gibrat’s law for (all) cities - narod.ru · 2013. 4. 3. · all cities and not just the upper...

Gibrat’s Law for (All) Cities

By JAN EECKHOUT*

Two empirical regularities concerning the size distribution of cities have repeatedlybeen established: Zipf’s law holds (the upper tail is Pareto), and city growth isproportionate. Census 2000 data are used covering the entire size distribution, notjust the upper tail. The nontruncated distribution is shown to be lognormal, ratherthan Pareto. This provides a simple justification for the coexistence of proportionategrowth and the resulting lognormal distribution. An equilibrium theory of localexternalities that can explain the empirical size distribution of cities is proposed.The driving force is a random productivity process of local economies and theperfect mobility of workers. (JEL D30, D51, J61, R12)

The law of proportionate effect will there-fore imply that the logarithms of the vari-able will be distributed following the[normal distribution].

—Robert Gibrat (1931)

The way the population is distributed acrossgeographic areas, while continuously changing,is not random. In fact, there is a strong tendencytoward agglomeration, i.e., the concentration ofthe population within common restricted areaslike cities. And while physical geography—riv-ers, coasts, and mountains—may have played acrucial role in early settlements, in the currentday and age, the evolution of the populationacross geographic locations is an extremelycomplex amalgam of incentives and actionstaken by millions of individuals, businesses,and organizations. Most people will agree thateconomic factors are the principal determinantof the dynamics of city populations. In the lastdecade, Detroit, for example, experienced a de-cline in population as the manufacturing indus-try in the area suffered a severe downturn. Atthe other extreme, when the high-technologyindustry was booming, villages, towns, and cit-

ies in the San Francisco Bay area experiencedhigher-than-average population growth. In-creased productivity due to technologicalprogress in the e-business sector led to the cre-ation of such new companies as Yahoo! and theexpansion of such existing companies as HPand Apple. This in turn increased labor demandand wages, which induced many individuals torelocate to the Bay area. No doubt an exodusfrom the Bay area has been at work since thetechnology market crashed at the beginning ofthe current decade. This confirms that agglom-eration and residential mobility of the popula-tion between different geographic locations aretightly connected to economic activity.

Given this direct connection between eco-nomic activity and population mobility, it haslong been recognized that fully understandinggeographic economic activity involves under-standing population mobility and economicdriving forces.1 A crucial first step is to providean accurate description of agglomeration andpopulation mobility. This involves accountingfor the way the population is distributed overdifferent geographic locations and accountingfor the evolution over time. Once populationmobility is understood, the second step involvesanalyzing the underlying economic mecha-nisms. Because economic factors are of para-mount importance in providing incentives forindividuals and businesses to move to differentlocations, being able to model the economic

* Department of Economics, University of Pennsyl-vania, 3718 Locust Walk, Philadelphia, PA 19104(e-mail: [email protected]; http://www.ssc.upenn.edu/�eeckhout/). I am grateful for discussion and commentsfrom Pol Antras, Jesus Fernandez-Villaverde, XavierGabaix, Gregory Kordas, Robert Lucas, Samuel Pessoa,Esteban Rossi-Hansberg, Stephen Yeaple, the editor, andtwo referees. Jesus Fernandez-Villaverde provided the codefor the nonparametric kernel estimator.

1 See, among others, George K. Zipf (1949), J. VernonHenderson (1974) and Paul Krugman (1996).

1429

forces is of direct importance, especially sincedifferent cities are subject to different types ofgovernment policies, both within a city andbetween cities. The motives for interventionoften depend on externalities (see Robert E.Lucas and Esteban Rossi-Hansberg, 2002, for adiscussion). Through their interventions, poli-cymakers affect economic factors, in particularequilibrium prices of land and labor and, there-fore, decisions by individuals and businesses onwhere to locate. For example, city-specificincome-tax incentives will affect after-taxwages and will make certain locations moreattractive than others. This in turn will lead to achange in the number of people deciding toestablish residence in certain locations. Otherexamples include transportation taxes and sub-sidies within and between cities (for exam-ple the subsidization of roads, railways, andairports),2 regional subsidies, and agriculturesubsidies that benefit companies in rural towns.An equilibrium theory of choice of geographiclocation (city, town, or village) driven by mar-ket wages and property prices is necessary forthe optimal design and evaluation of suchpolicies.

Unfortunately, the literature has faced sub-stantial difficulties in the description of popula-tion mobility. The difficulty derives from apuzzle caused by two robust empirical regular-ities. The first empirical regularity is that thelargest cities satisfy Zipf’s law. Despite the ap-parent chaotic evolution of city populations,surprising regularities have been observed in thesize distribution of cities. As early as 1682,Alexandre Le Maıtre observed a systematic pat-tern of the size distribution of cities in France.He describes how the size of Paris related to twogroups of cities, each of them proportionallysmaller than Paris. But it was not until 1913 thatFelix Auerbach, and then George Kingsley Zipfin 1949, formally established the first empiricalregularity. They show that within a country, thesize of the largest cities is inversely proportionalto their rank. For example, in the United States,New York City is roughly twice the size of LosAngeles, the second largest city, and about threetimes the size of Chicago, the third largest city.

The proportionality of rank and size implies thatthe upper truncated distribution is the Paretodistribution3 (or power distribution) with expo-nent equal to one. Zipf’s finding has beenshown to be robust, both over time and acrosscountries, though with varying Pareto expo-nents. The second empirical regularity is thatthe growth rate of city populations does notdepend on the size of the city. Even thoughgrowth rates between different cities vary sub-stantially, there is no systematic pattern withrespect to size, i.e., the underlying stochasticprocess is the same for all cities. This is labeledthe proportionate growth process. Empirical re-search4 has repeatedly shown that city growth isproportionate: larger cities on average do notgrow faster or slower than smaller cities.

While it is surprising that such regularitiesemerge from a highly intricate underlyingmechanism, there is also a puzzle: the two reg-ularities cannot easily be reconciled.5 In partic-ular, the proportionate growth process (thesecond regularity) gives rise to the lognormaldistribution, not the Pareto distribution (i.e.,Zipf’s law, the first regularity). This is a well-known proposition established by Gibrat (1931)and originally formulated by the astronomerJacobus C. Kapteyn (1903): a stochastic growthprocess that is proportionate gives rise to anasymptotically lognormal distribution.6 This isnot to say that a proportionate growth processplus “something else” cannot give rise to thePareto distribution or another distribution.There is a long tradition in the economics ofincome inequality starting with David G. Cham-pernowne (1953) and industrial organization

2 Transportation expenditure in 2002 was $62 billion(3.1 percent of total government outlays; 9 percent ofoutlays excluding transfers) (www.whitehouse.gov/omb/budget).

3 Vilfredo Pareto (1896) is credited with the discoverythat the distribution of individual income satisfies a powerlaw, the Pareto distribution.

4 See, among others, Edward Glaeser et al. (1996),Jonathan Eaton and Zvi Eckstein (1997), and Yannis M.Ioannides and Henry G. Overman (2003).

5 Krugman (1995) writes: “We have to say that therank-size rule is a major embarrassment for economic the-ory: one of the strongest statistical relationships we know,lacking any clear basis in theory.”

6 Kapteyn (1903) studies skew distributions, mainly inbiology, and establishes that they are driven by a simpleGaussian process. If a variable y is generated by additiverandom shocks which give rise to an asymptotically normaldistribution, then given a transformation y � f (x), thevariable x has a skew distribution, derived from the trans-formed stochastic process. One such transformation is y �ln x.

1430 THE AMERICAN ECONOMIC REVIEW DECEMBER 2004

(see John Sutton, 1997, for an overview andBoyan Jovanovic, 1982) studying the relationbetween proportionate growth and size distribu-tions different from the lognormal. With respectto the size distribution of cities, Xavier Gabaix(1999) and Aharon Blank and Sorin Solomon(2000) propose a resolution of the puzzle andshow that proportionate growth processes cangenerate Zipf’s law at the upper tail.7

The purpose of this paper is twofold. First,a new resolution of the puzzle is uncoveredregarding the two empirical regularities, thusproviding an accurate description of popula-tion mobility. While an accurate descriptionof population mobility per se may not be ofprimary interest, it does have fundamental im-plications for the underlying economic mecha-nism, which in turn drives the populationmobility. The second purpose is to propose andsolve an equilibrium theory of local externali-ties. The equilibrium theory provides an analy-sis of the underlying economic mechanisms thatis consistent with the empirically observed pop-ulation mobility. This approach of providing anempirically consistent theory is in line with thecentral thesis in this paper: population mobilityis driven by economic forces. Such an empiri-cally consistent equilibrium theory is novel be-cause heretofore the literature8 has focused onsolving the puzzle concerning population mo-bility. The main interest of this empirically con-sistent equilibrium theory is that it facilitates theevaluation of government policies that affectcitizens’ mobility decisions. Is it efficient toprovide federal subsidies to small cities to at-tract residents? What is the effect of govern-

ment-financed local transportation in largecities?

The breakthrough in the current resolution ofthe puzzle (the first purpose of this paper) de-rives from the availability of Census 2000 data.The new dataset is substantially larger thanthose of earlier censuses. The current data in-clude observations on the entire size distribu-tion of geographic locations, referred to in theCensus as “places.” For the year 2000, there areobservations on 25,359 places, including cities,towns, and villages,9 ranging in population from1 to over 8 million. Previously in the literature,only the truncated distribution, i.e., the uppertail of the distribution of the 135 largest cities,or metropolitan areas (MAs), was considered,i.e., 0.5 percent of the current sample and 30.2percent of the sample population. Using the newdata, it is shown that the size distribution of theentire sample is lognormal and not Pareto.Moreover, for those observations for which1990 data also exist, the growth rate of cities iscalculated, and the second regularity, thatgrowth is independent of city size, is confirmed.As a result, the growth process is shown to beproportionate. The proportionate growth pro-cess, together with the lognormality of the sizedistribution, establishes that when consideringall cities and not just the upper tail of thedistribution, Gibrat’s prediction concerning thestochastic process holds.

The second purpose of this paper is to ana-lyze an equilibrium model consistent with theempirically observed population mobility. Atheory of local externalities is proposed. Likethose in Lucas and Rossi-Hansberg (2002), thecities in this model are characterized by localexternalities—both positive production exter-nalities (spillovers from nearby factors of pro-duction) and negative consumption externalities(lost leisure time from traffic congestion).Those externalities are local, which means theyaffect the population within a city only, andtypically they depend on the size of the city’spopulation. In large cities, for example, firmsand workers benefit more from the availabilityof deep markets for employees and jobs, andthose cities also have larger “knowledge spill-

7 They consider random growth processes with “some-thing else”—the entry of new cities—and apply a processdeveloped in Champernowne (1953) and Harry Kesten(1973). While these processes do generate Pareto distribu-tions, Blank and Solomon (2000) point out that the detailsspecifying and enforcing the smallest size of the cities arecrucial, as are the rules for creating new entering cities.Whether or not the resulting limiting distribution is Paretowith exponent equal to one is very sensitive to this entryprocess. Moreover, testing whether the entry process satis-fies the exact and detailed requirements for the Paretodistribution is a challenging empirical endeavor (for themetropolitan areas in the United States, for example, therehas been no entry or exit in the set of MAs between 1990and 2000; the 276 MAs in 1990 are identical to those in2000). To date, no such evidence has been provided.

8 Some notable exceptions are discussed below.

9 In what follows, the term “city” will be used to indicatea place. When a city (as opposed to a town or a village) isreferred to, this will be made explicit.

1431VOL. 94 NO. 5 EECKHOUT: GIBRAT’S LAW FOR (ALL) CITIES

overs.” Information concerning new technolo-gies and products spills over faster in marketswith high degrees of local interaction, like thoseof large cities. Simultaneously, workers inlarger cities also impose negative externalitieson each other because commuting times arelonger. The economy differs from the one inLucas and Rossi-Hansberg (2002) because ofthe explicit mobility between cities, rather thanwithin cities. The aim is to capture the notion ofcompetition between geographic locations, i.e.,perfectly mobile citizens making location deci-sions between different cities. Local externali-ties within cities regulate the mobility ofcitizens between different cities (i.e., there areno externalities between cities). It is shown thatthe local externality model economy predictsbehavior that is consistent with the empiricalcity growth process.

The only remaining issue to resolve is how itis possible that Zipf’s law is repeatedly con-firmed in the literature, while the underlyingdistribution is lognormal. The Pareto distribu-tion is very different from the lognormal, so it isobvious that if the true distribution is lognor-mal, the entire distribution can never be fit to aPareto distribution at the same time. ConsiderFigure 1 with a plot of the density function ofthe lognormal and that of the Pareto distribution(both on a ln scale); observe that the lognormalon a log scale is the normal density function.The density of the Pareto distribution is down-ward sloping, whereas the lognormal density isinitially increasing and then decreasing (givensymmetry, half the observations are in the in-creasing part). If the underlying distribution islognormal, then goodness of fit tests will cate-gorically reject the Pareto distribution. Still,when regressing log rank on log size for the entiredistribution,10 the coefficient comes out signifi-cant. Estimating a linear coefficient when theunderlying empirical distribution is not Pareto(i.e., the relation is nonlinear) can obviouslyproduce a significant estimate. This regressiontest merely confirms that there is a relationbetween size and rank, but it does not provide atest for the linearity of this relation. As such,testing the significance of the linear coefficient

is not the equivalent of a goodness-of-fit test forthe Pareto distribution.11

More important though is that until now theliterature considered the truncated distribution(typically, the truncation point is at ln size equalto 12 on the horizontal axis, i.e., for only 135cities). At the very upper tail of the distribution,there is no dramatic difference between the den-sity function of the lognormal and the Pareto.Now both the truncated lognormal and the Pa-reto density are downward sloping and similar(the Pareto is slightly more convex). As a result,both the Pareto and the truncated lognormaltrace the data relatively closely. The problem is

10 This is the standard procedure in the literature toverify for Zipf ’s law.

11 See also Gabaix and Ioannides (2003) on the short-comings of OLS.

FIGURE 1. DENSITY OF LOGNORMAL (PANEL A) AND

PARETO (PANEL B) DISTRIBUTION


http://pubs.aeaweb.org/action/showImage?doi=10.1257/0002828043052303&iName=master.img-000.jpg&w=191&h=352

that the estimated coefficient on the Pareto dis-tribution is extremely sensitive to the choice of thetruncation point: as the truncation point increaseson the horizontal axis, the estimated Pareto coef-ficient increases, while the estimated lognormalcoefficients remain unchanged. Moreover, forlower truncation points, the Pareto fits the dataless and less well. In this paper, we show thatthese observed empirical changes in the estimatedPareto coefficient are theoretically consistent withthe comparative static of a changing truncationpoint of the lognormal distribution.12

Finally, there is a growing literature propos-ing equilibrium models of economic activitywith mobility of citizens that can account forZipf’s law.13 Rossi-Hansberg and Mark Wright(2004) propose a dynamic general equilibriumtheory with population mobility and balancedgrowth driven by industry-specific shocks. Whiletheir theory can explain Zipf’s law for the sizedistribution, the model can also explain deviationsof the empirical size distribution from Zipf’slaw.14 This attempt to account for empiricallyobserved differences from Zipf’s law usinggrowth theory is novel. The results they find con-cerning the truncated size distribution are consis-tent with those found in the current paper,confirming the importance of deviations fromZipf’s law.

This paper is organized as follows. In Section I,the Census 2000 data are described in detail. Thesize distribution is shown to be lognormal, and thegrowth process proportionate. In Section II, theimplications, both empirical and theoretical, forestimation of Zipf’s law are analyzed when thetrue underlying distribution is lognormal. In Sec-tion III, a theory of local externalities is proposed,consistent with Gibrat’s proposition that propor-tionate growth leads to a lognormal distribution.Finally, some concluding remarks are made inwhich the parallel is drawn between our resultsand findings in the exact sciences.

I. The Empirical Size Distribution of Cities

A. The Data

Newly available data from Census 2000 areused.15 The dataset for deriving the distributionof cities is novel. The units of account aredenoted by the Census Bureau as “places.”Places are either legally incorporated under thelaws of their respective state or are CensusDesignated Places (CDP). Incorporated placeshave political/statistical descriptions of city,town (except in New England, New York, andWisconsin), borough (except in Alaska andNew York), or village. People living in loca-tions that are not incorporated are legally resi-dent in the respective counties. Incorporatedplaces can cross county boundaries. Because aconsiderable fraction of the population lives inplaces that are not incorporated,16 the CensusBureau designates such places CDPs. Accord-ing to the Census Bureau, a CDP is a “statisticalentity that serves as a statistical counterpart ofan incorporated place for the purpose of pre-senting census data for a concentration of pop-ulation, housing, and commercial structures thatis identifiable by name, but is not within anincorporated place.”17 In the new census data,the CDPs are included for the first time withoutany restrictions.18 The data on places for allU.S. states (including Hawaii and Alaska, andthe commonwealth of Puerto Rico) will be used.In what follows, place (whether a city, town, orvillage) and city will be used interchangeably.

The main advantage of using these censusdata is that they cover the entire population sizedistribution. Moreover, with the inclusion of theCDPs in 2000, this new source of data repre-sents the entire geographic concentration of theU.S. population. In the year 2000, 208 millionof 281 million individuals (74 percent) wereliving in a total of 25,359 places. Table 1 reportson the population size in 2000 of the 10 largest“places.”

12 The sensitivity of the Pareto coefficient to the trunca-tion point has been observed in the literature (for an over-view, see Gabaix and Ioannides, 2003). Explanationsoffered for the sensitivity differ, however, from the expla-nation proposed here, i.e., that the underlying true distribu-tion is lognormal.

13 See, among others, Rossi-Hansberg and Wright(2004) and Gilles Duranton (2002).

14 Unfortunately, when predicting a size distribution thatis different from the Pareto distribution, their model nolonger satisfies proportionate growth.

15 Source: www.census.gov/main/www/cen2000.html.16 Very often, whether a place is incorporated depends

on state law. For example, under state law in Hawaii, thereexist no incorporated places.

17 Source: www.census.gov.18 For Census 2000, CDPs did not have to meet a

population threshold to qualify for the tabulation of censusdata.


A substantial portion of research into the sizedistribution of the U.S. population has beendone using the MA19 as the unit of measure-ment (see, for example, Krugman, 1996;Gabaix, 1999; Ioannides and Henry G. Over-man, 2003). An MA typically covers one (orseveral) large cities. The largest metropolitanarea is New York-Northern New Jersey-LongIsland, including the cities of New Haven, Con-necticut, Newark and Trenton, New Jersey, andseveral smaller towns in eastern Pennsylvania.The ten largest MAs and their population sizeare listed in Table 2.

The total number of MAs in the United Statesis 276, the smallest of which is Enid, Oklahoma,with a population of 57,813. In 2000, 80 percentof the entire U.S. population lived in MAs. Atfirst sight, it may seem surprising that 80 per-cent lived in the 276 MAs, while only 73 per-cent lived in 25,359 places. The reason is thatMAs cover huge geographic areas. For exam-ple, Trenton, New Jersey, is 64 miles from NewYork City and 144 miles from New Haven,Connecticut. As a result, MAs include a largepopulation living in rural areas which are notcounted as places. Consider, for example, Mer-cer County, New Jersey, in the MA of NewYork-Northern New Jersey-Long Island, whichincludes Princeton and Trenton. In 2000, MercerCounty had a population of 350,761, of whichonly about 31 percent lived in incorporated places.

B. The Size Distribution

Over the entire size distribution, the mediancity has a population of 1,338. Figure 2 plots theempirical density function on a natural logarith-mic (ln) scale, together with the theoretical log-normal density for the empirically observedmean and variance. Figure 3 plots the cumula-tive density function. The sample mean (in ln,standard error in brackets) is � � 7.28 (0.01)and the standard deviation is � � 1.75. Thetheoretical density function of the lognormalsize distribution is normal in ln S and given by�(�, �):

(1) ��, �� 1

��2�e��lnS � ��2/2� 2

.

A Kolmogorov-Smirnov (KS) test of good-ness of fit of the empirical density functionagainst the lognormal with sample mean � �7.28 and sample standard deviation � � 1.75generates the KS test statistic D � 0.0189, andthe corresponding p-value obtained is 1 percent.This is supporting evidence in favor of lognor-mality of the size distribution. Though the fit isremarkable, it is not perfect. There seems to besome skewness (third moment is 0.21) and themedian value is 7.20 (with mean of 7.28). Onthe other hand, there is hardly any kurtosis (thefourth moment is 0.03). Possibly there is somecensoring (most likely at the bottom of thedistribution). The data collected may be con-taminated by differences between state legisla-tion with respect to legal incorporation, inparticular for small places. In addition, since thedata contain CDPs, the decision procedure bythe Census Bureau to designate a nonincorpo-rated place may depend on the size of the placeand, as a result, it will affect the size distributionof places, in particular at the bottom end. Fur-thermore, given the extremely large sample sizeof n � 25,359, small deviations from the theo-retical distribution are exaggerated in goodnessof fit tests. It is surprising that, despite somepotential shortcomings of the data, the empiricalsize distribution fits the lognormal distributionthat well.

Before analyzing the properties of the citygrowth process, a fundamental issue remains:what is the appropriate economic unit thatshould be studied? As Tables 1 and 2 highlight,

19 According to the Census Bureau definition, an MA“must include at least one city with 50,000 or more inhab-itants, or a Census Bureau–defined urbanized area (of atleast 50,000 inhabitants) and a total metropolitan populationof at least 100,000 (75,000 in New England).”

TABLE 1—TEN LARGEST CITIES IN THE UNITED STATES

Rank City Population S SNY/S

1 New York, NY 8,008,278 1.0002 Los Angeles, CA 3,694,820 2.1673 Chicago, IL 2,896,016 2.7534 Houston, TX 1,953,631 4.0995 Philadelphia, PA 1,517,550 5.2776 Phoenix, AZ 1,321,045 6.0627 San Diego, CA 1,223,400 6.5468 Dallas, TX 1,188,580 6.7389 San Antonio, TX 1,144,646 6.996

10 Detroit, MI 951,270 8.419

Note: SNY /S denotes the ratio of population size relative toNew York.Source: Census Bureau, 2000.


cities and MAs represent different notions aboutthe corresponding theory of an economic unit.And depending on the definition, we are study-ing different objects and therefore different dis-tributions. As is the case with comparisons ofcountries, we do not have a perfect justificationfor using a particular unit of account whencomparing cities. In our theory below, we con-sider local externalities that do not affect agentsoutside the economic unit as the defining char-acteristic of a city. In reality of course, noexternality is purely local. One may thereforewant to interpret this assumption as a matter ofthe extent to which externalities do or do notaffect agents outside a given city. The danger isthat the partition into economic units is eithertoo fine or, at the other extreme, too coarse. Theexternalities for some agents in one part of agiven economic unit (say those living in NewHaven) may not have an impact on those livingin different parts of the same unit (say Prince-

ton). Moreover, different research objectivesmay call for the use of different units of ac-count. For example, if one is interested in ana-lyzing the economic impact of airports, the MAseems a natural unit of account, while citiesmay be more appropriate when studyingschools, public transportation, or waste collec-tion. In past research, both MAs and cities haveproven to be useful and relevant economicunits, and both have been studied extensively.

In this paper, cities are chosen for severalreasons. In addition to the fact that cities are anatural economic unit for studying the localexternalities that are modeled in Section III,there is a practical reason: the availability ofdata. We want to use data that cover the entirerange of the populations, in particular thesmaller ones. Because MAs are defined by theCensus Bureau only for large populations (MAsmust include “at least one city with 50,000 or

FIGURE 2. EMPIRICAL AND THEORETICAL DENSITY

FUNCTIONS FIGURE 3. EMPIRICAL AND THEORETICAL CUMULATIVE

DENSITY FUNCTIONS

TABLE 2—TEN LARGEST METROPOLITAN AREAS IN THE UNITED STATES

Rank MA Population S SNY /S

1 New York-Northern New Jersey-Long Island, NY-NJ-CT-PA 21,199,865 1.0002 Los Angeles-Riverside-Orange County, CA 16,373,645 1.2953 Chicago-Gary-Kenosha, IL-IN-WI 9,157,540 2.3154 Washington-Baltimore, DC-MD-VA-WV 7,608,070 2.7875 San Francisco-Oakland-San Jose, CA 7,039,362 3.0126 Philadelphia-Wilmington-Atlantic City, PA-NJ-DE-MD 6,188,463 3.4267 Boston-Worcester-Lawrence, MA-NH-ME-CT 5,819,100 3.6438 Detroit-Ann Arbor-Flint, MI 5,456,428 3.8859 Dallas-Fort Worth, TX 5,221,801 4.060

10 Houston-Galveston-Brazoria, TX 4,669,571 4.540

Note: SNY /S denotes the ratio of population size relative to New York.Source: Census Bureau, 2000.




more inhabitants”), the MA dataset does notcover the entire size distribution. And even ifthe dataset spans the entire domain of the sizedistribution of all cities, not all inhabitants livein cities, towns, or villages.20 Unfortunately,these restrictions do not allow for the possibilityof augmenting the dataset to include popula-tions that are currently not covered.21 It shouldbe noted that the current dataset of all cities hasalready been augmented to form the largestpossible dataset that is feasible, with the inclu-sion of the census-defined CDPs. This increasesthe number of cities by 31 percent, from 19,361to 25,359.

The fact that part of the population is notcovered is potentially a cause for concern, be-cause rather than capturing deep patterns ofpopulations and population dynamics, we maymerely be describing the idiosyncrasy of thejurisdictional formation in the United States.The population that is not covered may be dis-tributed in a completely different way from thelognormal distribution. And since we cannotassign that population to any geographic areacomparable to a city, there is no hope of know-ing how the remainder is distributed. The log-normality seems to be a strong regularity,however, from whichever perspective popula-tion dynamics is considered. First, while wehave no way of showing that the distribution ofMAs is lognormal given the truncation by def-inition, we show below that even for MAs,changes in the truncation point produce changesin the estimated Zipf coefficient that are consis-tent with the fact that the underlying upper tailis derived from the lognormal. Second, the sizedistribution of CPDs is pretty close to the entiredistribution of cities and hence the lognormal.And finally, in the Appendix we show the re-sults of further analysis using additional datathat are available from the Census. We plot thesize distribution of counties, which covers theentire U.S. population (see Figure A-1 andTable A-1 in Appendix A for the ten largest

counties). While it is hardly convincing to makea case for counties as the relevant economicunit, it is surprising that even the size distribu-tion of counties is close to the lognormal. Look-ing at population dynamics from the perspectiveof different economic units and including aslarge a fraction as possible of the U.S. popula-tion, there is a strong pattern that is consistentwith lognormality.

C. Proportionate City Growth

For the cities in the upper tail of the sizedistribution, population growth has repeatedlybeen shown to satisfy constant proportionategrowth.22 These findings can be extended be-yond those for the upper tail of the distribution.We therefore use the data on population size forplaces in the United States from both the 1990and 2000 Censuses. Unfortunately, 1990 Cen-sus data do not include the CDPs. As a result,the sample size is significantly smaller (19,361instead of 25,359). Figure 4 shows the scatterplot of growth against city size (on ln scales).Mere observation of the scatter plot seems to

20 All citizens belong to a county, which is the primarylegal division and the functioning governmental unit.

21 Those residual populations are included in the coun-ties, and after accounting for the cities, residual populationsvery often are located in different geographic areas, sepa-rated by cities. To make things even worse, many citiesextend over different counties, therefore guaranteeing thatparts of the residual populations are counted twice.

22 Glaeser et al. (1996) have shown this to be true for thelargest cities in the United States. Eaton and Eckstein (1997)have confirmed this for the largest cities in France andJapan. In a detailed investigation, Ioannides and Overman(2003) nonparametrically estimate the mean and variance ofgrowth rates conditional on size for the largest MAs in theUnited States. They accept the hypothesis that the city-sizegrowth rate is constant across cities of different sizes, i.e.,population growth is proportionate.

FIGURE 4. SCATTER PLOT OF CITY GROWTH AGAINST

CITY SIZE



support that growth is independent of size. Inwhat follows, the dependence relation of growthon size is analyzed in greater detail. We performboth nonparametric and parametric regressionsof growth on size.

First, we perform a nonparametric regressionof growth on size.23 The standard parametricregressions as performed below provide us onlywith an aggregate relationship between growthand size, which is constrained to hold over theentire support of the distribution of city sizes. Incontrast, the nonparametric estimate allowsgrowth to vary with size over the distribution.The regression relationship we model is there-fore

gi � m�Si � � �i

for all i � 1, ... , 19361. The objective is toprovide an approximation of the unknown rela-tionship between growth and size using smooth-ing, without making parametric assumptionsabout the functional form of m. Before estimat-ing m, we report the distribution of growth ratesfor each decile of the size distribution. Follow-

ing Ioannides and Overman (2003), we use thenormalized growth rate (the difference betweenthe growth rate and the sample mean divided bythe standard deviation). In Figure 5, the stochas-tic kernel density24 is plotted for each of the 10deciles. Fixing a particular decile in the distri-bution, we can observe the distribution ofgrowth rates within that decile. Figure 6 reportsthe contour plot of the same stochastic kernel,i.e., the vertical projection of the density func-tion. Both figures illustrate that the distributionof growth rates is strikingly stable over differentdeciles. The best illustration of the size inde-pendence is the fact that the contour lines areparallel. The distribution is slightly skewed (themode is just below zero), and the mode appearsfairly constant over different deciles. The sameis true for the variance. While the variance ofthe lowest decile seems to be somewhat higher(the contour lines fan out somewhat), thereseems to be little change in the spread of thedistribution for higher deciles.

We now proceed to estimate the regressionrelationship gi � m(Si) � �i, i � 1, ... , 19361,where gi is the normalized growth rate, i.e., the

23 This section on the nonparametric analysis followsclosely the analysis in Ioannides and Overman (2003). Wederive a sequence of results for our dataset of all citiessimilar to theirs, obtained for a time-series dataset on thelargest MAs.

24 Each stochastic kernel is calculated using the band-width derived with the automatic method corresponding tothe Gaussian distribution (see Bernard W. Silverman,1986).

FIGURE 5. SURFACE PLOT: KERNEL DENSITY ESTIMATION OF NORMALIZED GROWTH RATES

BY DECILE OF THE SIZE DISTRIBUTION



difference between growth and the sample meandivided by the sample standard deviation, and Siis the log of the population size of a city. Wewill approximate the true relationship by theregression curve m(s) for all s in the support ofSi. The estimate of m(s) will be denoted m(s)and is a local average around the point s. Thislocal average smooths the value around s, andthe smoothing is done using a kernel, i.e., acontinuous weight function symmetric around s.The kernel K used in the remainder of the paperwill be an Epanechnikov kernel.25 The band-width h determines the scale of the smoothing,and Kh denotes the dependence of K on thebandwidth h. With the kernel weights, we cal-culate the estimate of m using the Nadaraya-Watson method,26 where

m�s� �

n�1¥

i � 1

n

Kh �s � Si �gi

n�1¥

i � 1

n

Kh �s � Si �

.

In Figure 7 there is a plot of m(s) calculatedfor a bandwidth of h � 0.5 (see Silverman,1986). The Figure also shows the bootstrapped

95-percent confidence bands (calculated from500 random samples with replacement). In linewith the earlier results, the nonparametric esti-mate of the conditional mean is stable acrossdifferent population sizes, except for the verybottom of the distribution.27 The estimate seemsto exhibit some slightly inverted U-shape, withsomewhat higher growth rates in the middlerange of population sizes and lower growth atthe ends. If the underlying relation betweengrowth and size is constant, then the estimatewill lie in the 95-percent confidence bands. Thisseems to suggest that, except for some valuesnear the lower boundary, we cannot reject thatgrowth is independent of size. Observe thatbecause the kernel is a fixed function andboundary observations have support only onone side of the kernel, the kernel estimates nearthe boundaries must be read with caution.

In Table B-1 in Appendix B, some furtherdescriptive statistics are reported for growthrates over the entire support of the distribution.Consistent with the kernel estimates, averagegrowth rates seem to be constant, except at thevery bottom of the distribution. We also calcu-late the standard deviation and the InterquartileRange (IQR) of the growth rate. The IQR isdefined as the difference between the seventy-fifth and twenty-fifth percentiles (Q3 � Q1).This provides an indication of the variation ingrowth rates. For the largest 100 cities, growthrates vary less, whereas the smallest 100 citiesexhibit higher variation in growth rates. The

25 Results below have been replicated using the Gaussiankernel and reveal no differences with those using the Ep-anechnikov kernel.

26 See Wolfgang Hardle (1990).

27 At the bottom of the distribution there is also morevariation in growth rates (see IQR calculations below).Because the confidence bands impose a requirement overthe entire domain of the size distribution, the width of thebands is likely to be affected by the variation at the bottom.

FIGURE 6. CONTOUR PLOT: KERNEL DENSITY ESTIMATION

OF NORMALIZED GROWTH RATES BY DECILE OF THE

SIZE DISTRIBUTION

FIGURE 7. KERNEL ESTIMATE OF POPULATION GROWTH

(BANDWIDTH 0.5)




standard deviation of the growth rate of thelargest 100 cities is an order of 4 to 5 timessmaller compared to the entire sample (0.158versus 0.729). Also for the IQR there is a de-crease at the top of the distribution (0.154 ver-sus 0.199), but to a lesser extent than in the caseof the standard deviation. This seems to indicatethat the tails of the distribution of growth ratesof the top 100 cities are not as fat. For thesmallest 100 cities, the variation in growthrates as measured by the IQR increases28

2.5 times relative to the IQR for the entiresample (0.493 versus 0.199). For the remainderof the support of the size distribution, the IQRof growth rates is more or less constant for allsizes, except for the bottom decile of the sizedistribution. Figure B-1 in Appendix B plotsthe IQR for each decile. Observe the sharp in-crease in the IQR at the bottom decile of thedistribution (0.297 versus 0.199 for the entiresample).

The proportionate growth process that satis-fies Gibrat’s law and that gives rise to a log-normal distribution is also characterized by asize-independent variance. The kernel estimateof the variance �2(s) (see Hardle, 1990) is cal-culated as

�2�s� �

n�1¥

i � 1

n

Kh �s � Si ��gi � m�s��2

n�1¥

i � 1

n

Kh �s � Si �

.

As in Ioannides and Overman (2003) for MAs,we find that at the boundaries the variance ofgrowth rates of cities is dependent on size.29 Inparticular, for very small cities with populationsize around 10 inhabitants (with ln size between2 and 3) and for very large cities, the variationin growth rates is markedly different, as re-ported in the IQR calculations above. Figure 7plots the estimated variance30 (bandwidth 0.5)

for 95 percent of the cities in the sample, i.e.,excluding the top and bottom 2.5 percent. Thiscorresponds to all cities larger than 65 (ln is 4.1)and smaller than 56,000 (ln is 10.9). We findthat some outliers have an enormous impact onthe variance. For example, Eagle Mountain,Utah, the fastest growing city in the sample, hasgrown at a rate of 7,090 percent. These outliersalone cause spikes in the variance, which can beseen from observation of the dotted line, repre-senting the kernel estimate of the variance forall observations (for example, around ln sizeequal to 7; observe also that given the band-width of 0.5, the effect of the outliers is con-strained to a distance of 0.5). The solid linerepresents the kernel estimate of the variancefor all observations excluding 9 outliers (obser-vations have been dropped with growth ratesabove 1,000 percent). Without the outliers, thevariance is remarkably stable across differentsizes of cities.

Consider now the parametric growth re-gressions. For the entire size distribution, nosignificant effect of the size of a city is foundon the growth, as confirmed by the followingregression:

S00

S90� 1.102 � 3.75E��08�

S90 � S00

2�0.005� �7E��08��

(n � 19361), where S00/S90, the ratio of thepopulation size in 2000 and 1990, is the grossgrowth rate of the population, and S90 � S00/2is the average of the 1990 and 2000 populations.The coefficient on size is clearly insignificant(standard errors in parentheses). Note that theintercept—a net rate of 10.2 percent—is thecountry-wide growth for the entire sample pop-ulation between 1990 and 2000 and correspondsto an annual population growth rate of (1 �ga)10 � 1.103 or ga � 1 percent. The lack ofsignificance of city growth on size is furtherconfirmed when the dependent variable is thepopulation size in 1990:

S00

S90� 1.103 � 2.3E��09�S90

�0.005� �7.3E��08��

(n � 19361). Finally, also when using loga-rithm of gross growth between 2000 and 1990

28 Though this is not the case for the standard deviation.29 See also Hardle (1990) for a discussion on the reli-

ability of the estimates at the tails of the distribution wherethe density takes on very small values.

30 Observe in Figure 7 that the variance of the fullsample is equal to one. This is due to the fact that wecalculate the variance of normalized growth rates.


as the dependent variable, the coefficient on sizein 1990 remains insignificant. In the latter re-gression, the p-value is 7 percent. When re-gressing ln of the ratio of population sizes on lnaverage size, the coefficient comes out signifi-cant and positive: 0.0223 (0.001) and with anegative intercept �0.104 (0.007). As can bededuced from Figures 5, 6, and 7, this seems toindicate that the size dependence of growthrates at the very bottom of the distribution af-fects the nonparametric estimate.

In summary, all these results seem to providesupport for the fact that city growth is indepen-dent of population size. Some caution is due,however. Growth rates could be calculated onlyfor a sample of 19,361 cities, i.e., those citiesfor which there is an observation in 1990, andthose observations exclude all CDPs. The log-normal distribution in Figure 2 was derivedfrom the size distribution of 25,359 cities in2000, i.e., a distribution with an additional 31percent observations. This unfortunate limita-tion of the data does not permit us to make anydefinite statement about growth over the entiredistribution, most likely until the Census 2010data become available. If the size distribution ofCDPs in the 2000 data can provide any indica-tion (the distribution of CDPs is close to thedistribution of all cities), one may expect theCDP distribution of growth rates not to differtoo much from the rest of the cities.31

II. Zipf’s Law

The question remains: what is the relationbetween Zipf’s law for the truncated distribu-tion and the nontruncated lognormal distribu-tion? As argued in the introduction, the entiresize distribution cannot possibly fit the Paretodistribution. In what follows, the aim is to es-tablish Zipf’s law for the truncated distribution.It will be shown that the estimated coefficienton the Pareto distribution is systematically sen-sitive to the choice of the truncation point. Thiswill be confirmed to be consistent with the factthat the underlying distribution is lognormal.

In the literature on Zipf’s law, the truncationpoint has repeatedly been chosen around 135cities,32 i.e., the 135 largest cities out of the total25,359 cities are included in the census sample.This implies that 99.5 percent of the sample ofcities is dropped, and only the upper 0.5 per-centile of the size distribution is considered (thiscorresponds to 30.2 percent of the population inthe sample, and 22.4 percent of the U.S. pop-ulation).33 It is well known from the litera-ture that the upper tail of the distribution ofcities fits the Pareto distribution extremely well.The objective of this section is to investigatehow the estimated coefficient of the Paretodistribution changes as the truncation pointchanges. Consider therefore the following anal-ysis of Zipf’s law.

Zipf’s law for cities states that the populationsize of cities fits a power law with exponentapproximately equal to one: the population sizeof a city is inversely proportional to the rank ofthe size of the city. The law has been shown tohold for different definitions of cities, includingboth places and MAs. A city of rank r in the(descending) order of cities has a size S equal to1/r times the size of the largest city in thatcountry. For U.S. cities, the size S of Los An-geles, the second largest, should be 1⁄2 the sizeof New York. The tenth-ranked city, Detroit,should have a size 1⁄10 of New York. Above in

31 It is worth noting that even if growth is shown not tobe exactly proportionate, the limit distribution can still bethe lognormal. Michael Kalecki (1945) generalizes Gibrat’slaw for growth processes that are not exactly proportionate.

32 The 135th largest city in the Census 2000 sample isChattanooga, Tennessee, with a population of 155,554.

33 It is of interest to provide some further descriptivestatistics of the distribution of cities relative to the samplepopulation of 208 million. Half the sample population livesin the largest 647 cities, three quarters in the largest 2,678cities, 95 percent in the largest 10,255 cities, and 99 percentin the largest 17,425 cities.

FIGURE 8. KERNEL ESTIMATE OF THE VARIANCE OF

POPULATION GROWTH (BANDWIDTH 0.5)



Tables 1 and 2, S/SNY � r is reported for the tenlargest cities and MAs respectively. The impli-cation of Zipf’s law is that when the populationsize is plotted against their rank on a logarith-mic scale, an approximately straight line isobtained.

To see that a distribution that satisfies Zipf’slaw is the Pareto distribution, consider a vari-able S, distributed according to the Pareto dis-tribution. Then the density function p(S) and thecumulative density function P(S) satisfy

p�S� �aS� a

Sa � 1 , S S�

P�S� � 1 � �S�

S�a

, S S�

where a is a positive coefficient. Strictly speak-ing, Zipf’s law satisfies Pareto with a � 1. Notethat the rank in the empirically observed distri-bution is given by

r � N� � �1 � P�S��

� N� � �S�

S�a

where N� is the number of cities above the trun-cation point. Taking natural logs, we get thatrank is inversely proportional to size

ln r � K � a ln S

where K � ln N� � a ln S� is a constant.Typically, Zipf’s law is verified by regressing

ln r on ln S. For the upper truncated city sizedistribution, the regression gives a highly sig-nificant estimate of a equal to 1.354:

ln r � 21.099 � 1.354 ln S�0.144� �0.011�

(N� � 135, S� � 155, 554, R2 � 0.991). In Figure9, a scatter plot is presented of ln r against ln Sand, in addition, the linear regression line esti-mated above is plotted. This plot can be inter-preted as a transformation of the cumulativedensity function, where on the Y-axis we havethe natural logarithm of the survival function(1 � P(S)) multiplied by N� .

Before considering the sensitivity of the es-timated Pareto coefficient to the truncation inthe size distribution of cities, consider the sizedistribution of MAs. Performing the same re-gression on the truncated distribution of MAs,where the MA at the truncation point is Erie,Pennsylvania, with a population of 280,843, weget

ln r � 17.568 � 0.999 ln S�0.147� �0.011�

(N� � 135, S� � 280,843, R2 � 0.985). Observethat for MAs, the estimated coefficient a isnearly exactly equal to 1, as originally describedby Zipf (1949). Unfortunately, the fact that a isequal to 1 is highly sensitive to the choice of thetruncation point in either direction: for N� � 276(the entire MA sample and roughly double theoriginal), a � 0.850, and for N� � 67 (half theoriginal sample), a � 1.114). Figure 10 reportsa scatter plot of the MA size distribution and theregression lines for the different sample sizes.At the truncation point of N� � 135, the sam-ple ensures a perfect fit with Zipf’s originalobservation.34

34 A referee pointed out that the pervasiveness in theliterature of the truncation point at N� � 135 and the result-ing estimate of a power coefficient exactly equal to 1, aspredicted by Zipf’s law, is due to a remarkable historicalcoincidence. The literature, starting with Krugman (1996),used the Statistical Abstract of the United States, whichshows only the data for the top 135 cities.

FIGURE 9. CITY SIZE DISTRIBUTION AND LINEAR

REGRESSION LINE



The estimated coefficient on the Pareto dis-tribution is clearly sensitive to the choice of thetruncation point. Moreover, the dependence ofthe estimate is systematic: the lower the trunca-tion point (i.e., the larger the sample size), thelower the estimated coefficient of the Paretodistribution.35 The same is expected to be truefor the size distribution of cities. In what fol-lows, it is shown that a theoretical justificationfor the fact that the estimated Pareto coefficientis increasing for an increasing truncation pointis given by the fact that the underlying sample isdistributed lognormal.

Consider the lognormal density function ��as given in equation (1). To simplify notation,let x � ln S, and denote the normal cumulativedensity function by �(x). Now consider thetruncated lognormal distribution at truncationpoint x� � ln S� . Then the cdf of the truncatedlognormal is

��x� � ��x� �

1 � ��x� �.

As before, let N� be the sample size of thetruncated distribution. Then the rank can bewritten as

r � N� � �1 ��x� � ��x� �

1 � ��x� � �� N� � �1 � ��x�

1 � ��x� ��and taking logs

ln r � ln�N� � �1 � ��x�

1 � ��x��or

(2) ln r � lnN�

1 � ��x�� ln�1 � ��x��.

If the underlying true distribution is the log-normal, then from the last equation, the relationbetween ln r and ln S will not be linear. As aresult, the hypothesis that size is everywhereinversely proportional to rank (Zipf’s law) is notcorrect. In particular, ln(1 � �(x)) is not linearin x � ln S. Calculating the derivative of theterm that depends on x in equation (2) gives

d

dxln�1 � ��x��

��x�

1 � ��x�

� �h�x�

which is the negative of the hazard rate. It iseasily verified that the hazard rate for the cor-responding lognormal distribution with samplemean and variance � � 7.28, � � 1.75 isstrictly increasing over the entire domain (andpositive by definition). The plot of the hazardfunction h(x; �, �) is given in Figure 11.

A strictly increasing hazard rate implies thatthe second derivative of the term ln(1 � �(x))is strictly concave, i.e., d2/dx2ln(1 � �(x)) ��h(x) 0. Now, given a decreasing, strictlyconcave function in x, the linear estimate of thisfunction will systematically depend on the trun-cation point: the higher the truncation city size,the higher the estimate of the linear regression.Because an increase in the truncation size im-plies a decrease in the truncated sample popu-lation, the estimate will be decreasing as thesample population increases. This establishesthe following proposition:

PROPOSITION 1: If the underlying distribu-tion is the lognormal distribution �(x; �, �),

35 While the analysis below for cities as opposed to MAsmay be suggestive for the relation between the estimatedPareto coefficient and the truncation point, other expla-nations have been suggested in the literature. In particu-lar, see the review on MAs by Gabaix and Ioannides(2003).

FIGURE 10. MA SIZE DISTRIBUTION AND LINEAR

REGRESSION LINES FOR DIFFERENT TRUNCATION POINTS



then the estimate of the parameter a of thePareto distribution is increasing in the trunca-tion city size (da/dS�) � 0 and decreasing in thetruncated sample population (da/dN� ) 0.

Given this theoretical prediction, Table 3 isconsistent with the fact that the underlying em-pirical distribution function of city sizes (asestablished in the former section) is indeed log-normal. Estimated parameters are reported forthe regression

ln r � K � a ln S, N� , S� .

Not only are the estimates of a highly sensitiveto the choice of the truncation point, they are soin a systematic fashion, consistent with the factthat the underlying distribution is lognormal.For increasing S� (decreasing N� ), a is systemat-ically increasing.36

Finally, Figure 12 provides a plot of the datafor the entire size distribution (ln r against ln S),and the regression lines as obtained from thelinear regressions reported in Table 3.

III. A Theory of Local Externalities

The empirical analysis above supports thehypothesis that the underlying mechanism thatgoverns the evolution of the size distribution of

cities satisfies Gibrat’s proposition. Growthrates of cities are observed to be proportionateto city size, and the limiting size distribution islognormal. From an economics viewpoint, thequestion remains how economic forces can leadto such population dynamics. While there maybe many idiosyncratic reasons why individualsdecide to live in one city over another, orchoose to move between cities, it is hard to denythat economic forces are a major determinant inpopulation mobility. Cities like Detroit andPhiladelphia have seen a significant drop inpopulation, while at the same time experiencinga serious decline in their manufacturing indus-tries. In Silicon Valley on the contrary, citieshave seen higher-than-average population growthrates over the 1990s (and often equally lower-than-average rates since 2000). Cupertino City,home to technology companies like Apple andHP, saw a population growth rate of 25 percentbetween 1990 and 2000, two and a half timesthe national average. There is no doubt thatthe economic impact of the technology boomin those cities has contributed to attractingcitizens.

We therefore propose a general equilibriumtheory that incorporates those differences intechnological change across cities. In addition,the main reason for the existence of cities andthe determination of population boundaries isthe presence of local externalities within cities.Firms and workers locate in cities because thereare positive spillovers in production fromworkers,37 consumers, suppliers, and even com-petitors. Without those external benefits, firmswould locate in rural areas where propertyprices are much lower. At the same time though,land and space are in limited supply. All firmsand workers ideally want to locate as closelytogether as possible, but that tendency is slowedby a counteracting force. Not only does a higherpopulation lead to higher property prices (whichhas been experienced extensively in citieslike Cupertino), the presence of more inhabit-ants causes congestion. There is a negative ex-ternal cost due to increased commuting time.

36 The GI s.e. is the Gabaix-Ioannides (2003) correctedstandard error. They show that the nominal OLS s.e. under-estimates the true standard error due to the positive corre-lations between residuals caused by the ranking procedure.

They derive that for large N, the approximate true standarderror is a(2/N)1/2.

37 Guy Dumais et al. (1997) provide evidence that shar-ing a common pool of workers is the main reason whyindustries locate together.

FIGURE 11. HAZARD RATE FOR THE LOGNORMAL

h(x; �, �)


http://pubs.aeaweb.org/action/showImage?doi=10.1257/0002828043052303&iName=master.img-010.png&w=191&h=153

Citizens in large cities must devote part of theirleisure time to nonproductive but work-relatedcommuting.

The model, like Lucas and Rossi-Hansberg’s(2002) theory of the internal structure of cities,incorporates those two counteracting externalforces. The current model does not explicitlymodel internal geographic heterogeneity of thecity. Because in Lucas and Rossi-Hansberg(2002) citizens obtain the same utility over dif-ferent locations, it is without loss of generalitythat citizens within a given city are consideredidentical. The main objective is to understandeconomic and population differences betweencities, rather than within cities. The city is there-fore not considered in isolation, but rather ex-periences population mobility from and todifferent cities. The main aim is to extend thework in this literature on the internal structure

of cities and allow for competition betweencities of different sizes. The space in whichheterogeneous cities are considered is thereforethe size space rather than a given geographicalspace.

Define an economy with local externalities C.Time is discrete and indexed by t. Let there bea set of locations (cities) i � I � {1, ... , I}.Each city has a continuum population of sizeSi,t, and the total, country-wide population sizeS � ¥I Si,t. All individuals are infinitely livedand can perform exactly one job. Let Ai,t be theproductivity parameter that reflects the techno-logical advancement of city i at time t. The lawof motion of Ai,t is Ai,t � Ai,t�1(1 � �i,t). Eachcity experiences an exogenous technologyshock �i,t. Let �t denote the vector of shocks ofall cities. The city-specific shock is symmetricand is identically and independently distributedwith mean zero, and 1 � �i,t � 0.38 On ag-gregate, there is no growth in productivity.39

38 This law of motion implies that ln(Ai,t) follows a unitroot process. In empirical applications, the presence of aunit root often cannot be rejected. In the real business cycleliterature, for example, using the Solow residual to measureTFP, the point estimates found on the persistence parameter� in Ai,t � (Ai,t�1)�(1 � �i,t) cannot be rejected to bedifferent from 1 (see, for example, Robert G. King andSergio T. Rebelo, 1999).

39 Recent work by Rossi-Hansberg and Wright (2004)and Gilles Duranton (2002) has proposed different growthmodels that can explain Zipf’s law. Rossi-Hansberg andWright (2004), for example, have shocks at the industrylevel. The implication is that while industry size is persistentover time, the size of a given city is not related to that ofindustries, as industries and workers can relocate each pe-

TABLE 3—PARETO COEFFICIENT REGRESSIONS

Truncation point Estimates

R2N� S� City K (s.e.) a (s.e.) (GI s.e.)

135 155,554 Chattanooga (city), TN 21.099 1.354 0.99(0.144) (0.011) (0.165)

2,000 19,383 Lyndhurst (CDP), NJ 20.648 1.314 0.997(0.017) (0.002) (0.042)

5,000 6,592 Attalla (city), AL 18.588 1.125 0.985(0.019) (0.002) (0.023)

12,500 1,378 Fullerton (city), NE 15.944 0.863 0.961(0.014) (0.002) (0.011)

25,000 42 Paoli (town), CO 13.029 0.534 0.860(0.010) (0.001) (0.005)

Notes: Dependent variable: Rank (ln). s.e. standard error; GI s.e. Gabaix-Ioannides (2003)corrected standard error (a(2/N)1/2).Source: Census Bureau, 2000.

FIGURE 12. ENTIRE SIZE DISTRIBUTION AGAINST RANK

AND LINEAR REGRESSION LINES FOR DIFFERENT

TRUNCATION POINTS



Firms are identical, consist of one worker,and are infinitesimally small. The marginalproduct yi,t of a worker is composed of the city’sproductivity parameter and the positive localexternality a�(Si,t) from being in a city of sizeSi,t:

yi,t � Ai,t a� �Si,t �

where a�(Si,t) � 0 is the positive external ef-fect. It is increasing in Si,t which reflects the factthat larger cities generate bigger externalities inproduction. The city’s labor market is consid-ered perfectly competitive. Identical firms com-pete for labor of a representative worker, so thewage rate wi,t received by a worker is equal tothe marginal product yi,t. As a result of the factthat larger cities have a higher marginal prod-uct, they also have higher wages.

Workers are endowed with one unit of lei-sure, which can be employed as labor. Denoteli,t � [0, 1] as the amount of labor employed,and 1 � li,t as the amount of leisure. Unfortu-nately, not all labor employed is productive.Because of the negative commuting externality,out of the total amount of labor employed, afraction needs to be devoted to commuting. Asa result, productive labor Li,t � a�(Si,t)li,t,where a�(Si,t) � [0, 1] denotes the negativeexternal effect and a�(Si,t) 0. The larger thepopulation, the lower the fraction of time thatremains to be devoted to productive labor.

The amount of land in a city is fixed anddenoted by H. Land is a scarce resource, and itis assumed that the total stock of land availableis for residential use. The price of land is givenby pi,t. An individual citizen’s consumption ofland is denoted by hi,t.

Citizens have preferences over consumptionci,t, the amount of land (or housing) hi,t, and theamount of leisure 1 � li,t. The representativeconsumer’s preferences in city i at period t arerepresented by

u�ci,t , hi,t , li,t � � ci,t� hi,t

�1 � li,t �1 � � �

where �, , � � � (0, 1). Workers and firms

are perfectly mobile, so they can relocate toanother city instantaneously and at no cost.40

After observing the realization of the vector oftechnology shocks �t in each period t, citizenschoose location i to maximize the discountedstream of utilities. Because all citizens are iden-tical, each of them should obtain the same util-ity level. Moreover, because there is noaggregate uncertainty over different locations,and because capital markets are perfect, thelocation decision in each period depends onlyon the current period utility. The problem istherefore a static problem of maximizing cur-rent utility for a given population distribution,and the population distribution must be suchthat in all cities, the population Si,t equatesutilities across cities. In what follows, given apopulation size Si,t in city i, agents choose con-sumption bundles {ci,t, hi,t, li,t} in a Walrasianeconomy with local externalities. The “popula-tion market” clears if all Si imply that the equi-librium utilities are the same across cities.

Given Si, any individual maximizes utilityu(ci,t, hi,t, li,t) subject to the budget constraint(where the tradeable consumption good is thenumeraire, i.e., with price unity)

max�ci,t ,hi,t ,li,t

u�ci,t , hi,t , li,t ; Si� � ci,t� hi,t

�1 � li,t�1 � � �

s.t. ci,t � pi,thi,t � wi,tLi,t

where wi,t � Ai,ta�(Si,t) and Li,t � a�(Si,t)li,t. Acompetitive equilibrium allocation for thisproblem satisfies the first-order conditions(where � is the Lagrange multiplier)

�ci,t� � 1hi,t

�1 � li,t �1 � � � � � � 0

ci,t� hi,t

� 1�1 � li,t �1 � � � � �pi,t � 0

�1 � � � �ci,t� hi,t

�1 � li,t ��

� �wi,t a� �Si,t � � 0

which, after substituting for the market clearingcondition of the housing market (hi,tSi,t � H)

riod. Their theory therefore also predicts a particular sizedistribution of industries, in addition to that of cities.

40 In the real world, there are obviously transportationand relocation costs. Dumais et al. (1997) find, however,that transportation costs have become far less important.


and for the budget constraint, give the followingequilibrium prices

p*i,t � Ai,t a� �Si,t �a� �Si,t �Si,t

H

w*i,t � Ai,ta � �Si,t �

and the equilibrium allocation

c*i,t � �Ai,t a� �Si,t �a� �Si,t �

h*i,t �HSi,t

l*i,t � � � .

Observe that wages are higher in cities withpositive productivity shocks (higher Ai,t) andthey are also higher in cities with a larger pop-ulation (due to the externality a�(Si,t)). This isconsistent with the empirical fact that there is anurban wage premium (for an overview, seeGlaeser, 1998). Higher wages are in part offsetby higher property prices, which in equilibriumimplies that less hi,t is consumed, and in part bythe fact that more time must be devoted tocommuting in larger cities.41

Perfect mobility implies that upon realizationof the shocks, citizens must be indifferentacross different locations.42 As a result, in equi-librium, city populations will be such that citi-zens will obtain the same equilibrium utilityU43

u*�Si,t � � u*�Sj,t � � U

for all cities i and j and where u*(Si,t) � u(c*i,t,h*i,t, l*i,t; Si,t). This implies that

Ai,t � a� �Si,t �a� �Si,t �Si,t� /�

� Aj,t � a��Sj,t�a��Sj,t�Sj,t� /�

is equal to a constant for all cities. Denote�(Si,t) � a�(Si,t)a�(Si,t)Si,t

� /� the net local sizeeffect, so Ai,t � �(Si,t) is constant. Then providedthe inverse exists and ��1 is a positive powerfunction, we get

(3) ��1�Ai,t �Si,t � K

where K is a positive constant. After substitut-ing for the law of motion of technology Ai,t �Ai,t�1(1 � �i,t), we obtain

(4) ��1�Ai,t � 1 �1 � �i,t ��Si,t � K.

This expression now helps establish the follow-ing result:

PROPOSITION 2: Let ��1 be a positivepower function. If �(Si,t) is decreasing, i.e.,� 0, then (ex ante identical) cities withlarger shocks will have larger populations:(dSi,t /d�i,t) � 0.

PROOF:Apply the implicit function theorem to equa-

tion (4), then we get that

dSi,t

d�i,t� �

��1��Ai,t � 1

��Ai,t � 1�1 � �i,t��.

Since [��1]� � 1/��, dSi,t/d�i,t is positiveprovided � 0. This establishes the proof.

Consider the following example. Leta�(Si,t) � Si,t

� , and a�(Si,t) � Si,t�� then

��Si,t � � Si,t� � � � /�

� Si,t�

where � � � � � � /�. Note that ��1(Si,t) �Si,t

1/� is a positive power function. As a result, wewrite

41 The amount of time devoted to productive and non-productive labor is the same across city sizes. An argumentcould be made that in larger cities the total labor time islarger than in small cities. Unfortunately, our simple modelwith homothetic preferences cannot account for this. Amore sophisticated model with nonhomothetic preferences,or even with heterogeneous agents, may provide a way tointroduce it.

42 Interestingly, Gibrat (1931) himself discusses wageheterogeneity of a given profession (terrassiers) across dif-ferent cities (Saint-Etienne and Lyon) in the presence ofrandom shocks. Unlike the perfect mobility economy con-sidered here, he assumes the complete absence of mobilityof workers.

43 Like in Lucas and Rossi-Hansberg (2001), the locationchoice of an atomless agent does not change market equi-librium. For an analysis of the impact of individual locationchoices on market equilibrium, see an interesting model byEllison and Drew Fudenberg (2003).


�Ai,t � 1 �1 � �i,t ��1/� � Si,t � K.

From Proposition 2, bigger shocks will lead tolarger cities, provided � 0, i.e. � � � � /� 0. Observe that this condition requiresthat the positive knowledge spillover in produc-tion not be too large: � � � /�. If thepositive spillover is very large, an equilibriumwill involve a degenerate distribution of citieswhere all citizens live in the city with the largestproductivity shock.

From equation (3), it is immediate that

(5) Si,t � 1/��1�Ai,t � � K

for all t. Since Ai,t � Ai,t�1(1 � �i,t) and pro-vided ��1 is a power function, it follows that��1(Ai,t) � ��1(Ai,t�1) � ��1(1 � �i,t) andtherefore

Si,t � 1/��1�Ai,t � 1 � � 1/��1�1 � �i,t � � K

� 1/� � 1�1 � �i,t � � Si,t � 1

after substituting equation (5) evaluated at t �1. We now redefine 1/��1(1 � �i,t) � (1 � �i,t)to get

(6) Si,t � �1 � �i,t � � Si,t � 1 .

The latter equation is exactly what gives rise toGibrat’s law provided shocks are small.

Gibrat’s Law of Proportionate Growth.—Gibrat (1931) (following the discovery byKapteyn, 1903) establishes the law of propor-tionate effect. Consider a stochastic process{Si,t} indexed by place i � 1, ... , I and time t �0, 1, ... , where Si,t is the population size of aplace i at time t. Let �i,t be an identically andindependently distributed random variable44 de-noting the growth rate between period t � 1 andt for place i. If growth is proportionate, then

Si,t � Si,t � 1 � �i,t Si,t � 1

or

Si,t � �1 � �i,t � � Si,t � 1 .

Rewriting and taking the summation, we get

�t � 1

T Si,t � Si,t � 1

Si,t � 1� �

t � 1

T

�i,t

and since for small intervals

�t � 1

T Si,t � Si,t � 1

Si,t � 1� �

Si,0

Si,T dSi

Si� ln Si,T � ln Si,0

or equivalently between any two periods

ln Si,t � ln Si,t � 1 � �i,t .

As a result, it follows that ln Si,T � ln Si,0 ��i,1 � ... � �i,T. From the central limit theorem,ln Si,T is asymptotically normally distributed,and hence Si,T is asymptotically lognormallydistributed, provided the shocks are indepen-dently distributed and small (thus justifyingln(1 � �i,t) � �i,t). In other words, in line withGibrat’s proposition, a proportionate stochas-tic growth process leads to the lognormaldistribution.

As a result, the above establishes the mainproposition of the theory of local externalities:

PROPOSITION 3: Let C be an economy withlocal externalities, let ��1 be a positive powerfunction, and let �(Si,t) be decreasing. Then citysize satisfies Gibrat’s law: the populationgrowth process is proportionate and the asymp-totic size distribution is lognormal.

It is important to note that Gibrat’s law willstill hold for economies with local externalitiesthat in addition have economy-wide externali-ties. In fact, by introducing a technological pa-rameter A, common to all cities, economy-widetechnological progress can be captured whichresults from external effects. This typically de-notes an aggregate measure—most often themean or the max—of the economy-wide tech-nological progress. For Gibrat’s law to hold, it

44 Donald R. Davis and David E. Weinstein (2002) pro-vide some evidence for persistence in the population shocks,in particular when those shocks are extremely big. This wasthe case, for example, in the Japanese cities of Hiroshimaand Nagasaki after they were destroyed by the atom bombin August 1945. For small variance, i.e., “normal” shocks,nonpersistence is justified.


does need to be satisfied that this country-widetechnology parameter is independent of citysize. For the case of the size distribution offirms and not of cities, Eeckhout and Jovanovic(2002) provide evidence that spillovers acrossfirms are dependent on the size of firms. In fact,spillovers between firms are larger for smallerfirms. We do not find such evidence acrosscities. If the true model of the economy is theone proposed in this section, then the propor-tionate population growth process is consistentwith the fact that there are no net local spill-overs across cities of different sizes. That doesnot, of course, rule out the possibility that thereare local spillovers between cities of differentsizes that are geographically close, but the neteffect over the entire distribution cancels out.45

Our results provide no evidence in that direc-tion. Recent work on MAs by Linda H. Dobkinsand Ioannides (2001), however, establishes thatdistance from the nearest higher-tier city (i.e.,the nearest larger city in a higher tier) is notsignificant as a determinant of size and growth.

Ideally, further analysis of the data should bedone. In particular, one would like to analyzethe entire size distribution over time. Thiswould provide an exact description of the mo-ments of the distribution at different points intime which would allow for further verificationof the underlying statistical process. It wouldanswer questions concerning the limit variance,whether Gibrat’s law satisfies exactly a Geo-metric Brownian motion, thus pinning down thedetailed process that generates a limit log-normal size distribution.46 Unfortunately, dueto the lack of available data covering the entiresize distribution, those further analyses are notpossible at this time.

IV. Concluding Remarks

In this paper, a simple but robust underlyingmechanism of population dynamics of all cities

in the United States has been uncovered. Citiesgrow proportionately, i.e., at a stochastic ratethat is independent of city size, and this givesrise to a lognormal distribution of cities. Thisproperty of the stochastic process has beenknown at least since Gibrat (1931). At the sametime, this result can account for what for overhalf a century has been the benchmark stylizedfact of economic geography, that the upper tailof the city size distribution satisfies Zipf’s law.It has been shown that the results confirmingZipf’s law and the corresponding estimates ofthe power coefficient can be obtained even if thetrue underlying distribution is not the Pareto (orZipf) distribution. Estimated power coefficientsare sensitive to the choice of the truncationpoint and are consistently increasing in the trun-cation. Given a lognormal distribution, we haveproposed a simple resolution of one of the majorpuzzles related to the size distribution of citiesbased on Gibrat’s law.

This breakthrough can be made only nowbecause it hinges on the availability of new datain Census 2000 for the entire size distribution.The change in conclusion following the avail-ability of different data does not seem to be anisolated occurrence in science. A similar phe-nomenon has occurred in material sciences, inparticular in the measurement of the atmo-spheric aerosol size distribution.47 Atmosphericaerosols are particles of different componentsfloating in the air. When the measurement ofparticles is restricted to those with the largestsize (often due to the absence of measurementtechnology that can capture the distribution ofthe smaller ones), the resulting observed distri-bution is in fact the truncated distribution and isoften fit to a power law. With the advent ofadvanced measurement technology, however,smaller particles and hence the total size distri-bution can be measured. Knowledge of theentire atmospheric aerosol distribution is impor-tant mainly because, for humans, inhalation ofsmall aerosol is much more harmful than large.The latter get stopped in the nostrils and throatand never enter the lungs. For the entire sizedistribution of many aerosol types, the distribu-

45 Consider, for example, an economy with many pairsof cities, each pair with one large and one small city. If theshocks between the large and the small city are correlated,but shocks across the many pairs of cities are not, growthwill still be proportionate.

46 Kalecki (1945) extends the class of stochastic pro-cesses that lead to the lognormal distribution. This is moti-vated by his observation that Gibrat’s process leads to alognormal distribution with linearly and unbounded increas-ing variance.

47 I am grateful to Samuel Pessoa for pointing this out.See John H. Seinfeld and Spyros N. Pandis (1997), AminHaaf and Rainer Jaenicke (1980) and William Hinds (1982).


tion is actually lognormal, or a convolution ofdifferent lognormals.

The fact that Gibrat’s proposition is estab-lished concerning the population mobility ofcities is a necessary requirement for an empiri-cally consistent theory of the underlying eco-nomic activity. The second main purpose of thispaper is to propose and solve an equilibriummodel of local externalities where wages andprices guide citizens in their location decision.Consistent with proportionate growth and a log-normal size distribution, the model establishes amechanism of local productivity shocks in thepresence of local externalities and their effect,through worker mobility, on the population sizedistribution of cities.

APPENDIX A: THE SIZE DISTRIBUTION

OF COUNTIES

We investigate the size distribution of coun-ties. While counties may not necessarily be theright geographical unit that an economist isinterested in, they do have the major advantagethat the size distribution of counties comprises100 percent of the U.S. population, i.e., 281million in 2000. According to the Census, coun-ties are described as the primary legal divisionsof most states. For example, voting for mostelections is organized at the county level. Mostcounties are functioning governmental units,whose powers and functions vary from state to

state. Legal changes to county boundaries ornames are typically infrequent.

In 2000, there were 3,141 counties in theUnited States covering the entire population.The ten largest are listed in Table A-1. Thelargest, Los Angeles County, California, had9.5 million inhabitants and the smallest, LovingCounty, Texas, 67 inhabitants. The samplemean (in ln, standard error in brackets) is � �10.22 (0.02) and the standard deviation is � �1.41.

In Figure A-1 we plot the size distribution,together with the normal density �(�, �).

The size empirical density is remarkably simi-lar to the normal. There is somewhat more massnear the mode, and the distribution may be slightlyskewed, but the distribution of county size isnonetheless surprisingly close to lognormal.

TABLE A-1—TEN LARGEST COUNTIES IN THE UNITED

STATES

Rank City Population S SLA/S

1 Los Angeles County, CA 9,519,338 1.0002 Cook County, IL 5,376,741 1.7703 Harris County, TX 3,400,578 2.7994 Maricopa County, AZ 3,072,149 3.0995 Orange County, CA 2,846,289 3.3446 San Diego County, CA 2,813,833 3.3837 Kings County, NY 2,465,326 3.8618 Miami-Dade County, FL 2,253,362 4.2259 Queens County, NY 2,229,379 4.269

10 Dallas County, TX 2,218,899 4.290

Note: SLA/S denotes the ratio of population size relative toLos Angeles.Source: Census Bureau, 2000.

APPENDIX B: ADDITIONAL STATISTICS

OF CITY GROWTH

TABLE B-1—DESCRIPTIVE STATISTICS OF CITY GROWTH

Range of cities Growth rate (non-normalized)

N mean stdev IQR (Q3 � Q1)

All 19,361 0.103 0.729 0.199Top 100 100 0.108 0.158 0.154Bottom 100 100 �0.127 0.671 0.493P10 to P90 15,488 0.106 0.786 0.191

Source: Census Bureau, 1990–2000.

FIGURE A-1. EMPIRICAL AND THEORETICAL DENSITY

FUNCTIONS OF ALL COUNTIES



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