Zipf’s law and South African towns

An important question has exercised the minds of many economist over time: why are there differences in the sizes of different towns? For instance, why are all towns in a region not of equal size? The answer has to do with power laws.

There is no Hobbesian significance in the word ‘power’ – it is just a mathematical term. If the value of some quantity q depends on the value of another quantity x according to a power-law relationship, this means that each time x is doubled, y increases by some constant factor (Ball, 2005).

Vilfredo Pareto was probably the first person to introduce power laws into the social sciences (Ball, 2005). Later, George Kingsley Zipf, an American, collected extensive data about society and showed that many data sets featured probability distributions characterized by power laws. Although normal (Gaussian) distributions and related quantitative methods are still relevant for a significant portion of organizational research, the increasing discovery of power laws signifies that Pareto rank-frequency distributions are pervasive and indicative of nonlinear organizational dynamics (Andriani and McKelvey, 2009). Researchers ignoring Pareto distributions risk drawing false conclusions and promulgating useless advice to practitioners. Most managers face the extremes, not the averages.

Zipf’s law for cities is one of the most conspicuous empirical facts in economics, or social sciences in general (Gabaix, 1999). It has been observed in the city population data of many countries. To visualize Zipf’s law, cities are ordered by population and a graph of the log of the ranks on the y-axis and the log of the populations on the x-axis generate a straight line with slope usually close to -1. In essence, the higher ranked of two cities will have double the population (size) of the lower ranked city.

Does Zipf’s law apply to South African towns? The answer is affirmative as is shown in Figure 1 for the 40 largest towns in the Free State. The slope of the statistically significant line is close to -1, and Zipf’s law apparently applies.

FS Zipf

Figure 1. Free State towns and Zipf’s law

Despite the fact that many hypotheses have been advanced for the existence of Zipf’s law, Krugman (1996), a Nobel Prize winner in economics, lamented: “At this point we are in the frustrating position of having a striking empirical regularity with no good theory to account for it.” Eeckhout (2004) suggested that agglomeration and residential mobility of the population between different geographic locations are tightly connected to economic activity and that the evolution of population across geographic locations is an extremely complex amalgam of incentives and actions taken by many individuals, enterprises and organizations.

We still do not precisely know why Zipf’s law exist but we do know that the size distribution of towns is not haphazard, but a ‘fat-tailed’ orderliness exists in which concepts such as ‘average town’ has to be handled carefully. The warnings of Andriani and McKelvey should, therefore, be heeded by professionals advising district and local municipalities in the Free State, and elsewhere, about local economic development.

References

Andriani, P. and McKelvey B. (2009). From Gaussian to Paretian thinking: Causes and implications of power laws in organizations. Organization Science 20,6: 1053-1071.

Ball, P. (2005) Critical Mass: How One Thing Leads To Another. Arrow Books.

Eeckhout, J. (2004). Gibrat’s law for (all) cities, The American Economic Review: 94, 5, 1429-1451

Gabaix, X. (1999). Zipf’s law and the growth of cities. The American Economic Review: 89, 2, 129-132.

Krugman, P. (1996). Confronting the mystery of urban hierarchy. Journal of the Japanese and International Economies: 10, 399–418.

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