The third dimension of cities - relating building height, urban area, and population

TL;DR

This study analyzes the relationship between building height, urban area, and population using a global dataset, revealing that horizontal extent primarily drives population accommodation rather than vertical development.

Contribution

It introduces a Cobb-Douglas model to quantify the impact of building height and urban area on population, challenging assumptions about the benefits of vertical growth.

Findings

Population mainly driven by horizontal urban extent

Vertical development has limited impact on population capacity

Findings are consistent across all building types

Abstract

For decades, urban development was studied on two-dimensional maps, largely ignoring the third dimension. However, building height is crucial because it dramatically potentiates the interior space of cities. Here, using a newly released global building height dataset of 2903 cities across 42 countries in 2015, we develop a Cobb-Douglas model to simultaneously examine the relationship between urban population size and both horizontal and vertical urban extents. We find that, contrary to expectations, the residents of most urban systems do not significantly benefit from vertical dimension, with population accommodation being primarily driven by horizontal extent. The associations with country-level external indicators demonstrate that the benefits of horizontal extent are more pronounced in urban systems with more extreme size distribution (most population concentrated in few cities). Moreover, building classification tests confirm the robustness of our findings across all building types. Our findings challenge the intuition that building height and high-rise development significantly contributes to urban population accommodation, calling for targeted policies to improve its efficiency.

Equations

$$A\sim P^{{\beta }_A},$$

$$h\sim P^{{\beta }_h},$$

$$P\sim A^{{\alpha }_A}{h}^{{\alpha }_h},$$

$$\alpha_{A}=\frac{\partial\ln P}{\partial\ln A}\Big{)}_{h}\quad\text{and}\quad\alpha_{h}=\frac{\partial\ln P}{\partial\ln h}\Big{)}_{A},$$

$$\sigma =-\frac{{\alpha }_h}{{\alpha }_A},$$

$$A={\sum }_{i=1}^n{a}_i,$$

$$V={\sum }_{i=1}^n{a}_i\times h_i,$$

$$h=\frac{V}{A}=\frac{{\sum }_{i=1}^n{a}_i\times h_i}{{\sum }_{i=1}^n{a}_i}.$$

$$A\sim P^{{\beta }_A}\text{and}V\sim P^{{\beta }_V}.$$

$$P\sim A^{{\alpha }_A}{h}^{{\alpha }_h}\sim A^{{\alpha }_A}{\left(\frac{V}{A}\right)}^{{\alpha }_h}.$$

$$P\sim P^{{\beta }_A{\alpha }_A+{\alpha }_h({\beta }_V-{\beta }_A)}.$$

$$1={\beta }_A{\alpha }_A+{\alpha }_h({\beta }_V-{\beta }_A),$$

$${\alpha }_h=-\frac{{\beta }_A}{{\beta }_V-{\beta }_A}{\alpha }_A+\frac{1}{{\beta }_V-{\beta }_A}.$$

$$\ln A\sim -\frac{{\alpha }_h}{{\alpha }_A}\ln h.$$

Full text

1 Introduction

Urbanization is one of the most transformative global phenomena. In 1960, the urban population was 1.02 billion; by 2023, this number had increased to 4.61 billion, and it is projected to reach 6.68 billion by 2050 (source: https://ourworldindata.org/urbanization). This rapid growth has driven a surging demand for building stocks to accommodate urban residents. In response, cities generally grow in two distinct ways: horizontally and vertically. Understanding the relationship between these development patterns and urban population size is critical, as it influences the future vision for cities, providing insights into how cities can grow efficiently and sustainably [1, 2, 3, 4].

Horizontal development has been intrinsically linked to urbanization since the earliest settlements [5]. It typically involves expanding city boundaries to increase the urban area [6, 7]. While not exclusively associated with low-rise buildings, horizontal expansion often leads to a predominance of low-rise structures, constituting a significant portion of the urban fabric—not only at the periphery but also throughout the city. This development mode results in low-density urban environments. On the positive side, it can mitigate the urban heat island effect [8, 9] and provide more green space [10, 11, 12, 13]. However, the drawbacks include longer commuting distance [14] and higher carbon emissions [15, 16, 17, 18].

In contrast, vertical development emerged as complementary defining mode of urban development only after the invention of the steel frame and elevators, which enabled the construction and practical utilization of increasingly taller buildings [19, 20, 21]. As a response to the scarcity of space in dense urban centers, this mode multiplies available space by building upward [22]. While not exclusively residential, vertical development typically results in high-rise structures accommodating diverse human activities throughout urban cores. This architectural shift has fundamentally reshaped the face of modern cities. Today, high-rise buildings have become a defining feature of virtually all sufficiently large cities worldwide [23, 24, 25]. However, it also presents challenges such as blocking sunlight [26] and exacerbating infrastructure inequality [27].

While horizontal and vertical developments represent distinct modes, they are inherently interconnected, as both aim to meet increasing demands for urban space [28, 29]. Curiously, research on these two modes has largely remained separate. Extensive studies on horizontal development have established the concept of Fundamental Allometry [30, 31], which describes the nonlinear relationship between urban built-up area ($A$) and population size ($P$) in a cross-sectional manner (i.e., comparing different cities at a given time) using the scaling law [32, 31]:

[TABLE]

where $\beta_{A}$ is the scaling exponent. Typically, $2/3<\beta_{A}<1$ is reported for different regions [33, 34, 35], indicating that urban area increases less than proportionally with population size. In other words, larger cities tend to exhibit higher population densities.

The understanding of vertical development is constrained by the lack of three-dimensional building data. Existing research is largely confined to individual case studies. For example, Schläpfer et al. [36] adapted the established power-law model (Eq. 1) to examine the relationship between the average building height ($h$) and population size for 12 North American cities as:

[TABLE]

where they found the scaling exponent $\beta_{h}\approx 1/6$, indicating a relatively weak—yet statistically significant—increase in height with population size.

In this study, we develop a scaling model that integrates urban horizontal (total building footprint area) and vertical (average building height) metrics using a newly released global building height dataset [37]. This enables a systematic analysis of 2903 Functional Urban Areas (FUAs) within 42 countries in 2015 (see Section 4.1 in Methods for details on data; Extended Data Fig. 1 shows the geographical distribution of samples).

Figure 1 provides an overview of our study: (1) proposing the Cobb-Douglas model to integrate total building area and average building height (Fig. 1a); (2) measuring the marginal products of area and height in relation to urban population size (Fig. 1b); (3) analyzing the substitution rate between area and height to understand their interplay (Fig. 1c); and (4) applying a regression approach to explore how external metrics explain the observed patterns (Fig. 1d). It is important to note that our study is based on cross-sectional scaling analysis, which does not establish causality. Instead, it reveals how variations in urban population size are associated with differences in building area and height.

2 Results

2.1 From urban scaling law to Cobb-Douglas

We begin by explaining the rationale behind our model. While Eqs. (1)-(2) independently describe the relationships between population ($P$), total building area ($A$ in square meters), and average building height ($h$ in meters), they are insufficient to fully capture the relationships between all three variables. Empirically, we show that Eq. (2) fails to generalize beyond specific case studies (see Extended Data Fig. 2-3 for examples from Poland and Morocco; results for all countries are provided in Supplementary Figure S1). To achieve a more comprehensive understanding of average building height—an intensive variable, it is crucial to incorporate the extensive variable of total building area [38, 39]. Theoretically, as demonstrated in our previous study [40], integrating horizontal and vertical dimensions offers a natural approach to reconcile the opposing forces of urban growth. Accordingly, to capture the multivariate characteristics of urban development, we propose a Cobb-Douglas production function of the form

[TABLE]

where $\alpha_{A}$ and $\alpha_{h}$ are two independent exponents. Taking the natural logarithm of both sides yields a linear form $\ln P\sim\alpha_{A}\ln A+\alpha_{h}\ln h$, where we apply Ordinary Least Squares to estimate the exponents $\alpha_{A}$ and $\alpha_{h}$. To address differences in scale and measurement units between total area (extensive) and average height (intensive), we standardize all variables before regression. This allows for meaningful comparison between the estimated coefficients.

This approach, Eq. (3), is reliable, as tested by the Variance Inflation Factor, which indicates no multicollinearity (Supplementary Table S1). It supports our choice of using average height rather than total height, as total height is related to total volume, which is highly collinear with total area (see Supplementary Figure S1). The resulting fitted plane aligns well with the actual data points, as demonstrated using Poland and Morocco as examples (Fig. 2a-b). Supplementary Figure S1 extends to all countries analyzed. For a better representation of the data, interactive 3D visualizations are stored in https://zhangpprr.github.io/3D-visualization/.

2.2 Marginal product of area and height

The exponents $\alpha_{A}$ and $\alpha_{h}$ represent the marginal products of $A$ and $h$ in a logarithmic scale. Specifically, they are defined as:

[TABLE]

quantifying the extent to which additional $P$ can be accommodated by increasing either $A$ or $h$ (keep the other constant).

Figure 2c shows the estimated $\alpha$ for all analyzed countries. For $\alpha_{A}$, a consistent pattern emerges with $\alpha_{A}>0$, indicating that an increase in $A$ is associated with increase in population sizes. The majority of countries exhibit $0<\alpha_{A}<1$, suggesting that population size scales less than proportionally with building area. Notably, three countries—Egypt, Morocco, and Japan—show $\alpha_{A}>1$, suggesting a scenario where population increment outpaces increasing area.

In contrast, $\alpha_{h}$ reveals a surprising pattern. Most countries exhibit $\alpha_{h}\approx 0$, indicating that urban building height is not associated with population size. More strikingly, Egypt, Morocco, Bangladesh, and the Philippines show negative $\alpha_{h}$, suggesting that greater height tends to correspond to smaller populations. Other countries, such as Germany, exhibit $\alpha_{h}>0$, that is, taller buildings are associated with larger population sizes.

The negative relationship between $\alpha_{A}$ and $\alpha_{h}$ (Fig. 2d) indicates a negative relationship between horizontal and vertical dimensions, where gains from one often offset the other. Theoretically, this arises from a constraint imposed by total building volume, as area and height represent alternative ways of achieving similar volumetric capacity (see Section 4.2 in Methods).

2.3 Substitution rate of area and height

While we have shown how horizontal and vertical dimensions independently relate to population size by marginal products, the interaction between the two remains unclear. Here, we use the substitution rate ($\sigma$), defined as

[TABLE]

which quantifies how changes in area and height substitute for each other while maintaining a constant population size in a logarithmic scale (see Section 4.3 in Methods for derivation). There are three distinct substitution patterns:

• •

$\sigma>0$ indicates a complementary relationship between $A$ and $h$, where an increase in $h$ requires additional $A$ (Fig. 3a). Morocco exemplifies this pattern (Fig. 3d).

• •

$\sigma\approx 0$ indicates no interaction between $A$ and $h$ (Fig. 3b). Most countries exhibit this pattern (Fig. 3f).

• •

$\sigma<0$ indicates that $A$ and $h$ effectively substitute for each other, allowing cities to balance horizontal and vertical dimensions (Fig. 3c). The Great Britain is an example of this pattern (Fig. 3e).

2.4 Association with external metrics

The exponents $\alpha_{A}$ and $\alpha_{h}$ quantify how horizontal and vertical dimensions contribute to population accommodation, yet their variation across countries remains unexplored. To identify key factors influencing these differences, we examine the association of $\alpha_{A}$ and $\alpha_{h}$ with various geographic, economic, demographic, and urbanization metrics.

First, we compare countries of the Global North and Global South (Fig.4). Following classifications based on socioeconomic and geopolitical characteristics (source: https://unctad.org/), we define the Global South as comprising 29 countries (e.g., China, India, and Brazil), and the Global North includes 13 countries (e.g., the United States, Germany, and the United Kingdom).

Urban systems in the Global South generally exhibit higher $\alpha_{A}$ values, indicating a stronger reliance on horizontal development for population accommodation. In contrast, differences in $\alpha_{h}$ between these regions are minimal, suggesting that height plays a limited role in both contexts, with only a slight advantage observed in Global North countries.

Second, we employ a Lasso regression analysis to identify specific country-level metrics. For $\alpha_{A}$, the $\zeta$-exponent (exponent of the city size distribution in terms of urban population) is identified as the most influential factor, showing a clear positive relationship (Fig. 5a-b). A higher $\zeta$-exponent reflects a more uneven population distribution, where a few large cities dominate. This suggests that in countries with a more concentrated urban hierarchy, horizontal expansion plays a stronger role in increasing population. This finding is robust regardless of cut-off method (Extended Data Fig. 4).

In contrast to $\alpha_{A}$, we observe a negative relation between $\alpha_{h}$ with the $\zeta$-exponent. This indicates that while a higher $\zeta$-exponent tends to increase $\alpha_{A}$, it is associated with a lower $\alpha_{h}$, suggesting that urban concentration favors horizontal rather than vertical development. Notably, urban population share (i.e., the ratio of the total urban population to the country’s total population), which reflects the level of urbanization, shows a positive correlation with $\alpha_{h}$. This suggests that highly urbanized countries benefit more from building height. Economically, we find that per capita GDP is positively correlated with $\alpha_{h}$, but correlations are very weak (Fig. 5d).

2.5 The role of building functional types

While we considers all buildings within cities, their functional differences remain undifferentiated. Certain types, such as commercial or industrial buildings, tend to be taller [41, 42] but do not serve as residential spaces, potentially affecting our results. Here, we classify buildings based on land-use data from OpenStreetMap (OSM) [43].

We take Germany’s 44 FUAs as an example due to the high completeness of the OSM data [44]. By overlaying building footprints with land-use types, we assign functional types to individual buildings (Fig. 6a-b). We find that most buildings are residential (Fig. 6c). As expected, industrial and commercial buildings generally have larger areas and taller heights (Fig. 6d).

Next, we conduct regressions using building classes, where $A$ represents the total classified building area and $h$ the corresponding average height. Our results confirm that residential buildings best explain population accommodation, as indicated by the highest $R^{2}$ values (Fig. 6e-f). Despite this, the estimated $\alpha$ values remain within the same range as those derived from all buildings, indicating that total building stock serves as a reasonable proxy. This finding is further supported by evidence from the Philippines and Spain (Extended Data Figs. 5-6), which represent different substitution regimes in terms of $\sigma$. However, the absence of complete and accurate land-use data prevents broader verification across other countries.

3 Discussion

As urbanization continues, it is crucial to examine the fundamental factors shaping urban form, namely building area, height, and population size. Our results demonstrate that each dimension can be effectively captured by the proposed Cobb-Douglas model.

Contrary to the Fundamental Allometry, which suggests that larger cities tend to be denser, we find that, population size increases less than proportionally with total building area ($\alpha_{A}<1$) in most urban systems. This indicates that as urban areas increase, the rate of population accommodation declines, aligning with evidence from the evolution of individual cities [45, 24]. This pattern is likely driven by heterogeneous density distribution, where population density decreases from the center to the periphery [46, 47, 48, 49, 50]. Therefore, policies enabling higher densities also in the urban periphery, e.g. in the form of polycentric morphology, could improve land use efficiency. Meanwhile, in countries with high $\alpha_{A}$ (reliance on horizontal dimension), policies should prioritize “restricting horizontal expansion in mega-cities and incentivizing vertical development via floor area ratio bonuses.

Although vertical development might intuitively be seen as an efficient way to enhance population accommodation, our results show that its effect is weak ($\alpha_{h}\approx 0$). The reasons behind are complex. Taller buildings are often located in high land-price areas, where increased demand leads to higher prices rather than more housing units, as observed in Manhattan, New York City [51]. Public resistance to urban densification further limits vertical expansion [52]. Moreover, policy plays a crucial role. Building height restrictions—often imposed for historical preservation—force cities to expand horizontally instead, illustrated by Beijing [53]. Conversely, land-use policies, such as Germany’s net-zero land-take goal [54]—restrict urban horizontal development—aligning with our finding of higher $\alpha_{h}$ in Germany. Beyond these, our empirical regression analysis reveals the effects of broader national-level factors. For example, high GDP countries seem to benefit more from vertical structures. We suggest that in addition to relaxing building height restrictions, policies should also optimize the residential function of high-rises (e.g., limiting purely commercial skyscrapers) to enhance vertical space’s population capacity.

We find that horizontal and vertical dimensions offset each other (negative correlation between $\alpha_{A}$ and $\alpha_{h}$). This trade-off reflects both natural constraints and urban spatial dynamics. On the one hand, horizontal development is often constrained by natural barriers such as mountains, rivers, oceans, and national borders [55], making vertical development the only viable option. On the other hand, vertical form leads to higher density and urban centralization, increasing congestion for people commuting to centralized urban cores, as well as greater pressure on existing infrastructure systems. When further horizontal expansion occurs, it extends commuting distances for peripheral residents [56, 57], reducing the attractiveness to citizens [58, 59]. Beyond these local trade-offs, we find that the population distribution of the urban system seeks to play a strong role. Specifically, countries with higher $\zeta$-exponents—where population is concentrated in a few large cities—tend to benefit more from horizontal extent. It is intriguing as the horizontal vs. vertical development reflects local decision making while the $\zeta$-exponent emerges from regional planning. As we do not want to speculate, we need to leave the understanding of this finding for future work.

While we validate our findings using building classifications on a smaller scale, a broader verification remains necessary. This would require more comprehensive land-use data or advancements in building recognition technology. In addition, we use average building height as a proxy for vertical dimension, which does not fully capture the variation in height distribution across urban areas. To partially address this, we examine differences between higher-density urban centers and lower-density surroundings (Extended Data Fig. 7). While this analysis reveals variations in $\alpha$ rather than a consistent trend, it suggests that density plays a role. Future studies could refine this by incorporating more detailed height distribution information. Our analysis focuses on total building area and average height, but additional building characteristics, such as the number of floors, and envelope area, could further refine our understanding of urban form and population accommodation [60]. Lastly, due to data availability, our analysis is limited to a single-year cross-section. Incorporating time-series data in future work, employing an analogous approch could reveal the dynamic evolution of individual cities over time.

Despite these limitations, this study provides new insights into the three-dimensional development of cities, particularly the often-overlooked building height. While vertical development might leave the impression of efficiently densifying urban space to accommodate more population, our findings challenge this notion. In the end, this framework can be directly applied to explore how horizontal and vertical developments influence other urban indicators.

4 Methods

4.1 Data

In this study, we primarily use data on urban boundaries and buildings.

4.1.1 Urban boundaries

We define cities as Functional Urban Areas (FUAs) [61, 62], which include high-density urban centers and their surrounding commuting zones (where at least 15% of the population commutes to the center) in 2015. This definition is based on 2015 data, providing a standardized unit for cross-country comparisons. Population size data is associated with these FUAs.

To ensure robust statistical analysis, we conduct a series of data-cleaning steps to obtain a reliable and sufficiently large sample. First, we exclude countries with fewer than 30 FUAs to ensure meaningful comparisons at the national level. Next, we apply a population threshold cut-off to describe the power-law city-size distributions [63] to filter out cities below a minimum population size, ensuring that only sufficiently large urban areas are considered. After this refinement, we further exclude countries with fewer than 10 remaining FUAs to maintain statistical robustness.

As a result, our final samples comprise 2903 cities across 42 countries (Extended Data Fig. 1).

4.1.2 Urban buildings

To characterize urban building features, we use the newly released 3D-GloBFP dataset [37], which provides global 3D building footprint data. By overlaying this dataset with FUA boundaries, we first calculate the total urban building area ($A$ in square meter) as:

[TABLE]

where $a_{i}$ is the footprint area of building $i$, and $n$ is the total number of buildings within the city. Then, we calculate the total building volume ($V$ in cubic meter) as:

[TABLE]

where $h_{i}$ represents the height of building $i$. Finally, we derive the average building height ($h$ in meters) as:

[TABLE]

Thus, for a given city, we obtain its $A$, $V$, and $h$.

4.2 Theoretical relationship between $\alpha_{A}$ and $\alpha_{h}$

Urban scaling laws independently describe how $A$ and $V$ scale with $P$, both empirically exhibiting robust fits (see Supplementary Figure S1):

[TABLE]

Given that the Cobb-Douglas function integrates $A$ and $h$, we use the geometric relation (Eq.8) to obtain:

[TABLE]

Replacing $A$ and $V$ from Eq. (9) leads to:

[TABLE]

The scaling exponents are constrained by:

[TABLE]

leading to a relationship between $\alpha_{A}$ and $\alpha_{h}$:

[TABLE]

Empirical evidence consistently shows that $\beta_{V}-\beta_{A}=\beta_{h}>0$ (Supplementary Figure S2), reflecting a negative relationship between $\alpha_{A}$ and $\alpha_{h}$

4.3 Substitution rate

To quantify the trade-off between area and height under a constant population size, we define the substitution rate $\sigma$. This metric reflects how changes in $\ln A$ and $\ln h$ compensate for another within our Cobb-Douglas framework.

Under the condition of constant population size ($\ln P=c$), where $c$ is a constant, we obtain the following relationship between $\ln A$ and $\ln h$:

[TABLE]

Here, the slope $\sigma=-\alpha_{h}/\alpha_{A}$ defines the substitution rate. It describes how changes in one spatial dimension must be offset by changes in the other to maintain the same population level.

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Acknowledgements We thank P. Gabriel M. Ahlfeldt for useful comments. This work was supported by the National Natural Science Foundation of China (72242102 and 72288101). P. Z. acknowledges support from the program of China Scholarship Council (No. 202307090047). F. L. R. thanks CNPq (grant numbers 403139/2021-0 and 424686/2021-0) and Fapemig (grant number APQ-00829-21 and APQ-06541-24) for financial support. D. Rybski acknowledges financial support from German Research Foundation (DFG) for the projects UPon (#451083179) and Gropius (#511568027).

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Data availability Data used to achieve our results is available at https://doi.org/10.6084/m9.figshare.28601942.v1 .

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Code availability Code used to achieve our results is available at https://doi.org/10.6084/m9.figshare.28601942.v1 . Also for better visualization of the 3D plot, please refer to https://zhangpprr.github.io/3D-visualization/ .

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Competing interests The authors declare no competing interests.

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Author contribution P. Z., L. G., F. L. R. and D. R. contributed to study design, data collection, analysis, and preparation of the manuscript. All authors edited the manuscript. L. G., F. L. R., B. J., Z. G. and D. R. supervised the research. B. J. and Z. G. secure the funding.