Urban Systems: Rank-Size Rule and Primate Cities

An urban system is the set of cities in a country or region and the links between them, the rank-size rule states that the population of the city of rank n is roughly the population of the largest city divided by n, and a primate city is a city that is much larger than the next largest city, usually at least twice as large, and dominates the country’s economy and politics. In brief: if the largest city has 10 million people, the rank-size rule predicts that the second has about 5 million, the third about 3.3 million and the tenth about 1 million. Countries with a primate city, such as Thailand (Bangkok), France (Paris) and Argentina (Buenos Aires), have a much larger first city than the rule would predict, while countries such as the United States and India are closer to the rank-size pattern.

The size distribution of cities shows how a country’s economy, history and politics are organized in space. This guide explains the rank-size rule and how to test it, the idea of primacy and its causes, compares the two patterns in a table, works through calculations, outlines the history of the theories and corrects common mistakes.

What is the rank-size rule?

When cities in a country are ranked by population (1 for the largest), the population P_n of the city of rank n is often close to P₁ ÷ n. Plotted on a graph with logarithmic axes for rank and population, the points form a straight line sloping downward at an angle of about −1. The relationship is sometimes called Zipf’s law, after the linguist George Zipf, who noted similar patterns in word frequency and in city sizes in 1949. It is a statistical regularity and not a law of nature: many countries show a good fit, especially large countries with a long history of urbanization, such as the United States and India, while others deviate.

A rank-size distribution indicates a mature, well-integrated urban system in which cities fall in a smooth hierarchy with many small places, fewer medium-sized and a few large cities. Its link to central place theory is that each level in the hierarchy provides services to the area around it.

What is a primate city?

The geographer Mark Jefferson defined the law of the primate city in 1939: “A country’s leading city is always disproportionately large and exceptionally expressive of national capacity and feeling.” A primate city is usually at least twice the size of the second city, and often much more. It is generally the political capital as well as the economic, cultural and transport center. Examples include Paris (France), London (the United Kingdom), Bangkok (Thailand), Lima (Peru), Buenos Aires (Argentina) and Seoul (South Korea). Bangkok is many times larger than any other city in Thailand, and the United Kingdom’s second city, Birmingham, is only a small fraction of the size of London. An alternative is the four-city primacy index: the population of the largest city divided by the combined population of the second, third and fourth cities. A value above 1 indicates primacy.

Primacy has advantages, such as economies of scale, a strong global profile and efficient communication. It can also bring problems: congestion, high housing costs, regional inequality, and neglect of other regions, as the best jobs, services and universities are concentrated in one place, which draws migrants and leaves smaller cities behind. A binary (two-city) pattern, such as Rio de Janeiro and São Paulo in Brazil, Sydney and Melbourne in Australia or Mumbai and Delhi in India, has two leading cities of similar size.

What explains the patterns?

• Colonial history: colonial powers often developed one port or capital to export resources, as in many countries of Latin America, Africa and Southeast Asia, which gave a primate pattern.
• Political centralization: strong, centralized states concentrate administration, investment and elites in the capital.
• Size and age of the country: small or young countries tend to have a primate city, while large, old and diverse countries with several centers are more likely to follow rank-size.
• Economic development and transport: early industrialization with several resource centers, and a developed transport network, support a more balanced system. In developing countries, rapid migration to the largest city (for jobs, services and opportunities) increases primacy.
• Government policy: investment in secondary cities, decentralization and the relocation of capitals, such as Brazil’s building of Brasília (1960), may alter the pattern.

Rank-size and primate systems compared

Worked example: predicting and measuring

1. Predicting sizes: the largest city has 12 million people. The rank-size rule predicts P₂ = 12 ÷ 2 = 6 million, P₃ = 12 ÷ 3 = 4 million, P₄ = 12 ÷ 4 = 3 million and P₁₀ = 12 ÷ 10 = 1.2 million.
2. Comparing with real data: a country has the following city sizes: 12, 4.5, 2.5, 1.8 and 1.4 million. The predicted sizes are 12, 6, 4, 3 and 2.4. The actual sizes of the cities ranked 2 to 5 are lower than predicted (4.5 versus 6, 2.5 versus 4, 1.8 versus 3 and 1.4 versus 2.4), which suggests that the first city is relatively large for the system.
3. Primacy ratio: the two-city ratio is P₁ ÷ P₂ = 12 ÷ 4.5 = 2.7, which is above 2, so this is a primate pattern. A country with 12 and 6 million has a ratio of 2.0, consistent with rank-size.
4. Four-city index: 12 ÷ (4.5 + 2.5 + 1.8) = 12 ÷ 8.8 = 1.36, which is above 1, confirming primacy.
5. Share of urban population: if the five cities together hold 22.2 million of an urban population of 40 million, they are 22.2 ÷ 40 = 55.5% of it, and the first city alone has 12 ÷ 40 = 30%.
6. Fitting a slope: on a log-log graph, the points for rank 1 (12 million) and rank 10 (1.2 million) give a slope of (log 1.2 − log 12) ÷ (log 10 − log 1) = (0.079 − 1.079) ÷ 1 = −1.0, the slope expected for the rank-size rule.

How did the theories develop?

The German statistician Felix Auerbach noticed a regular pattern in the size of cities in 1913. Mark Jefferson wrote about the primate city in 1939, drawing on examples from many countries, and Zipf’s Human Behavior and the Principle of Least Effort (1949) popularized the rank-size relationship. Brian Berry’s study of 38 countries in 1961 found that most countries were neither perfectly rank-size nor primate, but intermediate, and that primacy was not clearly related to the level of economic development. Later work, for example by Paul Krugman and Xavier Gabaix, has sought to explain the pattern through models of random city growth. The rule is now used as a benchmark to compare urban systems and to guide discussion of urban policy.

Which mistakes are common?

Mistake 1: Treating the rank-size rule as a law that all countries must obey

It is a statistical tendency with many exceptions. Departures from it are informative about a country’s history and politics.

Mistake 2: Equating a primate city with the capital or the largest city

A primate city is the largest and also disproportionately large and dominant. A country’s largest city is not primate if the second city is of similar size, such as in Australia or Brazil.

Mistake 3: Assuming that primacy is always bad

Primate cities can provide scale and global connections. The problem is excessive concentration and neglect of the rest of the country.

Mistake 4: Using different definitions of “city” without noting it

Results depend on whether city limits, urban agglomerations or metropolitan areas are used, so comparisons should use consistent definitions.