Rank Size Rule

Rank Size Rule Formula

The specific formula for the rank size rule is 1/n^th, where n equals the ranking of the size of the city in the country. For example, Los Angeles, California is the second largest city in the United States. Therefore, its ranking would be two, and in the rank-size formula, n would equal two.

If we know what rank a city is in terms of the size of the population compared with other cities in the country, we could then see if the largest city corresponds to the ratio to see if the country follows the rank-size rule. Therefore, city sizes are inversely proportional to their rank.

There are arguments that the rank-size rule is more of a statistical phenomenon than a law or universal concept because the rule is sometimes present but certainly not consistently when looking at the distribution of populations among cities in different countries.

Although we usually talk about cities with the rank-size rule, it can be more widely applicable. The figure below shows the populations of countries that follow a pattern of exponential inverse regression as would be expected according to their rank based on the rank-size rule. China and India are anomalies, but every other country's population follows the expected population closely.

Fig. 1 - Rank-Size distribution

Rank Size Rule Example

Let’s look at the size of different cities in the United States. New York City is the largest city in the United States, with a population of around 8.5 million people.

$NYC=8.5LA=3.8\raisebox{1ex}{$8.5$}\!\left/ \!\raisebox{-1ex}{$2=4.25$}\right.4.25=50\%ofNYC\raisebox{1ex}{$3.8$}\!\left/ \!\raisebox{-1ex}{$8.5=$}\right.0.447\times 100=44.7\%$

Let us see if the trend continues in the United States.

Fig. 2 - Chicago skyline

Last one: the fifth largest city in the United States is Phoenix, Arizona.

There are some additional considerations to this rule for why it is best to interpret loosely rather than strictly.

Rank Size Rule Model

The rank-size rule can tell us a lot about a country. It may show us that a country has a higher level of development and inclusive institutions because power and wealth are fairly spread out compared to other models. Rapid growth, as is happening in many countries in Asia, can make following the rank-size rule difficult as a lot of power and investment is in one city, and not enough time has passed for urbanization and development to spread to the entire country.

The rank-size rule tends to work better in countries that have had large urban populations for many centuries, as this gives a lot of time for urbanization to spread out.

Rank Size Rule vs Primate city

The rank-size rule describes a descending order of progressively smaller but independently functional cities, whereas a primate city is overwhelmingly the largest city in a country and the center of most industry, power, and societal trends. If a country has just one major primate city, rather than a collection of cities that follow the rank-size rule, it may indicate that the country is less resilient; the primate city could have a detrimental impact on the rest of the country, whereas power and wealth are more spread out in countries following the rank-size rule.

An example of a country with a primate city would be Thailand, as Bangkok is by far the largest metropolitan area, with the next largest urban area being more than 30 times smaller. Primate cities are often a less desirable model than the rank-size rule, as primate cities typically are a reflection or cause of inequality, uneven development, and a lack of equity. The provinces around Bangkok may have as much as 8-10 times higher GDP per capita than many rural provinces in Thailand.⁴

Fig. 3 - Bangkok skyline

Primate cities tend to be in countries that are developing and experiencing rapid economic growth or countries that have had a large history of inequality and authoritarian rule that have concentrated wealth in the hands of a few, often in the center of political power. However, this is not always the case, and authoritarian countries may follow the rank-size rule as well.

Rank Size Rule Strengths and Weaknesses

The strengths of the rank-size rule are numerous. Most countries that follow the rank-size rule are overall stronger and more developed countries with a long history of urbanization, more even development, and less inequality. A country will be more resilient and secure with a diversity of large cities as it does not all put a majority of its resources and wealth into a single city.

Some weaknesses may be that there is no unified definition of where exactly a city should end and begin, almost making it possible to adjust city boundaries to fit the rule. Another weakness would be that it is a rough estimation of city sizes, and when dealing with large countries, this can mean that the measurement would be off by several hundred thousand people. Lastly, the rank-size rule is only sometimes applicable to some countries, as many countries have primate cities instead; therefore, it would be inaccurate to assume the size of other cities in any given country just because you know the rank and size of one city.