small, independent,random forces act on a variate in an additive manner; and a lognormalcurve can be generated if they act multiplicatively. In the present con.text this means that the determinants of the growth of firms tend tochange the size of firras by randomly distributed proportions....The first implication of this simple model is that large, medium, andsmall firms have the same average proportionate growth. The secondimplication is that the dispersion of growth rates around the commonaverage is also the same for large, medium, and small f rms. The thirdimplication is that the di~lf~~on of proportionate growth rates is alsolognormal. Thus if in any period arms on average stay the same size,so that average proportionate growth is unity, just as many f~ double
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as halve their sizes .... Since this applies to all firms, it follows that ifx percent of large firms double their size, x percent of small arms halvetheir size. Therefore a fourth implication of the simplest lognormalmodel is that the relative dispersion of the sizes of arms tends to in-crease over time. In this analysis, the disparity of the sizes of firmsincreases over time, in spite of the fact that large firms have the sameaverage proportionate growth as medium and small finn; this is so be-cause the 50% of large firms with above-average growth include armswhich were formerly among the smallest in the class of large firms butwhich enter the ranks of the very largest arms and overtake some ofthe former leaders
Thus a lognormal curve is circumstantial evidence that costs areconstant and that large firms have no particular efficiency advantages.The fact that several investigations have turned up highly skewed andin some cases lognormal distributions has been cited as evidence againstthe existence of economies of scale.
It is essential to note that this entire subject-both on the empiricalevidence and the theoretical explanations-is a matter of considerabledispute among economists. The wide range of empirical phenomena towhich such distributions can be fitted, however, together with the ac-curacy of the causal explanations in many cases where they can befully tested, indicates that the appearance of similar phenomena in inter-national politics would deserve very careful study, despite the skepticismwhich &dquo;curve-fitting&dquo; may initially provoke.** Short-run projections ofconcentration would of course be done more effectively through theaggregation of demographers’ single-country models of individual birthand death rates. The question to be investigated here is more generaland more fundamental.
THE SIZE DISTRIBUTION OF NATIONS
This discussion is relevant to the international system not simplybecause of any crude analogy between the distribution of firms andthe distribution of nations, but because some of the causal hypothesesadvanced about the size of firms and of nation-states are so similar.There is of course the proposition that only rather large nations are’efficient&dquo; and can provide necessary or desirable services to their peopleat minimum cost. Rather powerful are the arguments that recall econo-mists’ points not about e~ciencies, but about the market power of largefirms-big states, because of their power, can coerce smaller ones so asto obtain a more favorable cost-to-revenue ratio. Or it is contended that
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very large nations are inacient, that they must incorporate such diversepeoples, and incur so many bureaucratic costs from hierarchy, that theyare ill-suited to survive.
In fact, the distribution of nations by population size in three dif.ferent years, for which we have reasonably good data, is highly skewedand provides a very good f to the lognormal distribution. Figures la,lb, and ic show graphs for the size distribution for nation-states in 1938,1957, and 1967. The skewness has been removed by substituting log-arithms for the original figures for population in thousands. The actualdistribution is shown as the number of nations in each of a number ofuniform (by logarithms) size classes, and the curve superimposed illus-trates what the ideal normal distribution would be.7
Allowing for the effects of grouping, the fit is indeed very close, andfor each of the three years. Employing a Chi Square goodness of fit testto test the null hypothesis that the distribution is not normal, we obtainvalues of about 3.6, 7.0, and 6.6 respectively for 1938, 1957, and 1967.For 1967 this is statistically significant Qnly at the .88 level, meaningthat almost nine-tenths of the time we would correctly -reject the nullhypothesis in favor of the alternate hypothesis of normality. For 1938and 1957, when the range was narrower and the total number of nationswas less, I used fewer size classes, and hence there are fewer degrees offreedom. Their Chi Square values are both signxficant at the .73 level,meaning that one properly would reject the hypothesis of non-normalityonly about three-quarters of the time. But all three of these figurestogether indicate a strikingly good and consistent fit.8
The three separate years are given to show the effect of adding somany ex-colonies to the international system during the past threedecades. As is apparent, the effect has not been great. There has beenan increase in the dispersion (as the lognormal model of efficiency un-correlated with size would predict), and despite world population growththe mean size of nation has shifted slightly downward, from the logarithmfor almiost 7 million to just over 51h million. But the overall fit hasremained good throughout the period.
A problem arises, however, as to how we should treat colonial empires.Are we concerned with the growth or shrinkage of the national core, orof all of the peoples under centralized rule from that core? There arereasons for limiting attention to the nation-state itself-for instance, ifwe are concerned in some sense with power, colonial people are rarely asgreat an addition to the power of a nation as are those in the metropole-but doing so would omit some of the most interesting political (ratherthan biological) processes of growth and so lose important information.
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Accordingly it seems necessary to present the data both ways, and sopart Id of Figure 1 gives the data for the same nations in 1938, exceptthat for the ten (Belgium, Denmark, France, Italy, Japan, Netherlands,Portugal, Spain, United Kingdom, United States) that had colonial em-pires I have added the colonies’ population to that of the metropolitanterritory. This change weakens the fit somewhat, raising the Chi Squareto 4.2, significant at the .65 level. Apparently the random growth modelfits better for core-nations than for the dynamics of empire, although weshall continue to check our findings on both sets.
While actual events may not fit the model s assumptions precisely,it serves as a useful approximation. As lriji and Simon note, Galileolaw of the inclined plane, that the distance traveled by a ball rollingdown the plane increases with the square of the time
... does ignore variables that may be important under various cir-eumstar~ces: irregularities in the ball or the plane, rolling friction, airresistance, possible electrical or magnetic Belds if the ball is metal, vari-ations in the gravitational $eid-and so on, ad infinitum. The enormousprogress that physics has made in three centuries may be partly at-tributed to its willingness to ignore for a time discrepancies fromtheories that are in some sense substantially eorreet.~
For the moment accepting the hypothesis of a lognormal distribution,we have the circumstantial evidence that big countries typically growneither more nor less rapidly than smaller ones, and there is no overalltendency for either a decline or an increase in concentration in the inter-national system. This corresponds fully with what poor evidence wecited earlier for millenia-long periods. Of course if one thinks of nationsas growing in population only by the biological processes of birth anddeath among their citizenry, then such a finding is not surprising. Onthat basis, there would be few plausible hypotheses that big nationswould have an advantage except by some rather involved mechanismssuch as a correlation between size and per capita income and conse-quently faster (or slower) natural rates of population increase in thelarger and thus richer countries. But this hypothesis at least is poorlysupported empirically,10 and in any case natural life processes certainlydo not provide the only way that nations grow or shrink. Nations acceptimmigrants and provide emigrants. Furthermore, they annex other areas,fission, or in turn are annexed. When these processes are recalled,various alternative hypotheses about the advantages (or handicaps) ac-eraing to big countries in international competition are not so implausible.They just do not seem to meet with much support here.&dquo;
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Yet this rather neat fit to the lognormal distribution, with the con-sequent deductions we can make for speculating about how the distribu-tion might have been generated, is, unfortunately, complicated by somealternative model and hypotheses that cannot easily be rejected. HerbertSimon, for example, has suggested that if in addition to random growthrates one assumes a process of birth for new arms, the consequence isthe Yule distribution, which looks like the lognormal one except thatthere are a few more very large firms.
Let us assume that there is a minimum size, §~, of arm in an indus-try. Let us assume that for firms above this size, unit costs are con-stant. Individual firms in the industry will grow (or shrink) at varyingrates, depending on such factors as (a) In~t, (b) dividend policy,(e) new investment, and (d) mergers. These factors, in turn, maydepend on the ef~cieney of the individual ~, exclusive access to par-ticular factors of production, consumer brand preference, the growth ordecline of the particular industry products in which it specializes, andnumerous other conditions. The operation of all these forces will gener-ate a probability distribution for the changes in size of films of agiven size. Our first basic assumption (the law of proportionate effect)is that this probability distribution is the same for a~l. size classes ofiirms that are well above 8.. Our second basic assumption is that newfirms are being torn&dquo; in the smallest-size class at a relatively constantrate.... What distinguishes the Yule distribution from the lognormalis not the first assumption-the law of proportionate effect-but thesecond the assumption of a constant &dquo;birth rate&dquo; for new arms. If weassume a random walk of me arms already in the system at the be-ginning of the time interval under consideration, with zero mean changein size, we obtain the lognormal. If we assume a random walk, but witha steady introduction of new firms from below, we obtain the Yuledistribution. 1.
Simon has also developed an alternative model, allowing for the deathof small firms at the same rate as the birth of some new small anas, thatagain produces a Yule distr.i.bution.18 A distribution more like the Yulethan the lognormal can also be produced when there are some e~~advantages to be gained only by extremely large size.
One analyst has reconciled the basic proportionate effects model withwhat many economists have described em~r~ll~t as a p~ogMss~degree of concentration in quite a number of industries. He does it byassuming that the average firm does grow some, and by distinguishingbetween average growth rates and sunitx4 rates. He attributes lower
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survival rates to the smaller arms. The analysis recalls a variety ofphenomena in international politics-large nations’ attempts to securesafety through control over foreign sources of vital supplies, the coer-rive power of big countries in international politics, and the allegedgreater caution and &dquo;responsibility,’ or conservatism, which many ob-servers attribute to great powers.
The fact that morality decrease with size, that f4=-at least in freeenterprise conditions-are- subject to high infant mortahty but that verylarge finns rarely succumb, is well established.... Now if big £nmhave a better chance of survival than firms of smaller size, this winreadily explain why, in the course of time, they obtain more scope forthemselves, as a group, than the others....Simultaneously, the difference in the chance of survival can also offeran explanation of ... what becomes of the advantage of big firms inlarge-scale economies of og~rat~on-wbat do they do with it? They useit, not to earn more and to grow faster, but to survive better than the
’
smaller ~n. ~ .
It will be appreciated that a firm can use its advantages over its com-petitors in either of two ways : either to earn more, or to have moresafety-&dquo;4hat is, either to increase the mathematical expectation of theprofit rate or to decrease the variance of it.... A firm may increase itschances of survival at the expense of the rate of pront by holding morereserves of various kinds, such as financial reserves In the form of liquidassets, government bonds, etc., but in many cases also as stocks of rawmaterial. It may also acquire sources of raw materials, not very profit-able, but essential for operation in certain contingencies. Or again, itmay diversify its production programme in a way which will entail asacrifice in efficiency (higher cost) but will reduce the impact of afailure in any one line....Thus the greater profit and growth rate of ~1»gness&dquo; exists-at least toa large extent-only potentially, and is not realized owing to the pref-e~n~ of big firm for safety.1.
These models mean that there are several different mechanisms bywhich an increasing degree of concentration could arise in a system suchas the international system. They are relevant because in fact the Yuledistribution does fit our empirical distributions of nations at least aswell and perhaps a bit better than the lognormal one, and because ofour interest in f~diug the why behind the ~. From the graphs it is
apparent that the number of very big countries at the upper tail isslightly greater than predicted by the normal model. This evidencemust be taken with a good deal of skepticism, however, since it is basedon the deviations of only one or two countries from their &dquo;expected7
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point in the distribution. As some of the analogous controversies ineconomics have indicated, it is often virtually impossible to decide justwhich of several hypothetical distributions actually provides the best fitto a set of data and, even after obtaining a fit, to decide which variantof the law of proportionate effect offers the proper causal description.We cannot differentiate among them with purely static data.
. FOUR HYPOTHETICAL SOURCES OFC~~~~1~~’~ATIt~bfi
An international system is concentrated to the degree that a highproportion of the total population is contained in a small proportion ofthe countries. There are several ways in which increasing concentrationcould occur:
1. More rapid growth rates for the very largest nations than for smallerones.
2. The &dquo;death’* of small ~tiOf1S by annexation or merger at the same orgreater rate as new ones were being born, while large nations retained highsurvivor rates.
3. A great increase in the number of very small nations (births in thelowest size class).4. The birth of one or two very large nations, either by merger or byentry into the list of sovereign nations from a former state of colonialdependency.
The finding of a perfectly lognormal distribution would have beenstrong disc:o~ation of the first hypothesis, but the suggestion that aYule distribution fits equally well limits the confidence with which thathypothesis can be rejected. What is now required for further differen-tiating among these plausible alternatives, or for rejecting all in favor ofa finding that concentration is not increasing, is to look at the actualdata on observed growth rates as well as the size distribution. We mustalso have data on the births and deaths of nations. This can be done
successfully only with data spread over quite a long time period,measured in several decades as an absolute minimum. Unfortunately,such information is not very accurate even for the Western developednations, let alone for the underdeveloped world. Population censuseshave not been suflciendy reliable for a long enough time to permit thefine measurement really demanded here.
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We can, however, go a bit farther. First, we have nearly completeinformation, if of less than desirable accuracy, on national populationgrowth rates over the period ~~9-~~.m We can divide the ~tribn=~fl~a into several subsets and see if big countries have in fact grownfaster than smaller ones. The first row of Table 1 shows the averagegrowth rate for the countries in each quartile of the distribution. Thesecond row in the table allows for the decline of empires, giving the~uar~e means of the growth rates for the same units, except that forthe empires I have combined the population of the colonies with that ofthe metropolitan territory in the pre-war year, and in 1965 for those thatstill had colonies.
TABLE 1
MEAN GROWTH RATES OF NATIONS’ POPULATION, 1939-1965, BY QUAMILE
(1) The hypothesis that big nations have been growing faster thansmaller ones is unambiguously rejected. This finding is not merely anartifact of the grouping by quartiles, nor does it hide higher growth ratesfor the few nations at the very high end. Of the eight largest states,only China, and that barely (155%), had a growth rate higher than therate achieved by the average nation overall (154%). Size does makesome difference in gra~wth, but opposite from the way expected. Growthrates for the smallest quartile of nations have been well above the aver-age for nations as a whole. This is even more true if empires rather thannation-states are included in the computations, and there, reflecGng thebreakup of colonial foldings after World War II, the largest quartileshows by far the least growth.
Incidentally, there are other reasons for questioning whether the lawof proportionate effect is the source of the size distribution of nations.According to the second and third implications of the law as drawn byHart, the st~~r~ r~~ui~~~xa of growth rates should be the same for smallnations as for large ones, and the distribution of growth rates should itselfbe lognormal.
Investigation turns up little evidence of these properties. Table 2shows the standard deviations of nations’ 1939-1965 growth rates foreach quartile of the distribution. For the three lowest quartiles the dis-persion is virtually the same, but for the top quartile, whether one looks
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at nations only or at nations and colonial empires, the measure of dis-persion is much higher than for the other quartiles. Thus it not onlyappears that the proportionate-effects model does not strictly apply forthe largest states, but that big countries are prone to more risky be-bavior than are smaller ones. Apparently a nation or empire has totake chatices in order to grow big and to stay big, and is likely as aresult to expand or shrink by a greater proportional amount than mostcountries.1,6 Note, however, that this seemingly more risk-prone behaviorwas not reflected in lower ~Mfc~a~ rates for big countries. Shrinkage ratesare not so sharp (or, the &dquo;1u9hio<&dquo; available to a big country is largeenough) that over the time period studied any states above the smallestsize class actually dropped out of the International system of sovereignentities (see below).
TABLE 2
STANDARD DEMAT~ON OF CBOWTH RA1’E$ OF NATIONS’ POPULATION, 1939~1965,BY QUARTILE
_ As for the implication that the distribution of proportionate growthrates should be lognormal. it too is not bome out in the crude dataavailable. Figure 2 shows the actual distribution superimposed on thehypothetical normal curve. The curve is vaguely normal with bunch-mags toward the middle, but more nearly bimodaL The Chi Square forgoodness of fit is 9.8, significant at the .13 level We probably wouldy~ct the hypothesis of normality, though with a .13 probability oferror and hence without much confidence. (The curve for empires is aneven poorer fit [slightly skewed] and not shown.) It is true that the datafor time changes are rather gross for a procedure requiring fine measures,and also that they are incomplete in a way that may be relevant. Thereare four nations with missing data (Afghanistan, Liberia, Saudi Arabia,and Yemen) and from what crude information is available, it is clear thatthey would fall somewhere in the middle of the distribution. Shouldmost or all of them belong in the middle category, they would moder-ately improve the worst aspect of the fit. The significance level and theproblem of data error do leave open the possibility that some implicationsof the proportionate-effects model may not be so inapplicable as theyappear here.
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Fig. 2―Log to distribution of Nations’ populationgrowth rates, 1939-65
(2) Another of the four possible sources of concentration listed abovewas that large nations would have substantially lower death rates
(Steindl) or, ideally, that all &dquo;deaths&dquo; of nations would be in the smallestsize class (Simon). On the whole, the differential death rate modelssimply do not apply-only four permanent deaths by annexation ormerger have occurred since 1938. The Baltic states of Estonia, Latvia,and Lithuania were absorbed into Soviet Russia in 1940, and ~an~~ra~joined Tanganyika in 1664 shortly after achieving independence. It istrue that these few deaths all involved nations with populations underthree million, so they are not inconsistent with the hypothesis; but theywere so few compared with the number of births (<87) as to be essen-,iloy irrelevant to the characteristics of the distribution.
(3) We said tb~.t ~ great increase in the number of very small statescould produce a Yule distribution. That is a substantial oversimplifica-tiou of what has actually happened. True, a much larger proportion ofnew than of old states do fall in the very small size class. The smallestfour-Maldive Islan<k2 Iceland, Barbados, and Gambia-are all post-19Mnations, and of the 15 under one million, only Luxembourg was in ex-istence at the beginning of the period. But above that level, there is norelation between size and post-1938 independence. And the new sover-eign ~t~t~ ~ the past twenty years include India, Pa~taus and Indo-nesia, each now with ’over 100 million people. Thus, attributing part ofthe departure from lognomuùity to the introduction of new small states
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is plausible, but incomplete. In addition to the large new nations thatdo not fit the model, the process of introduction has not been &dquo;at a
relatively constant rate.&dquo; Forty percent of the countries that becameindependent over the entire 30-year period did so between 1960 and1962.
( 4 ) A single very big country-India-was born during the time periodin question. Its birth would, with the other birth and death patternsnoted above, in large part account for the departure from l~gnormalityin the upper tail observed earlier, and its position in the number tworanking has tended to increase the degree of concentration. But it isdifficult to generalize from the single case to any longer-term trends inthe systbm. India is the first state to join the community of nations nearthe top of the size distribution since Germany did so a century ago.
STABLE CONCENTRATION, 1938-1967
Having checked out the various mechanisms by which concentrationmight have increased in the post-1938 era, we can now return to theactual data on what did happen. Figure 3 below shows the Lorenzcurves for the distribution of nation-states by population in 1938 and1967. A Lorenz curve is drawn by ranking nations from smallest to largestand computing the percentage of the total populations held by variouspercentages of the total nurral~r of nations,, starting at the lower end.The further the curve is from the Line of Equality bisecting the squarefrom lower left to upper right, the greater the degree of concentration.&dquo;7
The two curves are similar but not identical with some indication ofan increase in concentration over the period. The Gini index of con-centration (which measures the area between the Lorenz curve and theLine of Equality) increased from .746 to .776. Virtually all that dif-ference occurred between 1957 and 1967, since the 1957 curve (notdrawn) is ahnost identical with the 1938 one and has a Gini index of.748. If we look at the extent of colonial empires as well as of nations,however, we find that the curve has been effectively unchanged. For1938 empires and nations together, the Gini index is .785, or veryslightly more concentrated than the distribution in 1967. By this inter-pretation, the post-World War II end of empire has not really varied thedistribution of a major power base, population, from what it was threedecades ago. The results of those events do not offer any support for aprediction of greater concentration in the’ future.
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i~ig.3~--L~ranz curves of populations concentration_
.
by Nations, 1’~3~ and 1961
SUMMA1BY
We examined the size distribution of nations At several points intime and explored several possible mechanisms by which that distribu-tion might change. Some, questions were resolved; others remain open.
(1) There is no evidence that the largest states are proportionatelyeither larger or smaller than 2000 years ago. -
(2) The size distribution for recent years is nearly lognormal, andhence consistent with a random growth model implying no correlationbetween size and proportionate growth.
(3) The size distribution also, however, departs slightly from log-normality in a way that leaves open the possibility that the very largeststates may have grown faster.
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(4) The size dwdbution also departs slightly from lognomía1ity iDa way that is consistent with a Yule distribution that could be generatedby various models employing random growth and certain birth anddeath rates.
( a ) Over recent decades large nations have not grown faster thansmaller ones, making less likely the possibility suggested by Item 3.
( ~ ) The dktribution of observed growth rates does not fit the re-quirements of the random growth model [Item 2)] well, but the dataare not highly reliable for the ~.~ measurements required.
(7) A combination of the death and especially the birth patternsfor nations in recent decades could have produced a departure fromlognormality roughly in the direction suggested by Item 4.
(8) The actual change in the degree of coticentratiots in the systemover the past three decades has been very small.
NOTBS
AUTHOR’S NOTE: The research for this article was in part supported bya grant from the National Science Foundation, and an earlier version wasprepared as P-3666 for the RAND Corporation. Any views expressed arethose of the author, and should not be interpreted as reflecting the viewsof RAND Corporation or the official opinion or policy of any govern-mental or private research sponsoring agency.
1. E.g., Morton A. Kaplan, System and Process in International Politics (N.Y.: Wiley, 1957), Part I; Karl W. Deutsch and J. David Singer, "Multipolar Power Sys-tems and International Stability," World Politics, XVI, 3 (April, 1964); Kenneth Waltz,"The Stability of a Bipolar World," Daedalus (Summer, 1964), pp. 881-909; andRichard N. Rosecrance, "Bipolarity, Multipolarlty, and the Future," J. Conflict Resolu-tion, X, 3 (Sept., 1966), 314-327.
2. J. David Singer and Melvin Small, "Alliance Aggregation and the Onset ofWar," in J. David Singer (ed.), Quantitative International Politics: Insights and Evi-dence: International Yearbook of Political Behavior Research, Vol. VI (N.Y.: FreePress, 1968).
3. W. S. Woytinsky and E. S. Woytinsky, World Population and Production (N.Y.:Twentieth Century Fund, 1953), pp. 33-34.
4. John Williamson, "Profit, Growth, and Sales Maximization," Economica, Vol.XXXIII, 1 (Feb., 1966), 1-16; and Oliver E. Williamson, "Hierarchical Control andOptimum Firm Size," J. Polit. Economy, LXXV, 2 (April, 1967), 123-138.
5. P. E. Hart, "The Size and Growth of Firms," Economica, XXIX, 113 (Feb.,1962), 30. For the basic paper, see G. Udny Yule, "A Mathematical Theory of Evolu-tion Based on the Conclusions of Dr. J. C. Willis, F.R.S.," Philosophical Transactionsof the Royal Society of London, CCXIII (1924), 21-87.
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6. I do not pretend to know this literature thoroughly, but in addition to thereferences cited above and in the following pages, see, among many others, StephenHymer and Peter Pashigian, "Turnover of Firms as a Measure of Market Behavior,"Rev. Economics and Statistics, XLIV, 1 (Feb., 1962), 82-87; Edwin Mansfield, "Entry,Gibrat’s Law, Innovation, and the Growth of Firms," Am. Econ. Rev., LII, 5 (Dec.,1962), 1023-1051; Richard E. Quandt, "On the Size Distribution of Firms," Am. Econ.Rev., LVI, 3 (June, 1966), 416-432 and T. R. Saving, "The Four-Parameter Lognormal,Diseconomies of Scale, and the Size Distribution of Manufacturing Establishments,"Internatl. Econ. Rev., VI, 1 (Jan., 1965), 105-114. Yuji Iriji and Herbert A. Simon,"Business Firm Growth and Size," Am. Econ. Rev., LIV, 1 (March, 1964), 79, haveshown that the law of proportionate effect does not require that the percentage changein size of a unit from one period to another be independent of the unit’s size, but onlythat the change in size of the totality of firms in each size stratum be independent ofstratum.
7. Data are from Bruce M. Russett et al., World Handbook of Political and SocialIndicators (New Haven: Yale Univ. Press, 1964), 18-20; United Nations, DemographicYearbook for the years 1966, 1962, and 1958 (New York: United Nations, 1967, 1963,and 1959, respectively) ; and World Almanac and Book of Facts, 1944. Some data areestimated. "Sovereign state" is defined in 1957 and 1967 to include all members of theUnited Nations, plus Switzerland and the nations excluded by cold-war politics whoseexistence is acknowledged, if not always recognized. Byelorussia and the Ukraine, how-ever, are not counted despite their formal UN membership. Mini-states that are notmembers of the UN e.g., Andorra, Monaco) are not counted because they are too small(below the minimum firm size) to carry on even the minimal activities of internationalpolitics that the small UN members do. The 1938 country list is compiled from BruceM. Russett, J. David Singer, and Melvin Small, "A Standardized List of Political Entitiesin the Twentieth Century," Am. Polit. Sci. Rev., LXII, 3 (Sept., 1968), applying UnitedNations memberships retroactively and including the Baltic states.
8. If significance levels of .88 and .73 don’t seem very impressive, bear in mindthat we are reversing the normal procedure where one takes a theoretical distributionsuch as the normal one, and sets the null hypothesis that the distribution is normal.One typically hopes to find evidence that the null hypothesis can be rejected and thealternate hypothesis of non-normality can be accepted, but to do so one must haverather high confidence that the departure from normality is not induced by chancealone. Since one would usually demand a significance level of .05 before rejecting thehypothesis of normality, levels of .88 and .73 look pretty good for the reverse procedure.
In calculating the Chi Square, I collapsed the last two size classes at each tail intoa single category, so as to raise the expected frequency in that class above one. Withexpected frequencies below unity, the Chi Square test is not appropriate. Some statis-ticians would insist on frequencies of at least 5, but this seems too conservative. SeeWilliam G. Cochran, "Some Methods for Strengthening the Common X2 Tests," Bio-metrics, X, (Dec. 4, 1954), 417-451.
9. Iriji and Simon, op. cit. note 6, p. 78.10. See Russett et al., op. cit. note 7, p. 277, where the correlation is shown to be
below .20.11. Data on nations’ size by measures such as total G.N.P. are, as indicated earlier,
less reliable than population data, especially for changes over time. A test of the dis-tribution of nations’ G.N.P. in 1957 nevertheless powerfully supports the major hypothesisof this article; it fits a lognormal distribution with a Chi Square of 4.5 at a significancelevel of .92, even higher than any test of the population data. This needs to be ex-amined further. Data from Russett et al., pp. 152-154.
12. Herbert Simon and C. P. Bonini, "The Size Distribution of Business Firms,"Am. Econ. Rev., XLVIII (Sept., 1958), 607-617. Note some of the analogies betweenreasons for the growth or shrinkage of firms and those for nations.
13. Herbert A. Simon, Models of Man (N.Y. : Wiley, 1957), chap. 9.
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14. Josef Steindl, Random Processes and the Growth of Firms (N.Y.: Hafner,1965), pp. 218-221. William J. Baumol suggests that the managers of firms may preferto avoid even moderate risk so as not to antagonize their shareholders. Losses attrib-utable to risk-taking may provoke immediate shareholder disaffection, and even
windfall gains from successful risk-taking may provoke ultimate dissatisfaction if theylead to expectations of continued high gain that cannot be met. Cf. Business Behavior,Value, and Growth (N.Y.: Macmillan, 1959), esp. chaps. 6, 7, 10. Governments similarlymay wish to avoid risk-taking for fear of reprisal from their constituents. MarshallHall and Leonard Weiss, "Firm Size and Profitability," Rev. Economics and Statistics,IL, 3 (Aug. 1967), 319-331, show that the rate of profit is higher for large firms thanfor smaller ones, even controlling for the degree of concentration of the industry.
15. The data are for 1939-1965 population growth, but it seemed relevant to dothe analysis for the political units that existed in January 1938, before the largelytemporary elimination of a number of independent states from the international sceneimmediately before and during World War II. Data are from United Nations, Demo-graphic Yearbook, 1958, Table 4, and Demographic Yearbook, 1965, Table 4. The pre-war population estimates for Afghanistan, Liberia, Saudi Arabia, and Yemen were toocrude to use here.
16. This finding in a way complements that of Stephen Hymer and Peter Pashigian,"Firm Size and Rate of Growth," J. Polit. Economy, LXX, 6 (Dec., 1962), 566-569, wholooked at the standard deviations of growth rates of firms in different size classes. Be-cause of the possibilities large firms have for diversification to spread their risks, theypredict a sharply inverse relation between size and the variation in growth rates. Al-though they do find a somewhat inverse relationship, the fact that the standard deviationfor large firms’ growth rates is greater than predicted (though less than for smaller firms)brings them to conclude that large firms are also prone to greater risk-taking behavior.While all the assumptions of their model are not appropriate for transferring an expecta-tion of an inverse relationship from firms to nations, the finding is nevertheless relevant.
17. Hayward R. Alker and Bruce M. Russett, "Indices for Comparing Inequality," inRichard L. Merritt and Stein Rokkan (eds.), Comparing Nations (New Haven: YaleUniv. Press, 1966).
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