Optimal City Size

Overview

Optimal city size refers to the population level at which two opposing spatial forces exactly balance: the increasing returns from spreading the fixed cost of local public goods over more residents, and the decreasing returns from rising marginal commuting costs as the city grows. At this optimum, the Henry George Theorem (HGT) holds as a corollary: aggregate differential land rent — urban land rent net of the opportunity cost of land in non-urban use — equals the aggregate expenditure on pure local public goods. The concept was formally derived by Richard Arnott and Joseph Stiglitz in their 1979 Quarterly Journal of Economics paper, which remains the canonical statement.

The Arnott–Stiglitz Derivation

Arnott & Stiglitz (1979) model a city with identical individuals and a single pure local public good. Population size determines two countervailing effects:

1. Increasing returns to scale in public goods. A larger population spreads the fixed cost of public goods over more taxpayers, reducing the per-capita cost.
2. Decreasing returns from commuting. As the city grows, residents at the urban fringe face longer commutes, raising the marginal cost of adding one more resident.

At the optimal population, these two forces exactly offset. As a corollary of the Product Exhaustion Theorem applied to a constant-returns economy, aggregate land rent at this optimum equals aggregate public-goods spending — the Henry George Theorem. The intuition is that public goods make a location more desirable, and that desirability is capitalised into land rent; collecting the rent therefore recovers precisely what the public spending created, a self-financing mechanism requiring no tax on labour or capital.

Generalization and Practicality: Arnott (2004)

Twenty-five years after the original derivation, Arnott published a retrospective assessing whether the HGT provides a practical empirical guide to whether real cities are over- or underpopulated. His 2004 paper, appearing in The American Journal of Economics and Sociology as part of a symposium on Henry George's economics, makes two major contributions:

The theorem generalizes broadly. Arnott shows the rent-equals-public-goods-spending result survives far beyond the identical-individuals textbook model. It holds with heterogeneous individuals and multiple goods (applying then to any Pareto-optimal allocation, not just a single optimal city size); with alternative sources of increasing returns such as Marshallian agglomeration economies and congestible facilities; with durable capital in dynamic settings; and even in the presence of distortions such as unpriced congestion — provided aggregates are measured at shadow prices rather than market prices.

But practical application faces specific obstacles. Arnott enumerates several problems that blunt the theorem's operational usefulness:

• Heterogeneity implies a system of differently-sized cities, not one number. With heterogeneous tastes or industries, "optimal population size is not well defined" for a single city; the theorem instead describes an optimal spatial unit of replication — potentially a multi-city cluster — making "optimal city size" for any one city an ambiguous target.
• The integer problem. A country's population will rarely be an exact multiple of the average-cost-minimizing spatial unit, so one cannot say a priori whether aggregate shadow profits are above or below zero. Arnott states that "the degree to which the integer problem causes the HGT to be violated has not been investigated empirically."
• Market vs. shadow prices diverge under real distortions. Once agglomeration externalities, congestion, land-use controls, and property taxation are present, "little can be said in general concerning how close the generalized Henry George Theorem, with the corresponding aggregates valued at market rather than shadow prices, comes to holding."
• Land heterogeneity (Ricardian differences) is unmodeled. The theorem assumes homogeneous land, but real locations differ in fertility, amenity, and natural accessibility; Arnott notes this question "has not been investigated in the literature."

Arnott's summary verdict: "Does the Henry George Theorem provide a practical guide to optimal city size? The jury is not yet in, but the approach is sufficiently promising to merit further exploration."

The Empirical Test: Kanemoto, Ohkawara & Suzuki (1996)

The first — and, per Arnott (2004), still essentially the only — empirical attempt to operationalize the HGT as a real-world test was conducted by Yoshitsugu Kanemoto, Toru Ohkawara, and Tsutomu Suzuki in a 1996 paper published in the Journal of the Japanese and International Economies.

The authors estimate aggregate production functions for 17 Japanese metropolitan areas to measure agglomeration economies, then test whether Tokyo — then the subject of a national policy debate about relocating the capital — is disproportionately overpopulated relative to other cities. A direct test of the HGT's rent-equals-spending identity is infeasible, they note, because "good land rent data are not available and we have to rely on land price data instead," and Japan's land price/rent ratio was extremely high and volatile (moving from 2.48× GNP in 1970 to 5.35× in 1990).

Instead, they compute the ratio of total land value to the total estimated agglomeration-economy benefit (the "Pigouvian subsidy") for each metro area and ask whether Tokyo's ratio is systematically higher than its peers'. Their finding: Tokyo's ratio came out slightly below the 17-city average, leading them to conclude that "there is no evidence supporting the hypothesis that Tokyo is too large" relative to other major Japanese cities. They caution that because the test is a ratio and Tokyo's population was more than double that of the next-largest metro area (Osaka), the same finding could coexist with a larger absolute gap between land rent and public spending for Tokyo.

Arnott (2004) gives this study a qualified endorsement, acknowledging "a long chain of questionable assumptions" but arguing that "quantification is an essential element of enlightened policy making, and if the procedure employed... is the soundest available, its conclusion should be respected until a sounder procedure is developed."

Relation to the Georgist Case

Optimal city size matters for Georgism because it is the setting in which the Henry George Theorem — the strongest theoretical result connecting land rent to public-goods finance — is derived. The theorem shows that under idealized conditions, a single tax on land rent suffices to fund public goods with no deadweight loss, providing a rigorous welfare-economics foundation for Henry George's intuition that public investment is capitalised into land value.

However, Arnott is explicit that George himself did not anticipate this theorem: "there is... no suggestion in George's writings that a confiscatory tax on land raises just the right amount of revenue to finance pure public goods in a city of optimal population." The theorem bears George's name because it formalizes his intuition, not because he stated or proved it.

The practical caveats identified by Arnott (2004) and the provisional, indirect nature of the Kanemoto et al. (1996) test mean that optimal city size remains primarily a theoretical construct. The HGT's rent-equals-spending identity is robust across many generalizations, but using it as an operational tool to determine whether any actual city is too large or too small — or to verify that observed land rent could fund observed public spending — remains an open research question.

See Also

• Henry George Theorem
• Agglomeration Economies
• Arnott & Stiglitz (1979): Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size
• Does the Henry George Theorem Provide a Practical Guide to Optimal City Size? (Arnott 2004)
• Agglomeration Economies and a Test for Optimal City Sizes in Japan (Kanemoto et al. 1996)
• Objection: Land value can't be assessed accurately

Sources

1. Richard Arnott & Joseph Stiglitz (1979), "Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size," Quarterly Journal of Economics 93(4): 471–500. PDF — used for the canonical derivation of the optimal-city-size condition and the Henry George Theorem.
2. Richard Arnott (2004), "Does the Henry George Theorem Provide a Practical Guide to Optimal City Size?" The American Journal of Economics and Sociology 63(5): 1057–1090. JSTOR · PDF — used for the generalized HGT, practicality caveats (heterogeneity, integer problem, market vs. shadow prices, land heterogeneity), the assessment of Kanemoto et al., and the quotation that George did not anticipate the theorem.
3. Yoshitsugu Kanemoto, Toru Ohkawara & Tsutomu Suzuki (1996), "Agglomeration Economies and a Test for Optimal City Sizes in Japan," Journal of the Japanese and International Economies 10(4): 379–398. DOI: 10.1006/jjie.1996.0022 · Free working-paper PDF — used for the empirical Tokyo test, the infeasibility of a direct HGT test, the land-value-to-Pigouvian-subsidy ratio findings, and the authors' conclusion that there is no evidence Tokyo is disproportionately overpopulated.