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Preface: This page contains the main parts of a paper which concluded a project that I did in 2006 at King's College under the dedicated supervision of Dr. Bruce Malamud, who then thought that it was worth developing further and submitting for publication. Nearly six years later, and after several attempts to complete it, I decided it is time to post something from it online, in case it could be useful for someone.

The main idea of the project was to apply a complex-systems approach to human migration, in order to develop methods for identifying organising principles for migration patterns, or proving that no such principles actually exist. This is useful, for example, to test the old rational-choice theory according to which people migrate to maximize their profits. As this paper shows, this theory finds no support in migration data.

The data used in this paper is old by now, and there are many things in it that I would write differently today, particularly regarding the implications of my findings. However, the statistical method might have value by itself as a tool for researching migration and possibly other complex phenomena. If there are any questions or comments on this paper, feel free to write to me: yoni.eshpar@gmail.com.


A Spatio-Economic Analysis of Migration

by Yoni Eshpar (2006)

Popular fears concerning unregulated immigration often have political and psychological roots, yet they are frequently cloaked by economic justifications; namely that open borders would result in unsustainable numbers of people moving from poor to rich countries. The common assumption is that while on the international scale migration must be regulated to avoid an economic and social disaster, on a national scale it can be (and must be, according to article 14 of the UN Declaration of Human Rights) left to its own self-balancing mechanisms. For the purpose of corresponding with this notion, this paper proposes a method of simplifying migration patterns for their analysis in relation to economic conditions. The method will be applied to free migration systems in the national and international levels, and will suggest that, on both levels, migration patterns have no apparent correlation with economic conditions.

Migration as a complex system
Human migration is a complex system within which individuals, families and groups relocate from one habitat to another. An extensive body of research has been dedicated during the last two decades to the study of this phenomenon using fractal mathematics and chaos theory. Wong & Fortheringham (1990) used such tools for the analysis of Rural-to-Urban migration, and Frankhauser (1998) used them for the study of urban agglomerations. Much attention has been dedicated to the application of chaos theory to population growth and to its relation to economic growth (Day 1994 and Paskwetz & Feichtinger 1995 in Blanchet 1998). However, those models were assuming a closed system and thus the phenomenon of migration was excluded from the analysis. Migration has been modeled, as were other group behavior phenomena, using viability theory (Aubin 1990 in Courgeau 1998), in an attempt to formulate the statistical probability that an individual would decide to migrate from one place to another based on a rational calculation of various factor. These methods of analysis, therefore, maintain the basic theoretical assumption, that of 'rational choice', that dominated the study of migration during the 1950s and 60s (Massey et al. 1993).
The 'rational choice' approach to migration emerged together with neoclassical economics. John Hicks, winner of a Nobel Prize for economics, stated as early as 1932 that "…differences in net economic advantages, chiefly differences in wages, are the main cause of migration" (Greenwood 1975 p.397). This idea is central to neoclassical economics and has provided the foundation for a large variety of economic theories, all of which have followed the notion that individuals make migration decisions according to universal economic principles of profit maximisation or risk minimisation. In this paper, this body of thought will be referred to as the 'rationalist thesis'.
It is, however, rather problematic to apply such actor-oriented formulae to migration considering the various considerations, preferences, and circumstances which determine migration decisions in each particular case. One person would choose to move to a warmer climate, while another would escape to a cooler place; one would migrate away from his or her family, while another would choose to move closer to them; and so on and so forth. The approach proposed in this paper, therefore, does not attempt to establish a universal formulation of the motivations to migrate, but rather to examine the extent to which the diverse, and often contradicting motivations that shape migration patterns serve as a self-balancing mechanism, i.e. enabling the movement of people to remain in levels that are socially and economically sustainable without the need for coercive regulation.

Migration as a multiscale phenomenon
Migration takes place on varying scales. Some moves may be over a distance of a few kilometers, others may be across oceans and continents. The available data on migration is arranged according to the hierarchy of formal administrative zones. For example, a person living in London belongs in a London borough, which belongs in the region of Greater London, which belongs in the United Kingdom, which belongs in the European Union. Each of these zones is a field of free migration by its own and a unit of reference in the field which contains it (see Figure 1).

Figure 1. Scales of Migration Fields.  The geographical space is divided into administrative units, each being a migration field by itself and a unit of reference for the migration field containing it. (Source: the author).

The term 'free migration' requires clarification. Freedom here is seen, following Isaiah Berlin's famous distinction, as a 'negative freedom' rather than a 'positive freedom' (Berlin 1969). Meaning, people are free from coercion to migrate or not to migrate, rather than necessarily being free to migrate if they should only choose to do so. It is not hard to imagine cases where a person wants to migrate, has no legal constraints on migration, but is not free to do so for personal, economic, or health-related reasons. The systems that were examined in this study have no legal restriction on the movement of people within them, and did not experience a war or a severe natural disaster on their territory during the timeframe to which the data relates.

The graphic representation of migration data
The basic analytical device for the method being presented here is an arrangement of migration data for a certain field in the form of a square chart with n rows and columns, when n is the total number of zones comprising the field. Figure 1 shows the migration chart of Brazil for 2005. The numbers on the X and Y axes represent the country's administrative zones as origins and destinations of migration respectively, arranged in ascending order according to a ranking criterion, which in this case will be their GDP per capita ratio for 2005. Zone 1, therefore, has the lowest ratio and zone 27 has the highest. The circle sizes in the chart represent the migration ratio M for each pair of origin O and destination D, with M being the total number of migrants mt who moved from O to D as their percentage out of the total population at the origin Po. And so:

(1) 2

In each chart, the circle representing the highest value of M will be marked by an arrow, and its value will be presented for reference. This method for plotting migration data will provide the base for the following analyses presented in this paper.

Figure 2.
The migration chart for Brazil during the period of 1995-2000. The relative sizes of the circles represent M - the percentage of the population who migrated to one destination, out of the total population at the origin in 1996. (Data source: IBGE 2006; image created by the author)

Frequency-size distribution analysis
To begin the analysis, the frequency-size distribution of M, represented by the circle sizes, will be investigated. If individuals are indeed driven by a universal principle of profit-maximisation, and considering they are legally able to move and settle anywhere within the examined field, then we may expected to see many medium or high values of M, especially in a country with such drastic economic gaps between its regions like Brazil. However, as shown in figure 3, the frequency density graph of the M values for Brazil, as well as for three other fields (see charts in Appendix I), prove that there are a greater number of instances (or circles) as M becomes smaller. Interestingly, the three fields follow a rather similar frequency density curve. The frequency density of M in the EU is closer to a power-law distribution; meaning that it has relatively fewer medium-range and more low-range M values compared to the two other fields. Although it would require a broader comparative study to arrive at any universal conclusions, these figures may still imply that migration rates across national boundaries, even when they are formally porous, is generally lower than within them.
Figure 3. Logarithmic frequency density graphs for migration ratio (M) values in the charts for the US (1995-2000), Brazil (1995-2000), and 12 countries of the European Union (2001). The M values have been normalized by dividing them by the length in years of the time period they refer to. The number n of instances in each bin of M has been divided by m - the total number of M values in each chart. (Data Sources: IBGE 2006; Eurostat 2006; UK National Statistics 2006; US Census Bureau 2006)

In search of directionality
So far the frequency of M values was examined, yet their positions in the chart were not included in the analysis. The frequency distribution of M does not inform us as to what extent migration patterns correlate with economic conditions. The rationalist thesis would suggest an organizing principle that would operate as follows: the better the economic conditions in a zone are, the less likely it would be for people to migrate away from it, and the more likely it would be for others to migrate to it from poorer zones. When applied to the chart, this principle would mean that statistically it would be more common for a circle to be larger than the circle to its right, and smaller than the one to its left. The same would apply vertically: it would be more likely for a circle to be larger than the circle below it, and smaller than the one above it.
Testing the rationalist thesis requires a method for calculating the statistical relationship between circles both horizontally and vertically. In order to do so, a horizontal and a vertical vector will be calculated for each circle, represented as V and W respectively. The horizontal vector of circle M[O,D] will be the average of the difference in value between the circle to its right and itself, and the difference in value between itself and the circle to its left; as shown in the following formula:

(2)        5

The vertical vector will therefore be calculated as follows:

(3)        6

For circles that have only one valid neighbor (either being located on one of the edges of the chart or next to a non-valid position where O=D) the vector will be calculated using the single valid neighbor. The sum of the horizontal vectors in the entire chart will then be divided by the total number of circles, to arrive at the average horizontal vector. The same procedure will be conducted with the vertical vectors, and a combined average vector for the chart will be calculated.
Figure 4 shows the combined average vectors of the same charts discussed earlier with the addition of Spain. It also includes a representation of a 'rationalist vector'. Clearly, neither of the cases presented here follow the rationalist thesis as an organizing principle. The chart for Spain has a 'rational' directionality as far as destinations are concerned, but is nearly balanced horizontally. Brazil, the US and the EU, however, show a rightward orientation, which implies that the probability of migration tends to increase in relation to higher GDP/capita at the origin. Brazil even has a strong pull downwards, very much in contrast with the rationalist thesis.

Figure 4. The average directionality vectors for the migration charts of the US, Brazil, the EU, and Spain. (Data Sources: IBGE 2006; Eurostat 2006; UK National Statistics 2006; US Census Bureau 2006)

The absence of a 'rationalist directionality' in the charts does not, however, refute the thesis altogether, but rather proves that it does not operate across the field as an organizing principle for the spatial distribution of M. It would still be possible to have a concentration of higher M values in the top-left area of the chart – the 'rationalist quarter' – without average directionality pointing in that direction at all. Another method of analysis would therefore be needed to test migration data against the rationalist thesis.

Spatial analysis using incremental aggregation
The purpose of the spatial aggregation analysis is to determine to what degree the positions of the existing M values of each chart correspond with the rationalist expectation that they would statistically be higher the closer they are to the top-left corner of the chart. The spatial aggregation method calculates the sum of all the M values in a square area of size r x r, which starts from the top-left corner of the chart. Variable r represents the number of values, or circles, in each side of the square, and therefore is a number between 1 and the total number of zones in the chart (n). As the value of r is increased incrementally by 1, as shown in figure 5, the sum S of the M values within the square will be plotted as a function of r. The top-left corner of the chart is the starting point of the process, and therefore when r = 1 then S1 = M[1,n]. When r = 2 the four top-left M values are summed up, and so:

(4)                         S2= M[1,n] + M[2,n] + M[1,n-1] + M[2,n-1]

Eventually, when r = n then Sn equals the sum of all the M values in the chart.

Figure 5. The incremental aggregation method.

The same procedure is applied to two other reference charts (see figure 6) that are based on the original data. The first represents an equal spatial distribution, and so all of its M values are equal to the average M value of the original chart – Ma. For this reference chart, S will always be equal to (Ma)r. The second reference chart represents a 'rationalist' distribution of the same M values as in the original, and so it has the highest M value at the top-left corner (1,n), and the following values in descending order located at: (1,n–1), (2,n), (1,n–2), (2,n–1), (3,n), etc.

9             10
Figure 6. The two reference charts: the 'equal' and the 'rational'.

To compare the spatial aggregation of the three charts – the actual, the equal, and the rational – the S values of each chart are plotted as a function of r. Figure 7 shows the spatial aggregation graph for the US. The aggregation of the 'actual' data is almost parallel to the exponential aggregation curve of the 'equal' reference chart. It certainly has no resemblance to the 'rational' line.

Figure 7.
Incremental aggregations plot for the migration chart of the US (1995-2000). Data source: US Census Bureau 2006.

It may be useful to reduce the findings of this method down to a single parameter which would help compare between the results from different migration fields. The parameter k would represent the ratio between the sum of all S values in the 'actual' curve and the sum of all S values in the 'equal' curve. If k > 1 than the 'actual' curve runs mostly above the 'equal' curve, which means that the spatial distribution of M values on the chart leans towards to top-left corner, i.e. signifying migration from poorer to richer zones. If k < 1 then the opposite is true; the 'actual' S values are mostly lower than the S values of the 'equal' curve, meaning that the chart's M values tend to concentrate closer to the bottom-right corner, i.e. signifying increased chances for migration from richer zones to poorer zones. The k value for the US chart is 1.02, meaning that its spatial pattern is spread quite evenly across the field. The k calculated in the same manner, but for the 'rational' curve, would be 1.92 – almost twice as much as the k of the 'actual' curve.

Table 1. A comparison between the 'actual' and 'rational' k values calculated by incremental aggregation for Brazil, Spain, the US, and 12 EU members.

Migration field    Time period    k (Actual) k (Rational)
Brazil   1995-2000 1.16 2.05
Spain  2005 1.03 1.90
US   1995-2000 1.02 1.92
EU 2001 0.68 2.17

Data Sources: IBGE 2006; Eurostat 2006; UK National Statistics 2006; US Census Bureau 2006.

Table 1 displays the k values of the actual and rational curves of four migration fields, and thus gives a clear idea as to the different spatial bias of each system. Brazil is the only field that has a significant bias towards the top-left corner; The US and Spain are quite balanced, and the EU leans strongly towards the bottom-right corner. All cases, however, are distant from a totally rational spatial arrangement of their M values.
That Brazil has the highest k value is not surprising, considering that Brazilians have much fewer options for emigration out of the country than Americans or EU citizens have. Many of them are forced by immigration regulations in other countries to remain in Brazil and try their luck in its wealthier cities, causing a severe problem of overpopulation in megalopolises such as Sãu Paulo. To demonstrate this on a different scale of migration, it is possible to assume that if citizens of Mississippi were not allowed to migrate to other states of the U.S., this would have created a problem of overpopulation in Jackson, its capital city. Restrictions on movement, therefore, should be suspected as major causes for population imbalances.

Methodological limitations
The method of analysis suggested in this paper has several limitations which should be noted before any conclusions can be extracted from it.
Data – the data used in this study is taken from different national statistics agencies, and therefore there may be discrepancies between the different definitions of a migrant, especially in the context of internal migration. Moreover, it is worth taking into consideration that there is a certain grey area in migration which is excluded from the data in cases where individuals do not actually reside in the place where they are registered, have more than one place of residency, etc.
Ranking criterion – The ranking of the zones in a migration field according to economic conditions is a crucial element in the analysis method. The indicator that was used in this study as the ranking criterion is GDP/capita, yet it may be argued that this indicator is less relevant for migration decision than, for example, average income, average consumption expenditure, public services spending, employment rates, health and education quality, and many more. Clearly, each of these indicators is highly relevant for migration, and it may be beneficial to formulate a composite ranking criterion that would include several of them.

Implications and suggestions for further research
Migration should be seen as a multiscale phenomenon where economic calculations undoubtedly play an important role in shaping behavior patterns. Yet the rationalist notion that economic conditions are the determining factor for migration cannot be supported by the empirical data, as was shown in this study. Firstly, by calculating the directionality of the migration pattern using a nearest-neighbor method, it was shown that economic conditions do not act as a general organizing principle of migration patterns. Secondly, using incremental aggregation it was shown that the spatial distribution of migration movement on migration charts has no substantial bias towards the rationalist behavior pattern, signified by the top-left area of the chart.
This paper has tried to prove in figures and numbers what most of us know from our everyday life experience; that laissez faire migration self-balances itself within the boundaries of a state without the need of coercive regulation. It also tried to provide some evidence that the same self-balancing forces, and even stronger ones, operate on the larger scale of the EU. Since there is no free international migration system in the world at present which contains both the very rich and very poor countries, any suggestion based on this paper regarding free international migration would be highly speculative. However, the findings in this study show that the migration pattern between EU member states showed a tendency for generally lower migration rates than internal migration within states, and a strong bias towards movements from the richer countries to the poorer countries. These results question the common notion that international migration would tend to be more precarious than internal migration if it was left unregulated. Migration across political boundaries often means moving to an alien culture and a foreign language, and often bares a high cost, not only of longer travel, but also of a longer adaptation period.
The approach presented here can be developed further in various ways. Similar migration charts can be plotted for many more migration fields of varying scales: states, counties, cities, etc. Such charts can also be produced using other ranking criteria, such as weather conditions, crime rate, or pollution levels. It may be illuminating to add a temporal dimension to the analysis in order to study the changes to migration patterns over time.
It is important to note at this point that excluding forced migration from the analysis definitely reduces its direct relevance to the far reaching vision of a world of open borders. War, famine, political persecution and natural disasters can clearly disturb any self-balancing mechanisms that may exist in the peaceful migration fields included in this study, and result in highly unsustainable waves of migrants. It should, however, be kept in mind that forced migrants account to only around 10% of the estimated total number of migrants around the world. The question of migration controls, therefore, remains mainly a moral issue. Should all human beings be free to move anywhere they wish, either away from danger or in pursuit of opportunities, a better standard of living, or even simply for the sake of adventurous exploration and the satisfaction of curiosity? In order to focus the debate on the moral aspect of this question, the economic justifications for immigration controls must be properly examined and, to a large extent, refuted.


Appendix I: Charts and graphs

Additional migration charts:

Chart 1. The United States, 1995-2000. The zones represent the fifty one states in the following acsending order of GDP/cap: Mississippi, Montana, West Virginia, Arkansas, Idaho, Oklahoma, Alabama, Maine, South Carolina, Vermont, Florida, North Dakota, Louisiana, Arizona, Kentucky, South Dakota, New Mexico, Utah, Tennessee, Kansas, Rhode Island, Indiana, Pennsylvania, Iowa, Wisconsin, Missouri, Oregon, North Carolina, Ohio, New Hampshire, Maryland, Nebraska, Georgia, Texas, Wyoming, Michigan, Virginia, Washington, Nevada, California, Colorado, Hawaii, Minnesota, Illinois, New York, Massachusetts, New Jersey, Alaska, Connecticut, Delaware, District of Columbia. Data source: Census 2000, US Census Bureau (2006)


Chart 2. EU, 2001. The zones represent the following states (in their order on the chart): Greece, Portugal, Spain, France, Belgium, Finland, Netherlands, United Kingdom, Ireland, Sweden, Denmark, and Luxembourg. (Data source: Eurostat 2006)



Additional incremental aggregation graphs:

Chart 3. Incremental aggregation graph, Brazil, 1995-2000. (Data Source: IBGE 2006)

Chart 4. Incremental aggregation graph, twelve EU countries (Greece, Portugal, Spain, France, Belgium, Finland, Netherlands, United Kingdom, Ireland, Sweden, Denmark, and Luxembourg), 2001. (Data Source: Eurostat 2006)



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