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Abstract

Large cities have benefits offset by costs. Since the inter-urban Rosen–Roback model, economists interpret the attractiveness of a city in terms of spatial equilibrium: a cost-benefit balance across locations. High incomes are counterbalanced by high prices or disamenities. A cross-sectional regression analysis on the ∼ 8000 Italian municipalities whose population sizes range from around 30 to 3 million, found the following association: by doubling the population of a settlement, holding constant its per capita income and touristic presences, its house price rises by 9%. House price increases are not offset by an increase in income. Bigger cities are more expensive than smaller. This opens reasoning upon indirect economic valuations of the value - or cost if you see it differently - (opportunities, services, vibrancy…) that urban size has for people who choose - or must choose - bigger settlements. At the best of my knowledge, this is the first work, in this form, finding a scaling coefficient for house price. It resulted around 0.13, namely, by passing from a town of 10˙000 inhabitants to a city of 1 million inhabitants (even if holding constant tourism and income), the average house price would almost double (+82%), almost tripled (+183%) from a village of 1˙000 inhabitants to a 3 million inhabitants’ city, and almost quadrupled (+282%) from a hamlet of 100 inhabitants to a 3 million inhabitants city.

Keywords

urban scaling Rosen–Roback model urban size estimate-urban-cost spatial equilibrium rural-urban spatial utility function residential decision

Introduction

An empirical investigation on the link between population size and house prices builds upon the theoretical framework of spatial equilibrium across cities (Glaeser, 2008a) and of urban scaling (West, 2017). It gives an inferential quantification of a type of cost which urban dwellers face when deciding the urbanicity level (rural settlement, village, town, small-medium-large city) of the place where to live.

Evaluating the household’s monetary costs of different urbanicity levels, indirectly offers an assessment of the cost-benefit trade-off that individuals are willing to accept, without taking into account those who do not have choices as low urbanicity levels might mean unemployment and lack of basic services and physical-social infrastructures (D’Acci, 2021).

Estimates of urban costs for dwellers include variables such as noise-light-air pollution (Chan and Yao, 2008; Dockery et al., 1993; Muzet, 2007; Navara and Nelson, 2007), congestions (Verbavatz and Barthelemy, 2019), crimes (Oliveira et al., 2017), and, to some extent, mental well-being (D’Acci, 2020) whose quantification might be subjective and intangible and whose effects on people’s quality of life can be so indirect that their actual economic translation, if not identification alone, becomes a challenging task.

Other types of urban costs are more straightforward such as house prices. Usually, the larger the city – in terms of population size – the greater the house prices (Combes et al., 2019; Hsieh and Moretti, 2019; Song and Zhang, 2020).

The inter-urban Rosen–Roback model (Roback, 1982; Rosen, 1979), interprets the attractiveness of a city in terms of the spatial equilibrium (Glaeser and Gottlieb, 2009) between wages, housing cost and urban amenities: ‘high incomes are offset by either high prices or disamenities’ (Glaeser, 2008b). The latter is based on the assumption that people are indifferent across space. A vast literature is available about across cities equilibrium from various angles and implicitly estimating amenities using a Rosen–Roback like approach (e.g. see in Albouy, 2016; Anenberg and Kung, 2020; Blomquist et al., 1988; Cheung and Minerva, 2024; Gabriel and Rosenthal, 2004; Oikarine et al., 2023; Rappaport, 2008; Shi et al., 2021; Weinstein et al., 2023).

Recently, Glaeser recognised the growing discontent generated by high housing prices and other urban disamenities and started asking ‘why has urban triumph produced acrimony rather than joy’? (Glaeser, 2020).

Among many, Henderson (1974, 1975), Fujita and Ogawa (1982), Fujita and Thisse (2002), Fujita (2009) rigorously formalised cities as a compromise between benefits and costs.

Bigger cities are usually attractive for their superlinearly higher wages (Batty, 2008; Bettencourt, 2013; Bettencourt et al., 2007; Lei et al., 2021) – though not always for the poorest income decile (Mora et al., 2021) and the less skilled (David, 2019) – but also for their amenities, aesthetics, and public services (Chen and Rosenthal, 2008; Florida, 2004; Glaeser et al., 2001; Hoogerbrugge and Burger, 2021). Notorious benefits of urban size derive from agglomeration economies (Combes et al., 2011; Puga, 2010; Rosenthal and Strange, 2004), but less is known about their costs (Combes et al., 2019).

House prices dynamics in temporal analysis studying their changes in relations to urban growth found large cities (Bogin et al., 2019) and rapidly growing cities (Himmelberg et al., 2005) as causes of higher price levels (Gray, 2021).

Cities with a greater diversity on their economic base (usually the larger the city the larger its economic variety) have less risk of bankruptcy or decline than cities heavily relaying on one particular economic base only. However, this depends from the risk related with the type of economic base and the national/international demand trend for it in relation with the capacity of other (or potentially new coming) national/international concurrent.

Despite Gibrat’s law of urban growth (Broxterman and Yezer, 2021; Eeckhout, 2004; Gray, 2021), a positive feedback (a recursive cycle reinforcing itself) might happen in large cities: the larger the city growths the larger its assortment of resources, such as infrastructures, and specialised services (train lines, airports, hospitals, universities, marketing, banking, …), and the larger the city growths. These specialised assets emerge as the demand from a large window of services emerge (agglomeration economies), and this is a characteristic phenomenon happening in large cities and megacities.

In this paper we will not face the forces determining city size and we will not deal with quantifiable and verifiable reasons about the link between city size and housing values; we will explore if, and if so how much, city size and housing value are linked. It will do so not dynamically involving rate of growths but statically comparing populations and house prices across cities in a cross-sectional analysis controlling for incomes and tourism.

The quantitative approach to urban sciences (Batty, 2021; Higham et al., 2017) and relatively recent studies about scaling law in cities (Bettencourt, 2013; West, 2017) stimulate researchers in understanding the allometric relationships between city size and some socioeconomic and infrastructural variables. They usually report socioeconomic factors super-linearly scaling with city size, while the infrastructural sub-linearly (D’Acci, 2025a).

One of those relationships we want to analyse is between city size and housing prices.¹ The literature is not abundant about this specific link. Regardless of the reasons and the direction of the causality (if any), investigating if a statistically significant association between urban size and housing prices appears, is of interest on its own, for both a research and policy perspective.

This is one of the first works finding the scaling coefficient of house prices of settlements. The closest works to this (Table 4) were in fact related to housing costs, or house prices, only in specific parts of the city (the centre or the Central Business District), or of the radial distribution of house prices from the city centre to its periphery (Combes et al., 2019; Laziou et al., 2024; Mora et al., 2021). Our sample covers the entire settlements of a country (around 8 thousand municipalities in this case, starting from the smallest, 29 inhabitant) rather than only the few biggest hundreds cities as usually done from previous works.

Our main research question is the following: is there any scaling law governing the average house price of settlements based on their population size?

Beside policies implications, the reply to this question would add empirical evidences to the growing urban scaling literature, keeping in mind the critics to the latter which we briefly list per points in S1 (supplemental material).

Data and methods

A database of 7˙751 municipalities² in Italy served for the estimation of the link between population size and house price across cities, in a same year, via a log-log multiple regression analysis.

Each individual of the sample is a municipality ranging from 29 inhabitants to around 3 million. Administrative criteria to define boundaries, as municipalities, are often no the best. However, data limitation does not allow us to use other criteria.

The dependent variable is the average house price per metre square in €/mq, called (H); the main independent variable of interest is city size in terms of residential population number, called (P); the control variables are the per capita income of individuals declared for taxes purposes – used as a proxy for average income – called (I), and the yearly number of tourists per 1˙000 inhabitants, called (T).

Controlling for the average income (despite not taking into account the inequality distribution of such average spread income, both at worker’s level and at family composition level), and for the tourism allows to at least partially consider the spatial equilibrium across cities (Glaeser, 2008a, 2008b) condition related to the balancing equilibrium among house costs, incomes, and a type of ‘pleasantness’ indicator of the place.

The year of the data is 2018 (just before the COVID-19 to avoid its unequal effect across regions and city sizes) and the source are ISTAT (National Institute of Statistics³) regarding populations, incomes and tourism, and ‘Il Borsino Immobiliare’⁴ concerning the house prices,⁵ which are the national official sources and the largest as well as most reliable data available.

The regressions occurred at two scales: national and regional.

The regional analysis clears up across-regional differences which in Italy are rather sharp in incomes among northern, central and southern regions. However, the latter regional contrast in richness is partially mitigated by the addition of income as a control variable in the multiregression.

The lack of availability of data constrained the number of control variables.

Results

The smallest Italian municipality of the house price (final) sample has 33 inhabitants and an average house price of 675 €/mq; the largest has 2˙743˙796 inhabitants and an average house price of 3˙532 €/mq.

The Italian municipality with the cheapest houses has an average house price of 228 €/mq and a population of 1˙600 inhabitants; the one with the most expensive houses has an average house price of 9˙983 €/mq and a population of 6˙093 inhabitants.

It is worth to note that the second most expensive (9˙963 €/mq) Italian municipality has a population of only 493 inhabitants, and the 50 most expensive municipalities are all smaller than 10˙000 inhabitants. We might have villages and small towns with very high average house prices, despite being small, for being very attractive (historically, naturally, scenically….) and/or with more than proportionally higher income for various reasons. As well as large cities with proportionally lower average house prices because of lower than expected income and/or amenities. This, in line with the spatial equilibrium across cities assumptions, imposes the necessity to add control variables enabling to filter the association between city size and house price holding constant crucial factors – others than urban size – altering house prices such as incomes and attractiveness.

Figure 1 shows a scatter plot of each couple of variables combination using their natural logarithms for a better visualisation due to the large variance of populations values. It also visualises the density distribution over the frequency histograms; the spearman correlations and the relative significance levels.

Figure 1.

Scatter plot of the natural logarithmic of the variables for all Italy (2˙967 observations); density distributions; frequency histograms; spearman correlations; significance of spearman correlations (*). The variables, from right to left, are average house prices, residential population, per capita income, and yearly touristic presences per 1 thousand inhabitants. At a national level is already visible a general pattern: the greater the settlements size the greater its average house prices, and the lower is tourism. The greater the tourism the greater the house prices. Income correlations are less clear. The multiple regression would help isolating its associations.

It is visible a positive (spearman) correlation between house prices and population size, per capita income and per capita tourism, but a negative correlation between tourism and population size. The multiple regressions would help in cleaning the house price link with each of them individually, and the regional analysis would check if congruent patterns appear across regions (Fig S1, S2 in the supplementary material).

Table 1 and Figure 2 show the relevant regression outputs. In average (keeping constant the other predictors of the model), a 1% increase in population is associated with a 0.130% increase in house prices (0.127% for the regression done in all Italy); a 1% increase in per capita income is associated with a 0.56% in house prices (0.53% for all Italy); and a 1% increase in tourists’ presence per 10˙000 inhabitants is associated with a 0.09% increase in house prices (0.13% for all Italy).

Table 1.

Output of the multiple double logarithmic regression.

LOG-LOG regression coefficients: log (house prices)∼log (population)+log (income)+log (tourism)
Region	Population (P)	p-value	Income (I)	p-value	Tourism (T)	p-value	Adjusted R2	Sample size
Val d’Aosta	0.12	***	0.77	***	0.11	***	0.45	59
Piemonte	0.12	***	0.37	***	0.10	***	0.32	337
Lombardia	0.09	***	0.34	***	0.08	***	0.28	412
Trentino Alto Adige	0.12	***	0.45	***	0.02	*	0.25	229
Friuli Venezia Giulia	0.16	**	0.28	*	0.08	***	0.63	104
Veneto	0.15	***	0.44	***	0.08	***	0.45	284
Liguria	0.16	***	0.85	***	0.21	***	0.57	142
Emilia Romagna	0.14	***	0.25	*	0.11	***	0.46	196
Umbria	0.08	***	0.26	.	0.02	.14	0.57	83
Toscana	0.12	***	0.77	***	0.11	***	0.45	231
Marche	0.12	***	0.29	*	0.09	***	0.66	117
Lazio	0.07	*	1.19	***	0.07	**	0.41	91
Abruzzo	0.12	***	0.09	.73	0.04	.17	0.57	22
Molise	0.13	***	0.13	***			0.48	132
Campania	0.17	***	0.47	**	0.21	***	0.52	114
Puglia	0.19	***	0.14	.34	0.10	***	0.49	147
Basilicata	0.23	***	0.33	.20	0.10	**	0.65	30
Calabria	0.11	***	0.41	***	0.03	*	0.39	113
Sicilia	0.06	**	1.06	***	0.07	***	0.44	154
Sardegna	0.15	***	1.19	***	0.15	***	0.65	107
AVERAGES	0.130		0.56		0.09		0.48	160
ITALY with control variables	0.127	***	0.53	***	0.13	***	0.40	2967
ITALY without control variables	0.061	***					0.03	2967
ITALY without control variables, entire national sample	0.12	***					0.13	7690

Dependent variable: city average house price (H); independent variables: city population (P), city declared income per capita (I), city tourism presence per 1˙000 inhabitants (T). Each individual of the sample is a municipality (‘cities’ of various sizes: from 33 to around 3 million inhabitants). The sample for the multiple regression covering the entire nation, has 2˙967 observations (rather than 7˙690 as for the national simple regression) because we have to keep only these observations having data for all the control variables. Variations of the coefficients among regions can be due to the level of disproportionateness of their municipality sizes and/or disproportionateness of their municipality key characteristics.

The averages are calculated only for the statistically significant values (p-value ≤.05).

Legend of p-value: 0 ‘***’ .001 ‘**’ .01 ‘*’ .05 ‘.’

Figure 2.

Population size (P) coefficients of the multiple double logarithmic regression of Table 1. The average P coefficient from each regression of each of the 20 Italian regions (black histogram with label ‘AVERAGES’) is calculated only for the statistically significant values (p-value ≤.05) visible on Table 1. The red histogram, labelled ‘ITALY’, is the P coefficient calculated on all Italian municipalities. The yellow histogram, labelled ‘ITALY only P and H’ is the coefficient calculated on all Italian municipalities without the control variables of income and tourism, but using only a simple double logarithmic regression having house price (H) as dependent variable, and population (P) as predictor.

The elasticity (scaling coefficient) of house price related to population size is then between 0.12 (considering a simple regression between Population and House price over the entire nation: 7690 observations) and 0.127 (from a multiple regression with all the control variables). Both results are very strongly statistically significant (p-value≈0). If instead we calculate the average of the coefficients for each region, the average scaling coefficient of house prices is 0.13.

The difference between the coefficients resulting by adding or not the control variables (respectively 0.127 and 0.061) over the same sample (2967 observations) reminds us the magnitude that the omitted variables bias⁶ can have: despite having the same sample size and p-value, after adding the control variables the elasticity more than doubled.

Even if both equally highly statistically significant (p-value≈0) we prefer to use the coefficient from the same sample but with the control variables as less invalidated by the omitted variable bias.

Mutlicollinearity seems not to be an issue as the variance inflation factor resulted low: log (P): 1.17; log (I): 1.03; log (T): 1.21.

Namely, by doubling the population of a city,⁷ its house price would rise by 9% (holding constant per capita income and per capita tourism, and using the average of the coefficients of the regions). Seeing it differently, by passing from a town of 10˙000 inhabitants to a city of 1 million inhabitants (even if holding constant tourism and income), the house price would almost double (+82%), and it would almost tripled (+183%) when passing from a village of 1˙000 inhabitants to a 3 million inhabitants’ city.

This association cannot claim any causality, which, if existent, could even be a reverse causality, or a mixed causality direction on a case to case base. Potential spatial autocorrelation and omitted variables bias might be present too.

These results are anyway in line with the expectations of having higher house prices in bigger cities, not entirely payed off by higher incomes.

The link between House Price and Income is also verified and in line with the expectations: the greater the income per capita the greater the House Price, although not with the same proportionality.

Table 1 also suggests that population size on its own is affecting house price, rather than the latter rising because income per capita rises with population size. In fact, the multiple regression holds constant the other predictors (Income and Tourism) while varying only population size, namely, isolating the association between house price and population size alone.

The reasons could be various. Holding constant everything else, economies of agglomeration and scale usually induce a larger availability of services, opportunities and amenities such as schools, universities, hospitals, restaurants, theatres, libraries, vibrant city centres, museums, sport facilities and shops, which attract residents and urban users, regardless of negative aspects such as crime, pollution and congestion. Larger cities would also provide a larger and more variate offer of jobs. This attractiveness is somehow reflected in the real estate prices. Even though, according to cases, greater population size can make the city overall better or worse.

Eq (1) represents the typical scaling model in its power law and logarithmic form:

\begin{array}{l} y = α X^{β} \\ l n y = \ln α + β l n X \end{array}

(1)

where y is the urban indicator we want to investigate, X is the resident population of the settlement, α is the proportionality constant and β is the scaling exponent.

Reading our results allometrically, within the relatively recent urban scaling framework (Bettencourt et al., 2007, 2020; Batty, 2008, 2012; D’Acci, 2025; Rybski et al., 2009), if we write it in a power law form, using the coefficients of Tables 1 and 2 related to the simple log-log regression of the whole Italy, we have the following:

House Price \sim {Population}^{0.12}

(2)

Total Income \sim {Population}^{1.000291}

(3)

Table 2.

Coefficients of the population, their p-value and adjusted R squared from three double logarithmic regressions.

LOG-LOG regression outputs
Variables	Coefficients	p-value	Adjusted R2	Sample
Total income∼population	1.000291	***	0.9428	7751
Per capita income∼population	0.000291	.917	−0.000128	7751
Total income∼population	0.99136	***	0.951	2967
Per capita income∼population	−0.008643	.0365*	0.001138	2967
Total income∼population + house price + tourism	0.966836	***	0.9604	2967
Per capita income∼population + house price + tourism	−0.033	***	0.192	2967

First two lines: dependent variables = total income and per capita income; independent variable = population size. Sample: the entire national sample of 7751 settlements with data. Other lines: dependent variables = total income and per capita income; independent variables = population size, average house price, tourism presence per 1000 inhabitants. Each individual of the sample is a municipality: from 29 to around 3 million inhabitants, for the 7751 observations of the first two lines; from 79 to around 3 million for the 2967 observations for the rest, the latter smaller sample is due to alignments among the available data for all the predictors. Legend of p-value: 0 ‘***’ .001 ‘**’ .01 ‘*’ .05 ‘.’ Regarding the number of significant digits, the original output of the software R appears.

The total⁸ Income scaling exponent for Italy, 1.000291, is essentially linear; lower than the one found in Australia, 1.03–1.05, (Sarkar et al., 2018, 2020), or in the USA, 1.07 (Mora et al., 2021), and in the UK (about wages), 1.022 (Batty, 2025); and much lower than the wage scaling found in the USA, 1.12–1.15, (Bettencourt et al., 2007; Lobo et al., 2013), and China, 1.18 (Lei et al., 2021, 2022).

This basically linear relation, rather than superlinear as usually found, is probably due to the fact to have considered the entire settlements sample rather than only the biggest one (Fig S3 in Supplementary Material) as usually happens in other studies, as well as to have decided not to remove outliers (Fig S4 in Supplementary Material) as providing useful inputs for the multiple regression estimation of the coefficients.

A positive effect of using the comprehensive sample by including all the settlements even the smaller, is to offer to the model the entire picture rather than just a small percentage of the variation of settlement sizes. An undesirable aspect is that we include small towns and even rural villages which benefit from spillover effects from nearby bigger cities, and vice versa. Therefore, even the smaller municipalities might appear to have a decent income (namely, a linear rather than superlinear elasticity) probably because most of their residents work in the bigger nearby municipality getting their typically higher wages.

Our isometric relation, rather than super linear or near linear as often found, could also have been influenced by the Italian case itself where various smaller towns and (mountain, sea, lake, picturesque) villages are rather richer than the biggest, for various reasons. However, the multiple regression regressors ‘Income’ and ‘Tourism’ mitigated this effect and have been included for this reason.

In our model, a 1% increase in population size is associated with a 0.12% increase in house price and 1.000291% increase in total income. Meaning that, whatever the reasons, doubling the population size you have a 9% (footnote 3) increase in house prices and a 0.04% more than proportional increase in total income (i.e. 100.04% increase of total municipality income) than expected if it were to be linear; namely, the former is superlinear while the latter is practically linear.

The per capita income coefficient is, as mathematically expected (coefficient of the total income, minus 1), 0.000291 but completely statistically insignificant. When the control variables enter in the regression, the per capita income becomes negative (−0.033) but both coefficients are still practically linear for their low absolute values.

This is also visible (Figure 3(c)) in the slight downward trend of the ratio between per capita Income and House Price when population size increases: the greater the population size the lower the ratio – although with lot of noise – namely, the more than proportionally expensive houses are in relation to income. Figure 3(a) also shows how the per capita Income measure is essentially linear (−0.009⁹) and much noisier – as reported since Shalizi (2011); Sarkar et al. (2020).¹⁰

Figure 3.

(a) Per capita income versus population size. (b) House price versus population size. (c) Population size and the ratio ‘per capita income’/’house price.’ All plots are completed with linear trend and confidence interval. Values are in logarithms. β is the exponent of the power regression and R2 the R squared. The sample is the same used for the multiple regressions of Tables 1 –3 (2967 observations) for data alignments among variables. The bigger the settlement the costlier in terms of income/house price proportion. The latter in fact decreases (i.e. higher house price in relation to income) when the settlement size increases.

Equalised dwellers’ utility across settlements

The Rosen–Roback concept, and subsequent spatial equilibrium across cities literature, is based on an elementary household’s indirect utility function ( $v$ ) involving wages ( $w$ ), housing prices (or rents) ( $h$ ), and amenities ( $a$ ) of the settlements (j); such utility – under certain conditions (e.g. no spatial arbitrage23) – is equalised across settlements, being $\bar{u}$ the equilibrium utility level:

\bar{u} = v (w_{j}, h_{j}, a_{j})

(4)

A vector of settlement amenities ( $a$ ) would include pleasant factors such as nice weather, services, cultural and recreational attractions, nice landscape, environmental quality, historical value, and so on, net of disamenities such as crime, pollution, congestion, stress, etc.

Higher wages, ceteris paribus (i.e. partial derivative, $\partial$ ), increase the household utility; idem for amenities, while higher housing prices (or rents) decrease it (eq. (5)):

\partial v / \partial w > 0; \partial v / \partial a > 0; \partial v / \partial h < 0;

(5)

Our model didn’t include settlement’ amenities to properly close the spatial equilibrium circle of Rosen–Roback alike approaches, but only the income-house price ‘couple.’

However, we indirectly took into account some type of amenities partially summarised within the touristic variable, referring mostly to recreational-cultural-natural amenities. Tourism demand is in fact linked with amenities and quality of life attributes (Marcouiller et al., 2004; Power, 1988); leisure tourists are attracted by special traits – natural, historic, cultural, recreational, beauty – which are also some of the very factors attracting households permanently and associated with quality of life24 (Carlino and Saiz, 2019).

We cannot say much about how population size on its own plays a role within Rosen–Roback alike framework, as this is one of the naturally emerging sub-question of our study: is a higher size a wanted or unwanted feature for the urban dweller utility?¹¹ Saying it differently: is urban size seen as an amenity or disamenity?

We cannot easily reply. We can only show statistically significant associations and related considerations basing on which we can suggest that urban size alone appears to be appealing¹² as being associated to higher house prices, regardless income and tourism.

Deeper considerations should be analysed in order to establish if such observed higher house prices of bigger settlements are actually due to the fact that bigger size is felt as an amenity (therefore deserving to pay more for it as increasing the urban dweller utility) or it simply is an obligated necessity (e.g. to get a job, or healthcare services, schools…), even if also in the latter circumstance we can still see it somehow through the spatial equilibrium lens.

To extrapolate how changes in tourism and population is associated with changes in per capita income, we run the following multiple regression from the same data (Table 3):

Table 3.

Output of the double logarithmic regression.

LOG-LOG regression coefficients: log (income)∼log (population)+log (house prices)+log (tourism)
	Population (P)	p-value	House (H)	p-value	Tourism (T)	p-value	Adjusted R2	Sample size
ITALY	−0,03316	***	0.312	***	−0.012	**	.192	2967
ITALY only P and I	0,000291	.917					∼0	7751

Dependent variable: per capita income (I); independent variables: residential population (P); average house price (H) (€/mq); tourism presence per 1000 inhabitants (T). Each individual of the sample is a municipality (settlements of any sizes: from 33 to around 3 million inhabitants). Legend of p-value: 0 ‘***’ .001 ‘**’ .01 ‘*’ .05 ‘.’ The greater the population the lower the income (−0.033% per capita Income for +1% population); the greater the Tourism the lower the income (elasticity of −0.012); the greater the house price the greater the income (elasticity: 0.312).

Multicollinearity is not an issue according to the low variance inflation factor: log (P): 1.24; log (H): 1.39; log (T): 1.58.

Figure 4.

Ratio income/house price versus tourism. The higher the tourism the lower the income/house price ratio, namely, the relative higher costs (more expensive houses in relation to incomes) are balanced by higher tourism, a proxy of some kind of appeal of the settlement.

The outputs of our multiple regressions (Tables 1 –3), read through Rosen–Roback lens25, are the following:

• From Table 1, holding constant the other variables, an increase alone in:

◦ (1) the population size (p) of the settlement;

◦ (2) the ‘per capita’ touristic presence;

◦ (3) the per capita income;

is associated with an increase of house prices which could be interpreted as an increase in the willingness to pay (in terms of house prices) to live, respectively, in:

◦ (1) bigger settlements, which could happen because the latter being (overall¹³) seen as an amenity (e.g. the greater the population the greater the number and level of services, opportunities, varieties, etc.): $\partial v / \partial p > 0$ ;

◦ (2) nicer settlements¹⁴: $\partial v / \partial t > 0$ ;

◦ (3) richer settlements where to get higher income: $\partial v / \partial i > 0$

• From Table 3, ceteris paribus, an increase in the per capita touristic presences, or in the population, is also associated with, respectively, a decrease of per capita income, and a near constant one (namely, the latter would mean a lower real income if taking into account the above mentioned increase of house prices); which could be interpreted as people willingness to get lower incomes to enjoy living in nicer places (if tourists are going there could generally mean, that is, a nice place for some reasons) and with more services-opportunities (bigger cities). Which is in line to the previous point and the spatial equilibrium across cities framework, where the equalised spatial utility arises from the following:

• If a settlement has higher per capita income, this advantage is mitigated by the disadvantage of higher house prices (Table 1). Likewise, greater population size is associated to greater house prices (Table 1) presumably for providing more and better services and opportunities (which, still presumably, would surpass the typical negative aspect of size such as crime, pollutions, congestions…), these advantages are counterbalanced by the disadvantage of lower per capita income (Tables 2 and 3) and, indeed, higher house prices (Table 1). Similarly, greater per capita tourism is likely linked with some pleasant factors of the settlement,¹⁵ an advantage which is offset by lower per capita income (Table 3) and higher house prices (Table 1) as Figure 4 and the multiple regression between the Income/house price ratio and the regressors Tourism and Population show.

Conclusion, comparisons, critics and perspectives

In the almost 8˙000 municipalities (hamlets, villages, towns, cities) analysed via multiple regression in Italy, average house prices increase by 9% when doubling population size (even if) holding constant per capita income and per capita touristic presences. This suggests that – regardless income and some form of place’s pleasantness, as both kept fixed by the multiple regression – population size itself acts as a statistically significant source of influence on house prices, and it does so with a relevant magnitude effect. It can only be a suggestion, as causality, and its directionality cannot be proven.

Depending from lifetime earnings, overall living costs, length of time lived in a city, level of income and house price, and type of use of eventual savings, larger cities may result overall costlier. This result is in line with Mora et al. (2021), Combes et al. (2019); Laziou et al. (2024). However – rather than ‘average’ house ‘prices’ as in our case – they talk, respectively, about aggregate housing ‘costs’¹⁶; house prices ‘at the CBD’; and house prices ‘at the city centre.’ Despite the latter, it is interesting to notice similarities with our results (Table 4): house costs scale faster than income (Mora et al., 2021); the scaling coefficients (or elasticities) of house prices – even though at the CBD or city centre rather than the average one of the entire settlement – are around 0.208 (Combes et al., 2019), 0.2 for the radial function¹⁷ of house prices, and 0.29¹⁸ for the house prices at the city centres (Laziou et al., 2024), against our result indicating a scaling coefficient of 0.12 but which refers to the average house price. Such numerical difference is not a surprise considering that house prices in the centres/CBDs are found to scale more than in the rest of the city (Laziou et al., 2024).

Table 4.

Elasticities from different studies.

	House variable
	Average house price	Median price at city centres	Homotheticaly rescaled housing price radial profiles from centres	Estimated price at the city centres			[per capita] average yearly rent, and/or average yearly owner costs	Estimated price of a representative short-term rental property at the city center
Elasticity	0.12, 0.13 (with control variables)	0.29	0.2	0.161	0.23	0.33	[0.11 (bottom decile), 0.29 (top decile)]	0.208
sample	7690, 2967	161		733			379	277
Type of sample	Municipalities >0 inhabitants	Cities >50k inhabitants		Cities >300k inhabitants			Cities >100k inhabitants	Urban areas >80k inhabitants (each urban areas contains in average 46 municipalities)
Country	Italy	France		World (regression with country fixed effects)	USA	Eurozone	USA	France
Source	D’Acci (2023), (2025)	Laziou et al. (2024)		Nobauer (2023)			Mora et al., (2021)	Combes et al., (2019)

Our result, therefore, is in line with the literature, despite different samples, variables, methods and countries. Other studies focused on the city centre prices which is where are more expensive, and, more importantly, disproportionally more expensive once cities get bigger (Figure 3 in Laziou et al., 2024) than in the rest of the city. That is probably also why they got higher elasticities than ours. Their results corroborate these of Mora et al. (2021), as the latter shows a less than half elasticity (0.11) for the cheapest houses related to the more expensive (0.29) which usually are in the centres (and in a few prestigious other areas). This (0.29) is in fact very similar to the elasticities found in the city centres from other studies (0.29, 0.23, 0.33, 0.208); while the elasticity for the poorest houses (0.11) is, as expected, slightly lower than ours (0.12, 0.13–13). In fact, if we consider that our results are related to the average house prices throughout all the city, and that the central part is the smallest proportion – namely, our average takes into account only a minor percentage of house prices in the centres – finding 0.12, 0.13 is reasonably in support to previous similar studies.

We prefer to consider 0.127 or 0.13 as elasticity because coming from the multiple regression (Table 1) cleaning up the effects of anomalous characteristics of the settlement such as over higher (or lower) tourism and/or income (both factors typically affecting house prices) related to the ordinary settlements of its size. Similar reasons which induced Laziou et al. (2024) to prefer their elasticities coming by lowering the weights of smaller settlements rather than the one without weights which could be ‘an artifact of the presence of cities with specific characteristics, because the fit provides a poor prediction for “ordinary” cities’ (Figure 4, Laziou et al., 2024).

From an evaluation perspective, if larger cities are overall costlier regardless per capita income and touristic appeal,¹⁹ larger settlements should provide a range of benefits (objective and subjective), beside relative higher incomes (as typically resulting from other studies, but not ours), for which their residents are willing to pay. These benefits might be related to features which typically increase with city size such as services, schools, hospitals, recreational activities, historic centres, shops, work places, stimulating careers, varieties and vibrant life style, at least for those being attracted by them.

This result is contributing to the existing literature on spatial equilibrium across cities, on inter-urban Rosen–Roback alike models, and on urban scaling. According to inter-urban spatial equilibrium expectations, higher incomes are offset by something else such as higher house price and some form of disamenities such as crime, pollutions, congestions, stress and climate.

Our results are in line with such expectations as show a link between higher income and higher housing price. In the simple log-log regression, they show that when doubling the settlement size, average housing price increases by 9% while total municipality income by 0.04%. However, when considering the per capita income in the multiple log-log regressions including control variables (tourism and house price), the latter even nearly decreases when city size increases (by 2% when doubling the population). Similarly, the ratio between per capita income and house price decreases when city size increases, indicating that larger cities have higher house price in proportion to per capita income.

The multiple regression results suggest (holding constant the city size and tourism) a 1% increase in per capita income associated with a ∼0.5% increase in house price (Table 1), or, the other way round, an increase of 1% in house prices is associated with a ∼0.3% increase in per capita income (Table 3), holding fixed population size and ‘per capita’ touristic presences.

Our results are also adding a case in the urban scaling literature by including housing price26 to it, which, as per our knowledge, has not being added yet when taking into account the average settlement house price and using as sample practically all²⁰ the municipalities of a country which in our case were around 8 thousands, ranging from 33 to millions of inhabitants.

However, this big sample was enabled by compromising certain aspects such as not considering (1) functional urban area and other types of aggregative settlement definitions; (2) not involving relative distances²¹ among municipalities, particularly from the closest main one which would have a relevant effect in terms of house prices especially within a functional urban area²²; and (3) not adding other variables and amenities affecting house prices.

The third point could instead be marginally included in the variable already present in the model related to the touristic presences but inevitably cannot take into account other important variables and amenities which are disconnected from touristic appeal despite being relevant for house price levels. It would be beneficial for the model to include some more direct indicators of overall urban quality of life for each settlement.

Regarding the outlier issue, we think that having added income and tourism helped cleaning up extreme case (e.g. smaller settlement but atypically attractive for touristic or financial reasons, or bigger cities but atypically economically a failure related to their size) being a precious source for the estimation of the coefficients.

Other critical points are these listed under the scaling laws section and the appendix regarding how selecting different lower bounds of the sample can alter, even dramatically, the regression coefficients, passing even from linear to superlinear. All these points would need further attentions in future researches.

In extreme synthesis, this study quantified the effect that settlement size has on housing prices, which, in line with others, shows the greater the size the greater the housing prices. This happens also when holding constant the per capita income and the touristic presences.

The latter result opens perspectives and quantifications about economically evaluate the costs – or the benefits if seen in other way around – to live in bigger cities rather than smaller settlements, and vice versa.

Understanding, with the support of quantitative analysis, how settlements size affect residents’ utility would be beneficial for regional and municipal policies in terms of beautification of settlements, favouring, for example, smaller settlements opportunities, or mitigating negative effects of urban size, and any type of decision making process requiring deep knowledge on urban scaling and related quality of life.

Supplemental Material

Supplemental Material - Allometry of housing price in urban scaling: Scaling Law of House Price versus Settlement Sizes

Supplemental Material for Allometry of housing price in urban scaling. Scaling Law of House Price versus Settlement Sizes by Luca S D’Acci in Journal of Environment and Planning B: Urban Analytics and City Science.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Luca S. D’Acci

Data Availability Statement

The datasets supporting this article is publicly available at Borsino Immobiliare: https://borsinoimmobiliare.it/ and ISTAT: .

Supplemental Material

Supplemental material for this article is available online.

Notes

Luca S. D'Acci, Associate Professor at the Polytechnic of Turin, has an educational path chronologically covering: Sciences of Cities and Architecture, Building Engineering, Economic Assessments (Polytechnic of Turin), and Anthropology (University of Oxford), with two degrees summa cum laude, a PhD, and a postdoc. At Erasmus University Rotterdam, he is an academic partner at EHERO, a former Asst. Professor at the School of Social and Behavioural Sciences, and the former Head of the Urban Environment, Sustainability and Climate Change unit. At the University of Birmingham, he is an Honorary Research Fellow in the College of Engineering and Physical Sciences. He was a visiting researcher at the University of Oxford, the University of Cambridge, and ETH Zurich. His work revolves around human settlements enhancing quality of life—intended as individuals and societies achieving healthier, happier, and longer lives—contributing from various angles to how our types of villages and cities can affect this goal.

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The allometry of housing prices in urban scaling laws and an equalised dwellers’ utility across settlement sizes

Abstract

Keywords

Introduction

Data and methods

Results

Equalised dwellers’ utility across settlements

Conclusion, comparisons, critics and perspectives

Supplemental Material

Supplemental Material - Allometry of housing price in urban scaling: Scaling Law of House Price versus Settlement Sizes

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data Availability Statement

Supplemental Material

Notes

References

Supplementary Material