The Mathematics of City Sizes Does Not Prove That Cities Are Nature’s Optimal Choice
Wendover Productions makes a convincing case for why businesses and workers benefit from living near one another. Agricultural surpluses made specialization possible, cities made trade easier, and concentrated labor markets helped businesses become more productive.
Then the video makes a much bigger claim: that cities are natural because billions of individuals independently select the best places to live, and that mathematical similarities between city populations, word frequencies and dolphin sounds demonstrate this.
That conclusion is not established by the evidence.
A mathematical pattern can describe how populations are distributed without explaining why they are distributed that way. It certainly cannot establish that the resulting arrangement is optimal, fair or even the product of unconstrained individual choices.
The distinction matters. Cities are shaped not only by workers and businesses, but also by political power, public investment, infrastructure, housing regulations and historical decisions. Wendover acknowledges some of these forces but gives them far less importance than its theory deserves.
A mathematical pattern is not proof of optimal human choices
The video's central mathematical argument relies on Zipf's law, a relationship in which the population of a city tends to decrease approximately in proportion to its ranking.
Wendover illustrates this with American cities: New York has approximately 8.5 million residents, Los Angeles 4 million, Chicago 2.7 million, Houston 2.3 million and Phoenix 1.6 million. Germany's Berlin, Hamburg, Munich, Cologne and Frankfurt provide another example.
These comparisons are interesting. They are not proof that people have independently discovered the most efficient places to live.
In a 1999 paper published in The Quarterly Journal of Economics, economist Xavier Gabaix demonstrated that Zipf's law can emerge when cities experience proportional growth processes. In other words, a recognizable population distribution can develop through the mathematics of growth itself. The model does not require proof that individuals have identified optimal locations.
There is another important qualification. Jan Eeckhout's 2004 paper, Gibrat's Law for (All) Cities, examined the entire distribution of American settlements using Census 2000 data. He found that the full distribution was lognormal rather than Pareto, the distribution associated with Zipf's law. The distinction matters because the familiar Zipf pattern describes the upper end of the distribution, not necessarily every settlement.
The statistical evidence is even more revealing when examined across different disciplines.
In a 2009 paper in SIAM Review, Aaron Clauset, Cosma Shalizi and M. E. J. Newman examined 24 datasets to test whether their distributions genuinely followed power laws. They found that city populations could plausibly follow a power law, but could also be explained by lognormal and stretched-exponential distributions. Word frequencies in Moby-Dick were the only dataset in their analysis for which the power-law explanation was truly convincing against the alternatives.
The dolphin comparison is not necessarily fabricated. A 1999 study in Animal Behaviour identified a Zipf-type distribution in bottlenose dolphin whistles, using 600 classified whistles from eight adult dolphins. However, the researchers treated this as a statistical measure of vocal repertoire structure, not proof of a universal mechanism governing every system that produces a similar distribution. They also emphasized the need for more extensive data to understand the internal organization of dolphin communication.
This is the elementary distinction Wendover misses: the same mathematical pattern can occur in different systems without demonstrating that those systems share the same underlying cause.
A distribution describes what happened. It does not automatically explain why.
Political power can manufacture urban growth
The most revealing contradiction in the video comes from its own examples.
Wendover argues that cities generally maintain their historical rankings unless an external influence intervenes. It then identifies three examples: British sovereignty helped Hong Kong grow, gold accelerated San Francisco's development, and the establishment of the US government transformed Washington, DC.
Washington is particularly revealing. The video itself describes the US government as a new industry that changed the city's position.
But government is not an incidental influence on urban development. Political institutions can concentrate employment, investment, infrastructure and opportunity in particular locations.
A particularly relevant study published in July 2026 provides new evidence.
In The Political Geography of Cities, Richard Bluhm, Christian Lessmann and Paul Schaudt examine hundreds of administrative and capital-city reforms between 1987 and 2018. They find that gaining capital-city status increases urban growth in the medium term, with benefits extending to nearby cities.
The researchers identify two mechanisms: educated people migrate toward newly designated capitals, and public and private investment increases there.
Read the 2026 American Economic Journal study.
This is not simply a theoretical possibility. It is a measurable relationship between political decisions and subsequent urban growth.
The political explanation is not new, either.
In their 1995 Quarterly Journal of Economics paper, Alberto Ades and Edward Glaeser examined urban concentration across 85 countries. They found that central cities in dictatorships were, on average, 50% larger than their counterparts in democracies. Their analysis concluded that political factors were an important cause of urban concentration, rather than merely a consequence of it.
Read Ades and Glaeser's research.
James Davis and J. Vernon Henderson reached another important finding in their 2003 Journal of Urban Economics study. Their research found that investment in infrastructure connecting different regions helps redistribute urban growth away from dominant cities toward smaller regional centres. Democratization and fiscal decentralization also contribute to this process.
Read Davis and Henderson's research.
These findings do not mean governments manufacture every city or that individual decisions are irrelevant. They establish something considerably more important: the distribution of urban populations responds to political institutions and deliberate government decisions.
Wendover treats these influences as exceptions to an otherwise natural process. The research shows why they belong much closer to the centre of the explanation.
People choose from a menu that governments help create
The video explains that businesses cluster because they want access to skilled workers, suppliers and customers. This is a sound economic argument.
But the location decisions of businesses and residents do not happen in an unrestricted market.
Governments determine land-use rules, approve developments, build transport networks, impose taxes and establish restrictions on what can be constructed.
An OECD study published in 2017 examined the relationship between local taxation, land-use regulation and urban development. Its findings are particularly relevant.
The research identifies several ways policy changes settlement patterns:
- Restrictive urban containment policies can contribute to housing shortages.
- Transportation network expansion can encourage suburbanization and reduce central-city populations.
- Minimum-lot-size restrictions can prevent development, raise property values and encourage urban sprawl.
- Local fiscal policies influence whether land is developed for residential or commercial purposes.
Consider what this means for Wendover's argument.
A worker might prefer to live near a major employment centre but be unable to afford the available housing. A family might live farther away because planning restrictions limit construction in the city. A business might establish itself in a particular location because public infrastructure makes that location commercially viable.
These are real economic decisions, but they are not evidence that people have independently selected the optimal settlement pattern.
They are decisions made within a system of rules, prices and available alternatives.
The distinction is especially important when the video describes people collectively selecting the best places to live. A person choosing between three expensive neighbourhoods has not necessarily chosen the best possible location. They have chosen among the locations they can access.
Even the definition of a city changes the mathematics
There is another problem with the video's population comparisons: what exactly counts as a city?
This sounds like a technical detail. It is not.
A city can be measured using its administrative boundaries, its continuously built-up area or its wider metropolitan labour market. These definitions can produce substantially different population figures.
The OECD and European Commission addressed this problem in their 2020 report, Cities in the World: A New Perspective on Urbanisation.
The report explains that national definitions of urban areas vary considerably. Population thresholds used to classify settlements as urban range from 200 inhabitants in Denmark to 100,000 in China. More than half of national urban and rural definitions rely partly or entirely on administrative designations.
The researchers developed harmonized global definitions to improve international comparisons.
Their findings show that 48% of the world's population lived in cities in 2015 under the report's standardized definition. The broader metropolitan measure also includes surrounding commuting zones. Under that measure, 71% of people in high-income countries lived in metropolitan areas, compared with 33% in low-income countries.
The broad relationship between wealth and urban concentration therefore survives a more consistent measurement approach. But the figures depend on what is being measured.
Read the OECD and European Commission report.
This matters for the video because it uses population rankings and international urbanization comparisons to support a general theory about cities.
Without clearly specifying the geographical units and definitions behind those comparisons, readers cannot determine how much of the observed pattern comes from actual population concentration and how much comes from the way settlements are classified.
A statistical rule is only as meaningful as the measurements used to establish it.
Efficient for whom?
The final problem is the video's use of the word efficient.
Wendover explains several genuine economic advantages of cities: businesses share labor markets, producers specialize, consumers access more goods, and concentrated distribution networks can reduce transportation costs.
But it never establishes that the resulting settlement pattern maximizes overall human welfare.
The video opens by acknowledging that urban life involves higher living costs, pollution, congestion, crime and disease risks. It later concludes that the advantages outweigh the disadvantages.
Where is that calculation?
The Amazon warehouse example measures the benefits of reducing delivery distances. It does not establish that the entire urban arrangement minimizes the combined costs of housing, transportation, health, pollution and infrastructure.
The industrial-clustering examples demonstrate why companies might benefit from locating together. They do not establish that every resident benefits equally from the resulting concentration of economic activity.
A city can be productive for businesses while remaining unaffordable for workers. It can generate substantial economic output while imposing congestion and environmental costs on people who do not receive an equivalent share of the benefits.
Those outcomes do not make cities inherently bad. They make the word efficient insufficient without specifying what is being optimized and whose costs are being counted.
Wendover's final proposition is especially ambitious:
"If you were to have a hand to pick up and organize every human into the most efficient pattern possible, this is what it would look like."
That is not a conclusion established by the preceding examples. It is an assertion about the optimal organization of humanity.
To establish it, one would need to define the objective, measure the competing costs, account for political and institutional constraints, and compare alternative settlement arrangements.
The video does none of these things.
The verdict: An emergent pattern is not an optimal society
I have no difficulty accepting the central economic insight of Why Cities Exist: density creates opportunities for specialization, trade, cooperation and productivity that are difficult to reproduce across dispersed populations.
But Wendover goes beyond that explanation.
It takes a mathematical regularity, interprets it as evidence of natural selection, and then treats the resulting urban arrangement as humanity's most efficient pattern.
The research does not justify that leap.
City populations respond to economic opportunities, but also to political status, public investment, infrastructure, housing regulation and institutional decisions. Even the statistical distributions used to explain their sizes have competing mathematical explanations.
People do make choices about where to live. That does not mean their choices are unconstrained, that the resulting distribution is optimal, or that the people bearing the costs of urban concentration are necessarily the people receiving its benefits.
The most important question is not simply why people gather in cities.
It is who determines where opportunities are concentrated, who can afford to access them, who benefits from the resulting productivity, and who pays for the costs.
Wendover explains why cities can be economically powerful. It does not prove that the cities we have are the cities humanity would choose under genuinely optimal conditions.
That is a much larger claim. And the mathematics does not get it across the finish line.
References
- Why Cities Exist — Wendover Productions, YouTube, 2018. The original video and its argument about agricultural surpluses, urban concentration, Zipf's law and economic efficiency. The transcript discloses Squarespace sponsorship; no evidence examined connects the sponsor to the argument.
- Zipf's Law for Cities: An Explanation — Xavier Gabaix, The Quarterly Journal of Economics, August 1999. Explains how proportional city growth can generate the observed upper-tail population distribution without establishing optimal location choices.
- Gibrat's Law for (All) Cities — Jan Eeckhout, American Economic Review, December 2004. Uses Census 2000 data to demonstrate that the full distribution of American settlements is lognormal rather than Pareto.
- Power-Law Distributions in Empirical Data — Aaron Clauset, Cosma Shalizi and M. E. J. Newman, SIAM Review, November 2009. Demonstrates why alternative statistical distributions must be tested before interpreting apparent power-law relationships.
- Quantitative Tools for Comparing Animal Communication Systems — Brenda McCowan, Sean F. Hanser and Laurance R. Doyle, Animal Behaviour, 1999. Establishes a Zipf-type pattern in bottlenose dolphin whistle repertoires while explaining the limitations of that statistical measure.
- The Political Geography of Cities — Richard Bluhm, Christian Lessmann and Paul Schaudt, American Economic Journal: Applied Economics, July 2026. Provides recent evidence that gaining capital-city status increases urban growth through migration and increased public and private investment.
- Trade and Circuses: Explaining Urban Giants — Alberto Ades and Edward L. Glaeser, The Quarterly Journal of Economics, February 1995. Documents the relationship between political regimes and urban concentration across 85 countries.
- Evidence on the Political Economy of the Urbanization Process — James C. Davis and J. Vernon Henderson, Journal of Urban Economics, January 2003. Examines how infrastructure investment, democratization and fiscal decentralization influence the distribution of urban populations.
- Local Taxation, Land Use Regulation, and Land Use — Hansjörg Blöchliger, Christian Hilber, Olivier Schöni and Maximilian von Ehrlich, OECD Economics Department Working Papers, March 2017. Documents how taxation, zoning, transport investment and local political incentives influence housing supply and urban development.
- Cities in the World: A New Perspective on Urbanisation — OECD and European Commission, OECD Publishing, June 2020. Establishes harmonized global definitions of cities and metropolitan areas and demonstrates how measurement choices affect international urbanization comparisons.
Define Zipf’s law more preciselySeparate the video’s claims more clearly