Artificial increasing returns to scale and the problem of sampling from lognormals

Andrés Gómez-Liévano, Vladislav Vysotsky, José Lobo

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


We show how increasing returns to scale in urban scaling can artificially emerge, systematically and predictably, without any sorting or positive externalities. We employ a model where individual productivities are independent and identically distributed lognormal random variables across all cities. We use extreme value theory to demonstrate analytically the paradoxical emergence of increasing returns to scale when the variance of log-productivity is larger than twice the log-size of the population size of the smallest city in a cross-sectional regression. Our contributions are to derive an analytical prediction for the artificial scaling exponent arising from this mechanism and to develop a simple statistical test to try to tell whether a given estimate is real or an artifact. Our analytical results are validated analyzing simulations and real microdata of wages across municipalities in Colombia. We show how an artificial scaling exponent emerges in the Colombian data when the sizes of random samples of workers per municipality are 1% or less of their total size.

Original languageEnglish (US)
Pages (from-to)1574-1590
Number of pages17
JournalEnvironment and Planning B: Urban Analytics and City Science
Issue number6
StatePublished - Jul 2021


  • Urban scaling
  • extreme value theory
  • heavy-tailed distributions
  • increasing returns to scale
  • lognormal distribution
  • urban wage premium

ASJC Scopus subject areas

  • Architecture
  • Geography, Planning and Development
  • Urban Studies
  • Nature and Landscape Conservation
  • Management, Monitoring, Policy and Law


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