Pollution and city size: can cities be too small?

Pollution and city size: can cities be too small? Abstract We study optimal and equilibrium sizes of cities in a city system model with pollution. Pollution is a function of population size. If pollution is local or per-capita pollution increases with population, equilibrium cities are too large under symmetry; with asymmetric cities, the largest cities are too large and the smallest too small. When pollution is global and per-capita pollution declines with city size, cities may be too small under symmetry; with asymmetric cities, the largest cities are too small and the smallest too large if the marginal damage of pollution is large enough. We calibrate the model to US cities and find that the largest cities may be undersized by 3–4%. 1. Introduction Urbanisation is rapidly increasing, especially in developing countries. According to the UN Population Division, urbanisation worldwide will increase from 51.6% in 2010 to 66.4% in 2050 and from 46.1% to 63.4% in the developing world. Some commentators are afraid that this urbanisation may have adverse environmental consequences. For instance, Seto et al. (2012) argue that the projected urbanisation until 2030 leads to significant loss of biodiversity and increased CO2 emissions due to deforestation and land-use changes. Urban economic activities such as manufacturing production, commuting and residential energy use also contribute to pollution. Figure 1 shows that over the last half century, urbanisation and CO2 emissions have moved together. Of course, this may not be a causal relation. Figure 1 View largeDownload slide World urbanisation and CO2 emissions 1950–2013. Source: UN Population Division, CDIAC. Figure 1 View largeDownload slide World urbanisation and CO2 emissions 1950–2013. Source: UN Population Division, CDIAC. In fact, some writers claim that large, densely populated cities produce lower per-capita emissions. Glaeser and Kahn (2010) show that in the USA, inhabitants of large, densely populated cities such as New York City and San Francisco tend to produce lower CO2 emissions from transport and residential energy use than those living in smaller and less densely populated cities, controlling for factors such as local weather. Glaeser (2011) writes about this Triumph of the City and in the subtitle succinctly states: ‘How our greatest invention makes us richer, smarter, greener, healthier and happier’ (our emphasis). This line of reasoning has prompted organisations such as the OECD and the World Bank to advocate high-density urban development to mitigate environmental pollution. Therefore, an important policy question is whether big cities are good or bad for the environment, especially in developing countries such as China, where new cities are springing up by the minute. While on the one hand, migrants flock to cities to take advantage of their economic opportunities, on the other hand, concern about congestion, environmental pollution and other side effects is mounting. So what is the optimal size of cities that are affected by environmental pollution? And what would be the unregulated equilibrium city size? In this paper, we build a simple model of a city system to study how the equilibrium and optimal city size distributions are affected by environmental pollution. We use a standard monocentric city model, where people work, consume goods and housing in cities. Agglomeration externalities make workers more productive in big cities. Pollution is related to city size since it is a by-product of urban production, commuting and housing. In line with reality, we assume that externalities arising from pollution are not internalised. We distinguish between pollution which is purely local, such as certain kinds of emissions from traffic, and pollution which spills over between cities, such as greenhouse gas (GHG) emissions. When cities are symmetric, we find that with local pollution, equilibrium cities are too large and there are too few of them, mirroring the classic result of Henderson (1974). By contrast, when pollution is global, we find that equilibrium cities may be either too small or too big. The former case can occur when per-capita pollution falls with city size. We also study the model with a given number of asymmetric cities. With local pollution, we find that the largest cities are too large and the smallest cities too small. With global pollution, if per-capita pollution decreases with city size and the marginal damage of pollution is large enough, the largest cities are too small and the smallest too large. We also quantify the extent to which cities may be undersized, using a calibrated version of the model. We use some standard parameter values from the literature, and, using data from Fragkias et al. (2013), we estimate the effect of the size of US metropolitan areas on CO2 emissions. We find that doubling city size reduces per-capita CO2 emissions by 13%. With asymmetric cities, in the case of global pollution, the largest cities may be undersized by 3–4% while the smallest cities are oversized by 6%. If pollution is local, the largest cities are oversized, but by only about 0.3%. Finally, we use an estimate of the degree of pollution spillovers (so pollution is neither completely global nor completely local) and find that the largest city is undersized by 3.5% and the smallest is oversized by 5.8%. Our paper is related to several strands of literature. First, the literature on city systems has studied equilibrium and optimal city sizes. Henderson (1974) first showed that in equilibrium, cities are too big. This finding also comes out of the models by Tolley (1974), Arnott (1979) and Abdel-Rahman (1988). Tolley (1974) considers local pollution and actually argues that it leads to cities being too big. We show that this argument depends on whether pollution spills over to other cities and how per-capita emissions change with city size. Abdel-Rahman and Anas (2004) review this literature and also discuss the role of externalities in city system models (though not externalities arising from pollution). On the other hand, some recent papers show that cities may be too small in equilibrium. Albouy et al. (2016) show that large cities may be too small due to federal taxation, local politics, wedges due to land ownership and the interaction between these. Eeckhout and Guner (2017) also show that spatially uniform taxation may lead to large cities being undersized and that the optimum spatial tax system taxes individuals in large cities less than the current US tax system.1 Like Albouy et al. (2016) and Eeckhout and Guner (2017), we show that cities may be too small.2 While these papers have found that cities may be too small, we differ from them by analyzing equilibrium and optimal city size in a city system model with pollution. Hence, the mechanism generating the divergence of optimum and equilibrium city sizes is different. Since pollution is high on the agenda of policy makers worldwide, we think this is an important topic. Second, there is a small but growing literature on cities and the environment more general. Related to this paper, Gaigné et al. (2012) and Borck and Pflüger (2015) study the interaction of agglomeration, pollution and welfare in models with a given number (two) of cities. The upshot from these papers is that pollution may rise or fall when population density increases or the city system becomes more agglomerated. They differ from our paper in that they theoretically show how pollution is affected by the location of mobile factors between two cities. They focus on the links between city size and pollution from commuting, goods transport, production and residential energy use. In a sense, they study in detail the link between city size and pollution, using microfounded models of urban structure and economic geography, whereas we, instead, model this link as a reduced-form relation in our theoretical model, and later estimate it as a basis for our numerical simulation. There are also some theoretical papers on urban structure and pollution, see Borck (2016), Borck and Brueckner (2018), Dascher (2014), Larson et al. (2012) and Tscharaktschiew and Hirte (2010). These papers, different from ours, study the interaction of urban structure and pollution within cities. Finally, Glaeser and Kahn (2010) and Larson and Yezer (2015) study empirically the relation between GHG emissions or energy use and city structure. Glaeser and Kahn (2010) find that large, dense cities in the USA produce fewer GHG emissions. Morikawa (2013) finds that dense cities in Japan produce lower per-capita energy consumption in the service sector and Blaudin de Thé and Lafourcade (2016) show that residents of low-density suburban households in France use more gasoline for driving. Larson and Yezer (2015) study the effect of city size on energy use in a simulation model, finding that per-capita energy use does not change with city size. A number of papers from other disciplines than economics also study the relation between city size and pollution empirically, with different results, see e.g., the study of Fragkias et al. (2013) using panel data from US cities and Sarzynski (2012) who use a sample of 8038 cities world-wide in 2005.3 Our paper is also concerned with the relation between pollution and city size, which is essential for the comparison of equilibrium and optimal city systems. Finally, the paper is related to a growing literature on equilibrium models with either exogenous or endogenous amenities, see, e.g., Diamond (2016).4 We proceed as follows. The next section introduces the model of a symmetric city system. Section 3 presents the modelling of pollution. In Section 4, we study the equilibrium and optimum size of cities with local and global pollution. In Section 5, we extend the analytical results to the realistic case of asymmetric cities. Section 6 contains a numerical simulation, to get a sense of the possible divergence of optimum and equilibrium city size. The last section concludes. 2. The model with symmetric cities There are m cities in the economy, whose total population is exogenous and denoted by N.5 For now, we assume cities to be identical. The population size in each city is endogenous and given by n=N/m. For simplicity, the city space is linear with unit width and the central business district (CBD) is a spaceless point located at x = 0, while the endogenous city border is denoted x¯ (we focus on the right side of the city for simplicity). All individuals commute to the CBD and have identical preferences given by u(s,z,E)=sαz1−αE−β, (1) and the budget constraint is w=z+rs+tx, (2) where s is the housing floor space (equivalently land consumption), z is the consumption of a composite non-housing good, E is pollution, w is wage income, r is housing rent per square meter, t is commuting cost per mile, x is distance from the CBD and 0<α<1, and β>0. Consumers choose s and z to maximise Equation (1) subject to Equation (2). From this we get optimal housing consumption s(w−tx,r)=α(w−tx)r. (3) Consumers are mobile within and between cities, and land is rented to the highest bidder. We can now solve for households’ bid rent, i.e., the maximum amount the household would be willing to pay per unit of land. Using Equations (3) and (2) in Equation (1) and solving u(z,s,E)=u¯ gives r(w−tx,E,v)=(w−tx)1/αE−β/αv−1/α, (4) where v≡α−α(1−α)−(1−α)u¯. The two equilibrium conditions in the representative city are: r(w−tx¯,E,v)=rA (5) ∫0x¯1s(w−tx,E,v)dx=n, (6) where rA is the agricultural land rent. Equation (5) states that at the city border, land rent just equals the agricultural land rent. Equation (6) says that the population n fits into the city between 0 and x¯. Suppose that there are external economies of scale at the city level, for instance because of gains from individual specialisation. Total city production is assumed to be Y=n1+γ, with α>γ>0 and the individual wage is w=nγ.6 The restriction α>γ is necessary to ensure that in the absence of pollution, utility is an inversely U-shaped function of city size, as shown in Figure 2, so that the optimal city size without pollution is finite [see Equation (11)]. Figure 2 View largeDownload slide Equilibrium and optimal city size with local pollution. Note: The figure shows optimal ( n^) and possible equilibrium (ne) city size with local pollution. Figure 2 View largeDownload slide Equilibrium and optimal city size with local pollution. Note: The figure shows optimal ( n^) and possible equilibrium (ne) city size with local pollution. Substituting Equations (3) and (4) into Equations (5) and (6) and solving gives the city border and indirect utility x¯=nγ[1−rAα(rA+tn)−α]t, (7) v=nγ(rA+tn)−αE−β. (8) Equation (7) shows that the city expands as population grows. It shows that x¯ is not directly affected by pollution although it is indirectly affected by population change through intercity migration. Equation (8) shows the standard tradeoff induced by an increasing city population: on the one hand, utility increases with n due to agglomeration forces, on the other hand, it decreases because of longer commutes and competition for land, which results in higher land rents. In the next section, we model pollution in order to study how it affects this fundamental tradeoff. In particular, the pertinent question is how reallocating population among cities affects the disutility from pollution. Note that we have assumed that land is owned by absentee landowners. As is well known, efficiency analysis requires returning differential land rents to city residents. We show in Appendix B, however, that our results hold qualitatively if land is owned by city residents.7 3. Pollution Pollution in city i is given by Ei(n)=e(ni)+δ∑j=1, j≠ime(nj), i=1,…,m, where e(ni) are local emissions and 0≤δ≤1 measures the degree of pollution spillovers. When δ = 0, pollution is purely local (for instance, some forms of particulate pollution which do not diffuse over long distances). Conversely, when δ = 1, pollution is purely global from the view of our city system, as is the case, for instance, for GHG emissions. Importantly, in the latter case, the environmental externality is independent of the individual’s location. An important issue in the coming analysis will be the relationship between emissions and city population, as captured by the function e(ni). We assume that city population affects local emissions through residents’ economic activities, such as commuting, housing and consumption of other goods whose production causes emissions. What do we know about this relation? In Section 6, we will try to estimate the population elasticity of pollution empirically, but here we briefly discuss theoretical an empirical studies that address this issue. Borck and Pflüger (2015) present a theoretical model in which urban pollution is driven by commuting, residential energy use, industrial and agricultural production and goods transport. They show that per-capita pollution from industrial production and residential energy use decreases with city size, while pollution from commuting and goods transport increases. The total effect of city population on urban pollution is ambiguous and depends on parameters. Some authors have estimated the relation between pollution and city population (or population density) empirically. Most of these papers estimate an equation of the form e=Bnθ, which we will also do in Section 6.8Lamsal et al. (2013) use cross-sectional cross-country data on NO2 and NOX pollution and find that the elasticity of pollution with respect to population density lies between 0.4 and 0.67. Gudipudi et al. (2016) study the effect of population density on CO2 emissions and find an elasticity around 0.6, so doubling population density would reduce per-capita emissions by 24%. Fragkias et al. (2013) also estimate the effect of population on CO2 emissions, using panel data from US cities. They find an elasticity of emissions with respect to population of 0.93. Rybski et al. (2016) conduct a meta-analysis of published articles that study CO2 emissions and city size, and find that in developed countries per-capita emissions decrease with city size while in developing countries per-capita emissions increase with city population. However, most of these estimates seem problematic. For instance, Lamsal et al. (2013) use cross-sectional OLS regressions to estimate the population elasticity of pollution. But this ignores potential confounders that are correlated with population density and pollution. Fragkias et al. (2013) use panel data, but they estimate the model with random effects, which assumes that any unobserved time-invariant heterogeneity between cities is not correlated with pollution. In Section 6, we present an alternative estimate of the population elasticity of CO2 emissions, using the same dataset as Fragkias et al. (2013). 4. Equilibrium and optimum number and size of cities The equilibrium city size in the city system is defined by the solution of vi=v∗ for all i. We focus on symmetric cities. Further, we require the equilibrium to be stable, which implies ∂v(n)/∂n<0. To study optimal city size, we assume a central planner who maximises aggregate welfare mnv(n) with respect to n and m. Using mn = N, this is equivalent to maximising v(n) with respect to n. Note that, from Equation (8) follows v(0)=0 so no one would ever want to live outside of a city. 4.1. Local pollution Suppose first that pollution is entirely local, i.e., δ = 0. Then migration is governed by the following utility differential v(ni)−v(nj)=v^(ni)e(ni)−β−v^(nj)e(nj)−β,where v^(ni)≡niγ(rA+tni)−α, (9) and optimum city size maximises v(ni)=v^(ni)e(ni)−β. We will assume that both v(n) and v^(n) are quasi-concave, which holds (in the neighbourhood of the equilibrium and social optimum) for the parameter values used in our numerical simulations. Moreover, we assume that locally produced pollution e(ni) satisfies e(0)=0 and de/dni>0. Since v(ni) can be shown to be inverted U-shaped, we get the standard result that equilibrium cities are too large, as in Henderson (1974). This can be seen by looking at Figure 2. The figure shows the optimal city size n^ and two potential equilibrium city sizes n˜ and ne.9 Any equilibrium with city size n˜<n^ is unstable: if the city population were to deviate slightly from n˜, migration in or out of the city would occur, as indicated by the arrows. Conversely, any equilibrium with ne>n^ is stable: as indicated by the arrows, a deviation from ne would induce migration flows which restore the equilibrium. Therefore, there is a continuum of equilibria with ne>n^ where n^=n∗ maximises v(ni). We summarise this as: Proposition 1 If cities are symmetric and pollution is purely local, cities are too large in equilibrium. The economic intuition for this result is that in a stable equilibrium, all cities are on the decreasing part of the indirect utility curve so that negative externalities dominate at the margin. Since migrants fail to internalise the consequences of their location choices, cities are too large in equilibrium. 4.2. Global pollution Now, let δ = 1 so that pollution is global from the viewpoint of the economy. Since pollution is global, we can drop the index i from pollution Ei and write the utility difference of living in city i versus j as v(ni)−v(nj)=E−β(v^(ni)−v^(nj)). (10) For E > 0, the individual migration decision is determined by the difference v^(ni)−v^(nj), so global pollution does not affect migration decisions. Let n^ denote the city size which solves max nv^(n). Setting v^′(n)=0 and solving gives n^=γrA(α−γ)t. (11) Then, by the same argument as in Henderson (1974), there is a continuum of stable equilibria with city sizes ne>n^. Figure 3 shows possible equilibrium city sizes. As before, any equilibrium with ne>n^ is stable. Figure 3 View largeDownload slide Equilibrium and optimum city size with global pollution. Notes: The figure shows optimal ( n*) and possible equilibrium (some n>n^) city size with global pollution. If equilibrium city size is on the thick red portion of the v(n) curve, it is smaller than optimal. Figure 3 View largeDownload slide Equilibrium and optimum city size with global pollution. Notes: The figure shows optimal ( n*) and possible equilibrium (some n>n^) city size with global pollution. If equilibrium city size is on the thick red portion of the v(n) curve, it is smaller than optimal. The optimum city size n∗ is found by maximising v(n)=v^(n)E(n)−β. The first-order condition can be written v^′(n)nv^(n)=βE′(n)nE(n). (12) At the optimum, the elasticity of (private utility) with respect to population size should equal the elasticity of total emissions, multiplied by the marginal damage of emissions. We know that ne≥n^ and that n^ maximises v^(n). Since β>0,E(n)>0 and v^(n) is quasi-concave, evaluating Equation (12) at n^ implies that n*<n^ if E′(n^)>0. Since E(n)=m·e(n)=Nne(n), we find cities are definitely too large if per-capita pollution is increasing in city size. Intuitively, in this case making cities larger increases pollution, which increases the disutility from pollution. This reinforces the argument in Henderson-style models which make cities too large. However, if per-capita emissions are decreasing in city size, we find n∗>n^. This opens up the possibility that in equilibrium, cities may be too small. However, since there is a continuum of equilibria with ne>n^, cities may also be too large. Summarising this discussion, we have: Proposition 2 Suppose that pollution is global, i.e., δ = 1. If per-capita emissions increase with n, cities are too large in equilibrium. However, if per-capita emissions decrease with n, cities may be either too small or too large in equilibrium.Figure 3 illustrates the case where pollution is global and per-capita emissions are decreasing with city size. The equilibrium city size is some ne>n^, where n^ is the maximum of the function v^(n). The optimum city size n* is the maximum of the v(n) curve.10 The thick (red) part of the v(n) curve shows the part where the possible equilibrium city size (with ne>n^) is smaller than the optimum size. However, the equilibrium city size may also be larger than n*. As Proposition 2 makes clear, in the case of global emissions whether cities are over- or undersized depends on how per-capita emissions change with city population. However, as already stated in Section 3, not much is known about this relationship. Therefore, we estimate this relationship in Section 6, where we use numerical simulation to gauge whether cities will be over- or undersized in equilibrium. 5. Asymmetric cities 5.1. Equilibrium and social optimum with asymmetric city sizes We now introduce asymmetric cities into the model. To do so, we assume that an individual living in city i obtains utility vi(ni)=Ainiγ(rA+tni)−αEi−β. (13) The variable Ai is a city-level amenity, which could be a consumption amenity such as good weather or a production amenity such as good infrastructure or a favourable geographic location. Without loss of generality, we assume A1=1 and Ai>Ai+1 for i=1,2,…,m−1. As before, pollution is given by Ei(n)=e(ni)+δ∑j≠ie(nj), with e=nθ, θ>0. We assume the number of cities m is fixed and then ask how the optimum allocation of population among these cities differs from the equilibrium one.11 Let v^ be the equilibrium utility level that is attained under free migration and let the equilibrium population vector be n^={n^1,…,n^m}}. The equilibrium city size distribution satisfies vi(n^)=v1(n^)=v for all i=1,…,m. Using Equation (13) and setting A1=1, we can then solve for the amenity levels that are compatible with a free migration equilibrium: Ai=(n^1n^i)γ(rA+tn^irA+tn^1)α(Ei(n^)E1(n^))β. (14) Note that our formulation implies that the amenity levels are uniquely identified by the equilibrium distribution of population up to the normalisation that A1=1. We want to compare the equilibrium city size distribution to the optimal distribution. To characterise the latter, we assume the social planner maximises the sum of utilities max n∑i=1mnivi(n) subject to the population constraint ∑i=1mni=N. Letting λ be the Lagrangean multiplier on the population constraint, the first-order conditions are given by12 vi+ni∂vi∂ni+∑j≠inj∂vj∂ni=λ, i=1,…,m. (15) The last term on the left-hand side of Equation (15) shows the pollution spillovers between cities. The sign of vi−vi+1 is important in the following analysis. While vi=vi+1 holds at the equilibrium, suppose vi>vi+1 holds for all i at the social optimum. Because ∂vi/∂ni is negative in the neighbourhood of a stable equilibrium for almost all i,13 it must be that the optimum ni is smaller than the equilibrium ni in large cities, whereas the optimum ni is larger than the equilibrium ni in small cities. The opposite is true when vi<vi+1. Therefore, we can now show the following: Proposition 3 Assume that θ<1. Then (i) if pollution is close to local, the optimal utility is higher in larger cities. Large cities are too large and small cities are too small at the equilibrium; (ii) if pollution is close to global and the marginal damage of pollution is sufficiently large, the optimal utility is lower in larger cities. Large cities are too small and small cities are too large at the equilibrium. Proof See Appendix A.   ▪ The intuition is as follows. Suppose that pollution is local, as might be the case, say, for NOX. Then, the indirect utility vi is a function of its city size ni only. As shown by Henderson (1974), the indirect utility is decreasing in ni at a stable equilibrium. Start from the equilibrium vi=vi+1 with ni>ni+1 and consider the effect of moving one person from the larger city i to the smaller city i + 1. The utility vi rises to vi+Δi whereas the utility vi+1 falls to vi+1−Δi+1 because vi decreases with ni. The rise Δi and fall Δi+1 are similar in magnitude when the one person is sufficiently small relative to total city size. Since there are more people in city i, however, the sum of niΔi exceeds the sum of ni+1Δi+1. Therefore, it is optimal to reduce the size of larger cities and raise that of smaller cities. As a result, the utility levels in larger cities are higher than those in smaller cities at the optimum. By contrast, if pollution is global, such as in the case of CO2, concentrating population in bigger cities decreases total emissions if θ<1, which benefits residents in all cities. When moving one person from a smaller city i + 1 to a larger city i, utility of city i residents falls while that of i + 1 residents rises. However, due to the global externality, utility of the residents of all other cities also rises. Therefore, as long as the marginal damage of pollution is large enough, social welfare rises. Examining Equation (A.3) in Appendix A, we can further say the following. Given sufficiently large δ (i.e., close to global pollution), large cities are more likely to be too small if the housing expenditure share α, the agricultural land rent rA and the commuting cost t are small. In this case, the crowding effects induced by commuting and tight housing markets in larger cities are outweighed by the beneficial effect of reduced pollution for all other cities. In order to correct the discrepancy between the equilibrium and optimal distributions of city sizes, the national government may impose location taxes and subsidies according to city size. In the case of global pollution with θ<1, in our setup, living in large cities should be subsidized to make them more attractive.14 We compute the optimal tax/subsidy scheme that achieves the socially optimal city size distribution under free mobility numerically below in Section 6.2. 6. Numerical simulation 6.1. Parameter values We now try to assess to what extent optimum and equilibrium city size may diverge, using numerical simulation. We present here the results for the asymmetric city case. The symmetric city case can be found in Borck and Tabuchi (2016). We use the following parameter values. We set the expenditure share of housing to α=0.24 following Davis and Ortalo-Magné (2011), and the agglomeration elasticity to γ=0.05 [see Combes and Gobillon (2015) for an overview].15 From Borck and Brueckner (2018), we set rA=$58,800, the annual land rent of agricultural land in the USA, and t=$503, the annual (monetary plus time) commuting cost per mile in the USA. As described in Appendix C, we calibrate β, using central estimates of the social cost of carbon (SCC) from the literature. Using the central estimate of USD 40.54 per metric ton CO2 for 2015 (assuming 3% discounting, value updated to 2015 USD) from the recent study by Interagency Working Group on SCC, US Government (2015), we find a value of β=0.022.16,17 The estimates of the SCC are surrounded by a lot of uncertainty and some controversy. Therefore, we also use a higher estimate for the SCC. In particular, we use the 95th percentile estimate of USD 118 for 2015 from Interagency Working Group on SCC, US Government (2015) (again at 3% discounting, in 2015 USD), which gives a value of β=0.064. Since much less is known about the emissions intensity θ than about the other parameters (see our discussion in Section 3), we estimate this parameter using US city data. Suppose that total emissions in city i in year t are eit=Bnitθ. Then, per-capita emissions decrease with population size if and only if θ<1. We can then estimate a linear regression of the form log eit=c+θ log nit+εit, (16) where c≡ log B is a constant and ε is the error term. We use data from Fragkias et al. (2013) to estimate CO2 emissions in US core-based statistical areas (CBSAs) (metropolitan statistical areas and micropolitan areas) from 1999 to 2008. The dataset contains CO2 emissions and population for 933 CBSAs. Emissions are based on data from the Vulcan Project, which quantifies US fossil fuel carbon dioxide emissions at 10 km × 10 km grid cells and at the scale of individual factories, power plants, roadways and neighbourhoods on an hourly basis. These are aggregated by Fragkias et al. (2013) to annual observations by CBSA. Table 1 shows the summary statistics. Figure 4 displays a binned scatter plot of log emissions against log population, where all data are pooled. Population varies from 12,340 in the smallest city to 18.7 million in the New York metro area. Per-capita emissions vary by a factor of over 200. Table 1 Summary statistics Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 aMillion metric tons. Source: Fragkias et al. (2013). Table 1 Summary statistics Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 aMillion metric tons. Source: Fragkias et al. (2013). Figure 4 View largeDownload slide CO2 emissions and city size. Notes: The figure shows a scatterplot of CO2 emissions by city size as well as a linear regression (solid line) and local polynomial smoothing (dashed). Source: Own calculations based on data from Fragkias et al. (2013). Figure 4 View largeDownload slide CO2 emissions and city size. Notes: The figure shows a scatterplot of CO2 emissions by city size as well as a linear regression (solid line) and local polynomial smoothing (dashed). Source: Own calculations based on data from Fragkias et al. (2013). We start by estimating Equation (16) by pooled OLS. Results are shown in Column (1) of Table 2. Standard errors are clustered at the CBSA level. The coefficient on population is 0.938 and it is significantly smaller than 1.18 According to this estimate, if population doubles, per-capita emissions would fall by −(2θ−1−1)×100=4.2%.19 Table 2 CO2 emissions and city size Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). Table 2 CO2 emissions and city size Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). This estimate may be biased due to omitted variables or reverse causality. If pollution were local, then our model would predict that individual migration decisions are based on city emissions, so population would be endogenous and OLS estimation would consequently be biased. Given that CO2 is a global pollutant, however, this is not a concern in the present setup, since migration should be independent of local emissions. Therefore, reverse causality may not be a big concern in the current setup. However, cities may still differ in unobserved factors that affect population size and emissions. To mitigate potential biases, we will add various fixed effects to our baseline regression. First, in Column (2), we include time-fixed effects to allow for any time varying factors that are common across CBSAs and affect emissions, such as national business cycles. If these cycles were correlated with population size (say because some cities grow more than others when the economy grows) and also affect CO2 emissions, the OLS coefficient would be biased. The coefficient in Column (2), however, is the same as in the model without time effects. In Column (3), we include CBSA-fixed effects. Some cities may have disproportionately many power plants that service larger geographic areas. Also, cities may differ in some unobserved dimension such as industry structure, climate or other factors that may affect population size and emissions at the same time. As long as this heterogeneity is time invariant, we can control for it by estimating a model with CBSA-fixed effects. As shown in Column (3), the coefficient on population drops to 0.83 once we control for CBSA- and time-fixed effects. Thus, the result that θ<1 does not seem to be driven by unobserved heterogeneity among CBSAs. Finally, there may be macroeconomic effects that affect regions differentially and may be correlated with city size and emissions. For instance, cities in industrial regions and those in regions with a large service sector will be differentially affected by business cycles, and so will their CO2 emissions. To deal with time varying regional differences of this sort, in Column (4), we include interaction effects between year and US census divisions (there are four census regions and nine divisions). As can be seen from the table, the coefficient on population slightly drops to 0.8 and it remains significantly smaller than one.20 Hence, doubling population would reduce per-capita CO2 emissions by −(20.8−1−1)×100=12.8%. This is our preferred estimate, since it controls extensively for time-varying regional heterogeneity, and we will use this value of θ for the numerical simulation. However, we will also use the higher value of 0.94 as a robustness check.21 6.2. Simulation results We now simulate numerically the equilibrium and optimal city size distribution with asymmetric cities. We assume a given number of cities, m, and given total population N. So we exclude the formation of new cities. For our simulation, we use the 180 largest US CBSAs. The total population is the sum of the population sizes of these 180 cities, n = 225,678,243. In Appendix D, we present an alternative simulation where we assume 180 cities with the same total population and a city size distribution which follows Zipf’s law. We then compare the equilibrium number of cities to the social optimum. 6.2.1. Results Out of all CBSAs in the year 2008, we keep the largest 180 cities (see Table 3 for the 10 largest and smallest CBSAs).22 These cities comprise 90% of the total population living in MSAs (252 million) and 80% of the population in CBSAs (283 million). Table 3 The 10 largest and smallest MSAs in the simulation Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Notes: The table displays population and emissions levels by MSA for 2008 from Fragkias et al. (2013). Amenity levels are computed from Equation (14); Δ refers to simulated population and emissions changes at optimum as described in the text. Table 3 The 10 largest and smallest MSAs in the simulation Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Notes: The table displays population and emissions levels by MSA for 2008 from Fragkias et al. (2013). Amenity levels are computed from Equation (14); Δ refers to simulated population and emissions changes at optimum as described in the text. We assume that the current distribution is an equilibrium. We can then compute the level of amenities that rationalise the equilibrium from Equation (14) (the values are in Table 3 for the 10 largest and 10 smallest cities, and in Table OA.1 in the Online Appendix for all cities). Figure 5 shows the equilibrium (light blue/gray) and optimal (dark red/gray) city size distribution. For better visibility, the figure plots the equilibrium and optimal distributions assuming the larger β value of 0.064. Zipf’s law holds fairly well except for the upper tail of the distribution. Note that the largest city in the sample, New York, has 18.7 million whereas according to Zipf’s law the largest city has more than twice that many inhabitants. Table 3 shows the equilibrium population level, emission level as well as the percentage change rates of the optimum relative to equilibrium levels (denoted by Δ). Figure 5 View largeDownload slide Optimal (dark red/gray) and equilibrium city size distributions (light blue/gray). Source: Own calculations based on data from Fragkias et al. (2013). Figure 5 View largeDownload slide Optimal (dark red/gray) and equilibrium city size distributions (light blue/gray). Source: Own calculations based on data from Fragkias et al. (2013). As shown in Figure 5 and Table 3, the three largest cities are undersized by 3–4%. At the social optimum, the largest city in the sample, New York, is undersized by 779,843 of its 18.7 million inhabitants, while the second largest, Los Angeles, is undersized by 450,180. The smallest city in the sample is oversized by about 6.2%. Note that out of the 180 cities, 23 are undersized and the other 157 are oversized at the equilibrium. In total, moving from the equilibrium to the optimal allocation would require moving 5.35 million people or 2.4% of the total population. The table also shows the changes in emissions, which amount to 80% of the changes in population levels. Total emissions fall from 3.74×108 MMT to 3.72×108, or by 0.6%. Mirroring the population changes, Table 4 shows changes in wages and housing rents.23 At the social optimum, the largest cities command higher wages and rents, but as shown in Proposition 3, at the optimum, the large city residents receive lower utility than small city residents. Table 4 The 10 largest and smallest MSAs in the simulation [Equation (2)] Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Note: Δ refers to simulated wage and rent changes at optimum as described in the text. Table 4 The 10 largest and smallest MSAs in the simulation [Equation (2)] Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Note: Δ refers to simulated wage and rent changes at optimum as described in the text. The welfare gain from moving to the optimal city size distribution is small, less than 0.1% of income.24 Small welfare gains from optimal policies are also found in similar models, e.g., by Eeckhout and Guner (2017) and Albouy et al. (2016). We can also compute the level of taxes that could be used by a social planner to support the social optimum. To do so, let the utility level in city i be v(ni,τi), where τi is a city-specific tax or subsidy. Denoting optimal population levels by ni*, the Pigouvian taxes/subsidies τi solve the linear system of equations v(ni*,τi)=v(nj*,τj) (17) ∑j=1mτj=0,  for i,j=1,…,m, i≠j. (18) When the optimal tax/subsidy scheme is applied, free migration will lead to the city size distribution that replicates the social optimum. The computation reveals that the optimum is supported by a subsidy on the 72 largest cities and taxes on the rest of the cities. Living in the largest city, New York, would have to be supported by a subsidy corresponding to 1.4% of the local wage. The tax levels for the small cities are all below 1% of the local wage. Finally, note that we have assumed that individuals are freely mobile. One could study a model with imperfect mobility, for instance by assuming heterogeneous location preferences. Depending on the distribution of these preferences, a social planner would want to move fewer individuals than in the absence of moving costs. 6.2.2. Sensitivity We now briefly describe how the results change when we vary some of the model parameters. First, we increase θ to 0.94 to reflect a potentially higher pollution elasticity. We find that the largest city is undersized by 1.6% and the smallest oversized by 2%. Conversely, when θ decreases to 0.75, the biggest city is undersized by 4.7% and the smallest city is oversized by 7.5%. Since the benefit of concentration is increased the more per-capita pollution decreases with city size, this finding is intuitive. Next, suppose that in line with a high estimate of the SCC, the emissions damage, β, increases from 0.022 to 0.064 (see Section 6). Now, the largest cities are undersized by 9–13% and the smallest oversized by 18%.25 When the agglomeration elasticity, γ, increases from 0.05 to 0.08, agglomeration becomes more efficient, and again, concentration increases at the optimum: the largest city is undersized by 4.8% at the equilibrium and the smallest oversized by 7%.26 Conversely, when we decrease γ to a lower value of 0.02, the largest city is undersized by 3.7% and the smallest one oversized by 5.5%. Lastly, we change the spillover parameter δ. As a first variation, we use completely local emissions, that is, δ = 0. As shown in Proposition 3, the biggest cities will then be oversized and the smallest cities undersized. However, our computations reveal small overall differences between optimum and equilibrium city sizes. Therefore, we conclude that when pollution is completely local, big cities are only slightly too big and the equilibrium city size distribution is close to optimal. A natural question to ask is, what is the actual degree of spillovers from local emissions? To approach this question, we borrow from Borck and Brueckner (2018) who consider optimal energy taxation with local and global emissions. Their results imply that local emissions make up 60.7% of emissions from commuting and 53% of emissions from residential energy use.27 Let us take the average of these values, 57%, so we set δ=1−0.57=0.43. We then find that the largest city is undersized by 3.5% and the smallest is oversized by 5.8%. In the past two decades local pollution has decreased relative to global pollution and this trend is likely to continue (see, e.g., Amann et al., 2013). Moreover, the damage from global warming is projected to increase over time, since GHGs accumulate in the atmosphere and warming is caused by the stock of pollution. Therefore, we tentatively conclude that using realistic parameters, the case for large cities being undersized and small ones oversized persists and will get stronger over time. On the other hand, increasing agricultural land rent rA to $100,000 per year increases the costs of agglomeration. The effect on the optimum size of cities is rather small, however. A very similar result obtains when the per mile commuting cost increases to $750 per year. 7. Conclusion The paper has analyzed the optimum size of cities in an urban model with environmental pollution. When pollution is purely local and cities are symmetric, we find that equilibrium cities are too large, mirroring the finding of Henderson (1974) and others. With asymmetric cities, this translates into the result that big cities are oversized and small cities undersized. However, when pollution is global and per-capita pollution decreases in population size, we find that in a symmetric city model, cities might be inefficiently small, contrary to the standard model. When cities are asymmetric, big cities are undersized and small cities oversized. Over the last decades, global pollution has increased relative to local pollution, and the damage from global warming increases over time. Hence, we conclude that for the future, a policy that favours big cities might actually be warranted. Some possibilities for future research suggest themselves. First, our analysis was based on one estimate of the population elasticity of pollution, which is a central parameter in the analysis. More robust evidence on this parameter clearly seems important. Second, we think it would be interesting to redo the quantitative analysis with data from different countries. For instance, there is a growing number of papers on Chinese cities (e.g., Au and Henderson, 2006). Since the properties of the equilibrium city system and pollution patterns in China and other developing economies are undoubtedly different from developed countries, studying equilibrium and optimum city systems in this context would seem to be relevant. Finally, we could include city governments that maximise residents’ utility. We conjecture that a city government would want a city size even smaller than the free mobility case. The reason is that in the symmetric case, the free mobility outcome is at least the n which maximises v^(n), the utility without pollution, whereas a city government would maximise roughly the utility v˜(n) which would correspond to v^(n) plus the disutility from local pollution, since the local government cares only for the locally produced pollution affecting its own citizens.28 This could magnify the welfare gain from imposing optimal city sizes. Supplementary material Supplementary data for this paper are available at Journal of Economic Geography online. Footnotes 1 Au and Henderson (2006) show that many Chinese are too small due to the migration restrictions of the hukou system. 2 See also Hsieh and Moretti (2017), who show that local housing supply inefficiently restricts migration of workers to high productivity cities such as San Francisco and San Jose. 3 See Section 3 for more on this literature. 4 Redding and Rossi-Hansberg (2017) survey related quantitative spatial economics models. 5 In contrast to Albouy et al. (2016), we don’t consider a rural sector in the economy. 6 Duranton and Puga (2004) show that several different mechanisms lead to the same functional form, such as gains from specialisation, matching, sharing intermediate inputs or learning. 7 Albouy et al. (2016) study cross-city externalities arising from landownership requirements on migrants. 8 Some papers not reviewed here estimate other functional forms, where, for instance, pollution is assumed to be a quadratic function of population. 9 Note that there is a continuum of equilibria, so all that can be said in general is that ne>n^, but the exact location of the equilibrium is indeterminate. 10 The functions have been rescaled so that v(n*)=v^(n*) for better visibility. 11 This differs from, e.g., Albouy et al. (2016) who study a city system with an endogenous number of asymmetric cities. However, looking at a varying number of cities in our context may not be reasonable. When we find that big cities are too small, we might want to close some small cities. Conversely, when big cities are too large, we might want to add more small cities. However, both exercises are not possible to implement in our numerical model, since we do not have data on the universe of all cities, so there is no “smallest” city. 12 We assume an interior solution where 0<ni*<N for all i and that the second-order conditions hold. 13 Tabuchi and Zeng (2004) show that a stable equilibrium requires ∂vi/∂ni<0 for at least m – 1 cities. 14 See also Eeckhout and Guner (2017) for an analysis of taxes related to city size. 15 On the one hand, recent papers have found values of γ lower than 0.05 (Combes and Gobillon, 2015), on the other hand, accounting for dynamic externalities (De La Roca and Puga, 2017) or consumption benefits (Tabuchi and Yoshida, 2000; Glaeser et al., 2001), a value of 0.05 or higher may seem reasonable. We use both higher and lower values of γ later in our sensitivity checks. 16 The study reports averages over three different integrated assessment models. We use their average value across the three models for 2015 at 3% discounting, USD 36 as our central value. 17 In Appendix C, we calibrate β to match the lower SCC value for the USA instead of the SCC for the world used by Interagency Working Group on SCC, US Government (2015). Interestingly, since US population is smaller and per-capita income higher, the resulting value of β=0.0266 does not differ much from this baseline. 18 Using Japanese data for 105 metropolitan employment areas in 2005, we obtain a similar result with a coefficient of 0.902, which is significantly smaller than one. 19 To check for a possibly nonlinear relation between the log of emissions and log population, we also include the square of log population. When we do so, the coefficient on log population drops slightly to 0.9, but the coefficient on the square term is insignificant. Figure 4 compares a linear regression to local polynomial smoothing. The latter shows that the relation between log emissions and log population may be slightly nonlinear, but this nonlinearity seems restricted to the upper and lower tail of the city size distsribution. 20 The p-values for the test for θ<1 are: 0.0001 in Column (1), 0.0001 in Column (2), 0.045 in Column (3) and 0.0499 in Column (4). In all cases, the hypothesis that θ<1 cannot be rejected at the 5% significance level. 21 An interesting question is whether the effect we estimate is due to population size or population density, since the two may affect emissions differently. In Appendix Table A1, we add the log of population density as a regressor. In the OLS regressions (Columns 1 and 2 of Table A1), the coefficient on population increases to 1.04, while the coefficient on density is −0.16. On this count, one could argue that population density, rather than population size, drives the result that population increases emissions less than proportionately. In the fixed effects regressions (Columns 3 and 4 of Table A1), however, the coefficient on population drops to 0.6–0.65 once we add density, while the coefficient on density is positive, although not significant. On this count, one could argue that density does not influence emissions, while population size increases emissions less than proportionately. However, since our model predicts that density is itself a function of population, the results should be interpreted with caution. 22 The full sample of cities is displayed in Table OA.1 in the Online Appendix. 23 The wage rate in city i is assumed to be Bniγ, where we set B to target the average wage in the USA in 2008, $52,029. In order to compute rents, we choose units of housing to match average dwelling size in the USA, 2196 sq ft. [see Borck and Brueckner (2018) for details]. 24 When population is efficiently allocated, as noted, total emissions fall by 0.6%, relative to the equilibrium. Note, however, that the welfare gain from efficiently allocating population in the absence of pollution would also be small with our parameters. 25 For completeness, we also report results for a low SCC value. When β is 0.012 (corresponding to a low SCC value of USD 20), the largest cities are undersized by around 2% and the smallest oversized by 3.3%. 26 In fact, Tabuchi and Yoshida (2000) show that agglomeration externalities from consumption in Japanese cities are about the same size as productive externalities. 27 It is assumed that local pollution is measured in units such that the same marginal damage value can be applied to global and local pollution. 28 This assumes global pollution. With local pollution, a city government maximises v(n) so the allocation would be efficient in the symmetric case. See, e.g., Abdel-Rahman and Anas (2004) and the literature therein. 29 We note that this is partly due to the assumption that the value of β is the same for local and global pollution, but local pollution is much smaller than global pollution. Therefore, for a more realistic simulation, the marginal damage of local emissions relative to global emissions should probably be increased. Acknowledgements We thank two referees, Stefan Bauernschuster, Jan Brueckner, M. Morikawa, T. Morita, Michael Pflüger and M. Yano as well as participants at MCC Berlin, Tinbergen Institute, University Duisburg-Essen, Free University Berlin, in Minneapolis (UEA), Lisbon (EMUEA), Münster and Dresden (VFS), Tokyo (RIETI), Osaka (spatial economics conference) and Pau (Location choices and environmental economics workshop) for comments and suggestions. The first author thanks the German Science Foundation (DFG) and the second author thanks RIETI for financial support. References Abdel-Rahman H. M. ( 1988 ) Product differentiation, monopolistic competition and city size . Regional Science and Urban Economics , 18 : 69 – 86 . Google Scholar CrossRef Search ADS Abdel-Rahman H. M. , Anas A. ( 2004 ) Theories of systems of cities. In Henderson J. V. , Thisse J.-F. (eds) Handbook of Regional and Urban Economics , vol. 4 , pp. 2293 – 2339 . Amsterdam : Elsevier . Albouy D. , Behrens K. , Robert-Nicoud F. , Seegert N. ( 2016 ) The optimal distribution of population across cities. NBER Working Paper No. 22823. Amann M. , Klimont Z. , Wagner F. ( 2013 ) Regional and global emissions of air pollutants: recent trends and future scenarios . Annual Review of Environment and Resources , 38 : 31 – 55 . Google Scholar CrossRef Search ADS Arnott R. ( 1979 ) Optimal city size in a spatial economy . Journal of Urban Economics , 6 : 65 – 89 . Google Scholar CrossRef Search ADS Au C. C. , Henderson J. V. ( 2006 ) Are Chinese cities too small? Review of Economic Studies , 73 : 549 – 576 . Google Scholar CrossRef Search ADS Blaudin de Thé C. , Lafourcade M. ( 2016 ) The carbon footprint of suburbanization: Evidence from French household data. Mimeographed. Borck R. ( 2016 ) Will skyscrapers save the planet? Building height limits and urban greenhouse gas emissions . Regional Science and Urban Economics , 58 : 13 – 25 . Google Scholar CrossRef Search ADS Borck R. , Brueckner J. K. ( 2018 ) Optimal energy taxation in cities . Journal of the Association of Environmental and Resource Economists , 5 : 481 – 516 . Google Scholar CrossRef Search ADS Borck R. , Pflüger M. ( 2015 ) Green cities? Urbanization, trade and the environment. IZA discussion paper 9104. Borck R. , Tabuchi T. ( 2016 ) Pollution and city size: can cities be too small? CESifo Working Paper No. 6152. Combes P.-P. , Gobillon L. ( 2015 ) The empirics of agglomeration economies. In Duranton G. , Henderson J. V. , Strange W. C. (eds) Handbook of Regional and Urban Economics , vol. 5 , pp. 247 – 348 . Amsterdam : Elsevier . Dascher K. ( 2014 ) City Silhouette, World Climate. Available at http://ssrn.com/abstract=2250673. Davis M. A. , Ortalo-Magné F. ( 2011 ) Household expenditures, wages, rents . Review of Economic Dynamics , 14 : 248 – 261 . Google Scholar CrossRef Search ADS De la Roca J. , Puga D. ( 2017 ) Learning by working in big cities . Review of Economic Studies , 84 : 106 – 142 . Google Scholar CrossRef Search ADS Diamond R. ( 2016 ) The determinants and welfare implications of US workers’ diverging location choices by skill: 1980–2000 . American Economic Review , 106 : 479 – 524 . Google Scholar CrossRef Search ADS Duranton G. , Puga D. ( 2004 ) Micro-foundations of urban agglomeration economies. In Henderson J. V. , Thisse J.-F. (eds), Handbook of Regional and Urban Economics , vol. 4 , pp. 2063 – 2117 . Amsterdam : Elsevier . Eeckhout J. , Guner N. ( 2017 ) Optimal spatial taxation: Are big cities too small? CEMFI Working Paper No. 1705. Fragkias M. , Lobo J. , Strumsky D. , Seto K. C. ( 2013 ) Does size matter? Scaling of CO2 emissions and U.S. urban areas . PLoS One 8 ( 6 ): e64727 . Google Scholar CrossRef Search ADS PubMed Gabaix X. ( 2016 ) Power laws in economics: an introduction . Journal of Economic Perspectives , 30 : 185 – 206 . Google Scholar CrossRef Search ADS Gaigné C. , Riou S. , Thisse J.-F. ( 2012 ) Are compact cities environmentally friendly? Journal of Urban Economics , 72 : 123 – 136 . Google Scholar CrossRef Search ADS Glaeser E. ( 2011 ) Triumph of the City: How Our Greatest Invention Makes Us Richer, Smarter, Greener, Healthier, and Happier . New York : Penguin Press . Glaeser E. L. , Kahn M. E. ( 2010 ) The greenness of cities: carbon dioxide emissions and urban development . Journal of Urban Economics , 67 : 404 – 418 . Google Scholar CrossRef Search ADS Glaeser E. L. , Kolko J. , Saiz A. ( 2001 ) Consumer city . Journal of Economic Geography , 1 : 27 – 50 . Google Scholar CrossRef Search ADS Gudipudi R. , Fluschnik T. , Garcìa Cantú Ros A. , Walther C. , Kropp J. P. ( 2016 ) City density and CO2 efficiency . Energy Policy , 91 : 352 – 361 . Google Scholar CrossRef Search ADS Henderson J. V. ( 1974 ) The sizes and types of cities . American Economic Review , 64 : 640 – 656 . Hsieh C.-T. , Moretti E. ( 2017 ) Housing constraints and spatial misallocation. NBER Working Paper No. 21154. Interagency Working Group on Social Cost of Carbon [SCC], US Government . ( 2015 ) Technical Support Document: Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis under Executive Order 12866 . Washington, DC : US Government Printing Office . Lamsal L. N. , Martin R. V. , Parrish D. D. , Krotkov N. A. ( 2013 ) Scaling relationship for NO2 pollution and urban population size: a satellite perspective . Environmental Science & Technology , 47 : 7855 – 7861 . Google Scholar CrossRef Search ADS PubMed Larson W. , Yezer A. M. ( 2015 ) The energy implications of city size and density . Journal of Urban Economics , 90 : 35 – 49 . Google Scholar CrossRef Search ADS Larson W. , Liu F. , Yezer A. ( 2012 ) Energy footprint of the city: effects of urban land use and transportation policies . Journal of Urban Economics , 72 : 147 – 159 . Google Scholar CrossRef Search ADS Morikawa M. ( 2013 ) Population density and efficiency in energy consumption: an empirical analysis of service establishments . Energy Economics , 34 : 1617 – 1622 . Google Scholar CrossRef Search ADS Nordhaus W. D. ( 2017 ) Revisiting the social cost of carbon . Proceedings of the National Academy of Sciences of the United States of America , 114 : 1518 – 1523 . Google Scholar CrossRef Search ADS PubMed Redding S. J. , Rossi-Hansberg E. A. ( 2017 ) Quantitative spatial economics . Annual Review of Economics , 9 : 21 – 58 . Google Scholar CrossRef Search ADS Rybski D. , Reusser D. E. , Winz A.-L. , Fichtner C. , Sterzel T. , Kropp J. P. ( 2016 ) Cities as nuclei of sustainability? Environment and Planning B , 44 : 425 – 440 . Sarzynski A. ( 2012 ) Bigger is not always better: a comparative analysis of cities and their air pollution impact . Urban Studies , 49 : 3121 – 3138 . Google Scholar CrossRef Search ADS Seto K. C. , Güneralp B. , Hutyra L. R. ( 2012 ) Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools . Proceedings of the National Academy of Sciences of the United States of America , 109 : 16083 – 16088 . Google Scholar CrossRef Search ADS PubMed Tabuchi T. , Yoshida A. ( 2000 ) Separating urban agglomeration economies in consumption and production . Journal of Urban Economics , 48 : 70 – 84 . Google Scholar CrossRef Search ADS Tabuchi T. , Zeng D.-Z. ( 2004 ) Stability of spatial equilibrium . Journal of Regional Science , 44 : 641 – 660 . Google Scholar CrossRef Search ADS Tolley G. S. ( 1974 ) The welfare economics of city bigness . Journal of Urban Economics , 1 : 324 – 345 . Google Scholar CrossRef Search ADS Tscharaktschiew S. , Hirte G. ( 2010 ) The drawbacks and opportunities of carbon charges in metropolitan areas—a spatial general equilibrium approach . Ecological Economics , 70 : 339 – 357 . Google Scholar CrossRef Search ADS Appendix A. Proof of Proposition 3 Since ni∂vi∂ni=γvi−αtnirA+tnivi−βθniθviEi−1 ∑j≠inj∂vj∂ni=−βθδniθ−1∑j≠injvjEj−1, we have vi+ni∂vi∂ni+∑j≠inj∂vj∂ni−λ=(1+γ−αtnirA+tni)vi−β(1−δ)θviniθEi−1−βδθZniθ−1−λ=Xivi−βδθZniθ−1−λ=0, (A.1) where Xi≡1+γ−αtnirA+tni−β(1−δ)θniθEi−1 and Z≡∑jnjvjEj−1 is constant across cities. Since Expression (A.1) is the same for i and for i + 1, we can eliminate λ as follows: Xivi−βδθZniθ−1=Xi+1vi+1−βδθZni+1θ−1, which can be rewritten as vi=1Xi(Xi+1vi+1+βδθZniθ−1−βδθZni+1θ−1). Thus, the utility differential is Δv≡vi−vi+1=1Xi[(Xi+1−Xi)vi+1+βδθZ(niθ−1−ni+1θ−1)]=ΔVa+ΔVb, where ΔVa≡αtrAvi+1(ni−ni+1)(rA+tni)(rA+tni+1)ΔVb≡βθ[(1−δ)vi+1(niθEi−1−ni+1θEi+1−1)+δZ(niθ−1−ni+1θ−1)]. While ΔVa>0, the sign of ΔVb is indeterminate. However, the first term of ΔVb is positive whereas the second term of ΔVb is negative because niθEi−1−ni+1θEi+1−1=1EiEi+1(niθEi+1−ni+1θEi)=δEiEi+1(niθ−ni+1θ)∑jnjθ>0 and niθ−1−ni+1θ−1<0,∀θ∈(0,1). Let δ = 0. Then, ΔVb>0 and thus Δv>0. By continuity, this also holds for δ positive but close to zero. Let δ = 1. Solving Δv<0 for β, we have Δv<0⇔β>β˜≡αtrAvi+1(ni+1−ni)θZ(rA+tni)(rA+tni+1)(niθ−1−ni+1θ−1)>0. (A.2) By continuity, Δv<0 holds for sufficiently large β when δ is close to but smaller than one.   ▪ B. Local landownership Suppose that all land in a city is owned by residents, so the total differential land rent is distributed equally to all residents. Let income be given by y=w+R/n, where R=∫0x¯(r(x,v)−rA)dx (A.3) is the total differential land rent. Rewriting Equations (8) and (7) gives x¯=(w+R/n)[1−rAα(rA+tn)−α]t. (A.4) v=(w+R/n)(rA+tn)−αE−β. (A.5)  Substituting from Equation (A.5) into r(x,v)=(w+R/n−tx)1/αE−β/αv−1/α with w=nγ gives r(x,v)=(rA+tn)(nγ+R/n)−1/α(nγ+R/n−tx)1a. Using this in Equation (A.3) and solving gives R=n1+γ[rA1+α−(rA+tn)α(rA−αtn)](rA+tn)1+α−rA1+α. (A.6) Finally, substituting in Equation (A.5) gives v=(1+α)tE−βn1+γ(rA+tn)1+α−rA1+α, (A.7) which is also inverted U-shaped in n. We then redo the simulation exercise from Section 6.2. For the city size distribution described by Zipf’s law with δ = 1, we find the largest city is undersized by 3.6% and the smallest is oversized by 7.3%, so results are very close to the baseline simulation. Varying δ shows that this also holds for local pollution and for the intermediate case δ=0.43. C. Calibration of β We now calibrate β using central estimates of the SCC from Interagency Working Group on SCC, US Government (2015). The MRS between pollution and (non-housing) consumption is MRS=−∂u/∂E∂u/∂z=βz(1−α)E. (A.8) Substituting optimal consumption, z(y)=(1−α)(w−tx) gives MRS=β(w−tx)/E, and integrating over the city gives citywide MRS MRS¯=∫0x¯β(w−tx)E1s(x)dx (A.9) =βw[tn+rA−rA1+α(tn+rA)−α](1+α)tE, (A.10) where we have substituted the optimal s(x) and used Equations (7) and (8). Finally, letting M be world population and n be city population, we get the SCC SCC=βMw[tn+rA−rA1+α(tn+rA)−α](1+α)tnE. (A.11) We use the following parameters: world population in 2015 was M = 7.35 billion (source: UN World Population Prospects, http://esa.un.org/unpd/wpp/Download/Standard/Population/), world per-capita income in 2015 was w=$10,743 (source: UN National Accounts Main Aggregates Database, http://unstats.un.org/unsd/snaama/dnllist.asp) and total CO2 emissions were E = 34,649 million metric tons CO2 in 2011 (source: World Bank, World Development Indicators, http://data.worldbank.org/indicator/EN.ATM.CO2E.KT/countries). We set n = 750,000 and from Borck and Brueckner (2018), we use t=$503.53,α=0.24 and rA=$58,800. The target value for the SCC is USD 40.54 per ton CO2, using the central value from Interagency Working Group on SCC, US Government (2015) (converted from 2007 to 2015 USD). Using the stated parameters, setting Equation (A.11) equal to 40.54 and solving gives β=0.022. The other values in the text are solved likewise using different values for the SCC. C.1. Calibration using SCC for USA An alternative approach would be to calibrate β to target the SCC for the USA, on the assumption that national policy makers care about national welfare, not the welfare of world citizens. Therefore, we recalibrate β to target the SCC value for the USA computed by Nordhaus (2017). His central US estimate is $4.78, much lower than the $31.21 value for the world. We therefore now use a target value of $5 instead of the world value of $40. Setting income per capita for the USA at $56,516 and population at 324 million and solving Equation (A.11) gives a value of β=0.0266. Interestingly, therefore, the results do not differ much when we use an SCC value for the USA instead of the world SCC. Table A1 CO2 emissions and city size (including population density) Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). Table A1 CO2 emissions and city size (including population density) Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). D. Simulation assuming Zipf’s law In this exercise, we assume that the city size distribution follows Zipf’s law. As is well known, this is a good approximation for city systems in most countries, except at the very top and bottom of the distribution (Gabaix, 2016). We continue to assume n = 225,678,243 and distribute this population to 180 cities so the city size distribution exactly follows Zipf’s law. The largest city has 39 million inhabitants (more than twice the New York City metro area), the second largest 19.5 million and the smallest city has 217,180 inhabitants. We compute the amenity levels from Equation (14) for these given population sizes. We first assume δ = 1 so pollution is global. We find that the biggest city is undersized by 4.8% (1.9 million) while the smallest one is oversized by 7.8%. When pollution is purely local, δ = 0, we find that the divergence between optimal and equilibrium city sizes is small, as in the real cities sample.29 The largest city is oversized by 0.01% and the smallest city is undersized by 0.06%. Finally, for an intermediate value, δ=0.43, the largest city is undersized 3.6% and the smallest oversized by 7%. © The Author(s) (2018). Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Economic Geography Oxford University Press

Pollution and city size: can cities be too small?

Loading next page...
 
/lp/ou_press/pollution-and-city-size-can-cities-be-too-small-KGsufy5p57
Publisher
Oxford University Press
Copyright
© The Author(s) (2018). Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com
ISSN
1468-2702
eISSN
1468-2710
D.O.I.
10.1093/jeg/lby017
Publisher site
See Article on Publisher Site

Abstract

Abstract We study optimal and equilibrium sizes of cities in a city system model with pollution. Pollution is a function of population size. If pollution is local or per-capita pollution increases with population, equilibrium cities are too large under symmetry; with asymmetric cities, the largest cities are too large and the smallest too small. When pollution is global and per-capita pollution declines with city size, cities may be too small under symmetry; with asymmetric cities, the largest cities are too small and the smallest too large if the marginal damage of pollution is large enough. We calibrate the model to US cities and find that the largest cities may be undersized by 3–4%. 1. Introduction Urbanisation is rapidly increasing, especially in developing countries. According to the UN Population Division, urbanisation worldwide will increase from 51.6% in 2010 to 66.4% in 2050 and from 46.1% to 63.4% in the developing world. Some commentators are afraid that this urbanisation may have adverse environmental consequences. For instance, Seto et al. (2012) argue that the projected urbanisation until 2030 leads to significant loss of biodiversity and increased CO2 emissions due to deforestation and land-use changes. Urban economic activities such as manufacturing production, commuting and residential energy use also contribute to pollution. Figure 1 shows that over the last half century, urbanisation and CO2 emissions have moved together. Of course, this may not be a causal relation. Figure 1 View largeDownload slide World urbanisation and CO2 emissions 1950–2013. Source: UN Population Division, CDIAC. Figure 1 View largeDownload slide World urbanisation and CO2 emissions 1950–2013. Source: UN Population Division, CDIAC. In fact, some writers claim that large, densely populated cities produce lower per-capita emissions. Glaeser and Kahn (2010) show that in the USA, inhabitants of large, densely populated cities such as New York City and San Francisco tend to produce lower CO2 emissions from transport and residential energy use than those living in smaller and less densely populated cities, controlling for factors such as local weather. Glaeser (2011) writes about this Triumph of the City and in the subtitle succinctly states: ‘How our greatest invention makes us richer, smarter, greener, healthier and happier’ (our emphasis). This line of reasoning has prompted organisations such as the OECD and the World Bank to advocate high-density urban development to mitigate environmental pollution. Therefore, an important policy question is whether big cities are good or bad for the environment, especially in developing countries such as China, where new cities are springing up by the minute. While on the one hand, migrants flock to cities to take advantage of their economic opportunities, on the other hand, concern about congestion, environmental pollution and other side effects is mounting. So what is the optimal size of cities that are affected by environmental pollution? And what would be the unregulated equilibrium city size? In this paper, we build a simple model of a city system to study how the equilibrium and optimal city size distributions are affected by environmental pollution. We use a standard monocentric city model, where people work, consume goods and housing in cities. Agglomeration externalities make workers more productive in big cities. Pollution is related to city size since it is a by-product of urban production, commuting and housing. In line with reality, we assume that externalities arising from pollution are not internalised. We distinguish between pollution which is purely local, such as certain kinds of emissions from traffic, and pollution which spills over between cities, such as greenhouse gas (GHG) emissions. When cities are symmetric, we find that with local pollution, equilibrium cities are too large and there are too few of them, mirroring the classic result of Henderson (1974). By contrast, when pollution is global, we find that equilibrium cities may be either too small or too big. The former case can occur when per-capita pollution falls with city size. We also study the model with a given number of asymmetric cities. With local pollution, we find that the largest cities are too large and the smallest cities too small. With global pollution, if per-capita pollution decreases with city size and the marginal damage of pollution is large enough, the largest cities are too small and the smallest too large. We also quantify the extent to which cities may be undersized, using a calibrated version of the model. We use some standard parameter values from the literature, and, using data from Fragkias et al. (2013), we estimate the effect of the size of US metropolitan areas on CO2 emissions. We find that doubling city size reduces per-capita CO2 emissions by 13%. With asymmetric cities, in the case of global pollution, the largest cities may be undersized by 3–4% while the smallest cities are oversized by 6%. If pollution is local, the largest cities are oversized, but by only about 0.3%. Finally, we use an estimate of the degree of pollution spillovers (so pollution is neither completely global nor completely local) and find that the largest city is undersized by 3.5% and the smallest is oversized by 5.8%. Our paper is related to several strands of literature. First, the literature on city systems has studied equilibrium and optimal city sizes. Henderson (1974) first showed that in equilibrium, cities are too big. This finding also comes out of the models by Tolley (1974), Arnott (1979) and Abdel-Rahman (1988). Tolley (1974) considers local pollution and actually argues that it leads to cities being too big. We show that this argument depends on whether pollution spills over to other cities and how per-capita emissions change with city size. Abdel-Rahman and Anas (2004) review this literature and also discuss the role of externalities in city system models (though not externalities arising from pollution). On the other hand, some recent papers show that cities may be too small in equilibrium. Albouy et al. (2016) show that large cities may be too small due to federal taxation, local politics, wedges due to land ownership and the interaction between these. Eeckhout and Guner (2017) also show that spatially uniform taxation may lead to large cities being undersized and that the optimum spatial tax system taxes individuals in large cities less than the current US tax system.1 Like Albouy et al. (2016) and Eeckhout and Guner (2017), we show that cities may be too small.2 While these papers have found that cities may be too small, we differ from them by analyzing equilibrium and optimal city size in a city system model with pollution. Hence, the mechanism generating the divergence of optimum and equilibrium city sizes is different. Since pollution is high on the agenda of policy makers worldwide, we think this is an important topic. Second, there is a small but growing literature on cities and the environment more general. Related to this paper, Gaigné et al. (2012) and Borck and Pflüger (2015) study the interaction of agglomeration, pollution and welfare in models with a given number (two) of cities. The upshot from these papers is that pollution may rise or fall when population density increases or the city system becomes more agglomerated. They differ from our paper in that they theoretically show how pollution is affected by the location of mobile factors between two cities. They focus on the links between city size and pollution from commuting, goods transport, production and residential energy use. In a sense, they study in detail the link between city size and pollution, using microfounded models of urban structure and economic geography, whereas we, instead, model this link as a reduced-form relation in our theoretical model, and later estimate it as a basis for our numerical simulation. There are also some theoretical papers on urban structure and pollution, see Borck (2016), Borck and Brueckner (2018), Dascher (2014), Larson et al. (2012) and Tscharaktschiew and Hirte (2010). These papers, different from ours, study the interaction of urban structure and pollution within cities. Finally, Glaeser and Kahn (2010) and Larson and Yezer (2015) study empirically the relation between GHG emissions or energy use and city structure. Glaeser and Kahn (2010) find that large, dense cities in the USA produce fewer GHG emissions. Morikawa (2013) finds that dense cities in Japan produce lower per-capita energy consumption in the service sector and Blaudin de Thé and Lafourcade (2016) show that residents of low-density suburban households in France use more gasoline for driving. Larson and Yezer (2015) study the effect of city size on energy use in a simulation model, finding that per-capita energy use does not change with city size. A number of papers from other disciplines than economics also study the relation between city size and pollution empirically, with different results, see e.g., the study of Fragkias et al. (2013) using panel data from US cities and Sarzynski (2012) who use a sample of 8038 cities world-wide in 2005.3 Our paper is also concerned with the relation between pollution and city size, which is essential for the comparison of equilibrium and optimal city systems. Finally, the paper is related to a growing literature on equilibrium models with either exogenous or endogenous amenities, see, e.g., Diamond (2016).4 We proceed as follows. The next section introduces the model of a symmetric city system. Section 3 presents the modelling of pollution. In Section 4, we study the equilibrium and optimum size of cities with local and global pollution. In Section 5, we extend the analytical results to the realistic case of asymmetric cities. Section 6 contains a numerical simulation, to get a sense of the possible divergence of optimum and equilibrium city size. The last section concludes. 2. The model with symmetric cities There are m cities in the economy, whose total population is exogenous and denoted by N.5 For now, we assume cities to be identical. The population size in each city is endogenous and given by n=N/m. For simplicity, the city space is linear with unit width and the central business district (CBD) is a spaceless point located at x = 0, while the endogenous city border is denoted x¯ (we focus on the right side of the city for simplicity). All individuals commute to the CBD and have identical preferences given by u(s,z,E)=sαz1−αE−β, (1) and the budget constraint is w=z+rs+tx, (2) where s is the housing floor space (equivalently land consumption), z is the consumption of a composite non-housing good, E is pollution, w is wage income, r is housing rent per square meter, t is commuting cost per mile, x is distance from the CBD and 0<α<1, and β>0. Consumers choose s and z to maximise Equation (1) subject to Equation (2). From this we get optimal housing consumption s(w−tx,r)=α(w−tx)r. (3) Consumers are mobile within and between cities, and land is rented to the highest bidder. We can now solve for households’ bid rent, i.e., the maximum amount the household would be willing to pay per unit of land. Using Equations (3) and (2) in Equation (1) and solving u(z,s,E)=u¯ gives r(w−tx,E,v)=(w−tx)1/αE−β/αv−1/α, (4) where v≡α−α(1−α)−(1−α)u¯. The two equilibrium conditions in the representative city are: r(w−tx¯,E,v)=rA (5) ∫0x¯1s(w−tx,E,v)dx=n, (6) where rA is the agricultural land rent. Equation (5) states that at the city border, land rent just equals the agricultural land rent. Equation (6) says that the population n fits into the city between 0 and x¯. Suppose that there are external economies of scale at the city level, for instance because of gains from individual specialisation. Total city production is assumed to be Y=n1+γ, with α>γ>0 and the individual wage is w=nγ.6 The restriction α>γ is necessary to ensure that in the absence of pollution, utility is an inversely U-shaped function of city size, as shown in Figure 2, so that the optimal city size without pollution is finite [see Equation (11)]. Figure 2 View largeDownload slide Equilibrium and optimal city size with local pollution. Note: The figure shows optimal ( n^) and possible equilibrium (ne) city size with local pollution. Figure 2 View largeDownload slide Equilibrium and optimal city size with local pollution. Note: The figure shows optimal ( n^) and possible equilibrium (ne) city size with local pollution. Substituting Equations (3) and (4) into Equations (5) and (6) and solving gives the city border and indirect utility x¯=nγ[1−rAα(rA+tn)−α]t, (7) v=nγ(rA+tn)−αE−β. (8) Equation (7) shows that the city expands as population grows. It shows that x¯ is not directly affected by pollution although it is indirectly affected by population change through intercity migration. Equation (8) shows the standard tradeoff induced by an increasing city population: on the one hand, utility increases with n due to agglomeration forces, on the other hand, it decreases because of longer commutes and competition for land, which results in higher land rents. In the next section, we model pollution in order to study how it affects this fundamental tradeoff. In particular, the pertinent question is how reallocating population among cities affects the disutility from pollution. Note that we have assumed that land is owned by absentee landowners. As is well known, efficiency analysis requires returning differential land rents to city residents. We show in Appendix B, however, that our results hold qualitatively if land is owned by city residents.7 3. Pollution Pollution in city i is given by Ei(n)=e(ni)+δ∑j=1, j≠ime(nj), i=1,…,m, where e(ni) are local emissions and 0≤δ≤1 measures the degree of pollution spillovers. When δ = 0, pollution is purely local (for instance, some forms of particulate pollution which do not diffuse over long distances). Conversely, when δ = 1, pollution is purely global from the view of our city system, as is the case, for instance, for GHG emissions. Importantly, in the latter case, the environmental externality is independent of the individual’s location. An important issue in the coming analysis will be the relationship between emissions and city population, as captured by the function e(ni). We assume that city population affects local emissions through residents’ economic activities, such as commuting, housing and consumption of other goods whose production causes emissions. What do we know about this relation? In Section 6, we will try to estimate the population elasticity of pollution empirically, but here we briefly discuss theoretical an empirical studies that address this issue. Borck and Pflüger (2015) present a theoretical model in which urban pollution is driven by commuting, residential energy use, industrial and agricultural production and goods transport. They show that per-capita pollution from industrial production and residential energy use decreases with city size, while pollution from commuting and goods transport increases. The total effect of city population on urban pollution is ambiguous and depends on parameters. Some authors have estimated the relation between pollution and city population (or population density) empirically. Most of these papers estimate an equation of the form e=Bnθ, which we will also do in Section 6.8Lamsal et al. (2013) use cross-sectional cross-country data on NO2 and NOX pollution and find that the elasticity of pollution with respect to population density lies between 0.4 and 0.67. Gudipudi et al. (2016) study the effect of population density on CO2 emissions and find an elasticity around 0.6, so doubling population density would reduce per-capita emissions by 24%. Fragkias et al. (2013) also estimate the effect of population on CO2 emissions, using panel data from US cities. They find an elasticity of emissions with respect to population of 0.93. Rybski et al. (2016) conduct a meta-analysis of published articles that study CO2 emissions and city size, and find that in developed countries per-capita emissions decrease with city size while in developing countries per-capita emissions increase with city population. However, most of these estimates seem problematic. For instance, Lamsal et al. (2013) use cross-sectional OLS regressions to estimate the population elasticity of pollution. But this ignores potential confounders that are correlated with population density and pollution. Fragkias et al. (2013) use panel data, but they estimate the model with random effects, which assumes that any unobserved time-invariant heterogeneity between cities is not correlated with pollution. In Section 6, we present an alternative estimate of the population elasticity of CO2 emissions, using the same dataset as Fragkias et al. (2013). 4. Equilibrium and optimum number and size of cities The equilibrium city size in the city system is defined by the solution of vi=v∗ for all i. We focus on symmetric cities. Further, we require the equilibrium to be stable, which implies ∂v(n)/∂n<0. To study optimal city size, we assume a central planner who maximises aggregate welfare mnv(n) with respect to n and m. Using mn = N, this is equivalent to maximising v(n) with respect to n. Note that, from Equation (8) follows v(0)=0 so no one would ever want to live outside of a city. 4.1. Local pollution Suppose first that pollution is entirely local, i.e., δ = 0. Then migration is governed by the following utility differential v(ni)−v(nj)=v^(ni)e(ni)−β−v^(nj)e(nj)−β,where v^(ni)≡niγ(rA+tni)−α, (9) and optimum city size maximises v(ni)=v^(ni)e(ni)−β. We will assume that both v(n) and v^(n) are quasi-concave, which holds (in the neighbourhood of the equilibrium and social optimum) for the parameter values used in our numerical simulations. Moreover, we assume that locally produced pollution e(ni) satisfies e(0)=0 and de/dni>0. Since v(ni) can be shown to be inverted U-shaped, we get the standard result that equilibrium cities are too large, as in Henderson (1974). This can be seen by looking at Figure 2. The figure shows the optimal city size n^ and two potential equilibrium city sizes n˜ and ne.9 Any equilibrium with city size n˜<n^ is unstable: if the city population were to deviate slightly from n˜, migration in or out of the city would occur, as indicated by the arrows. Conversely, any equilibrium with ne>n^ is stable: as indicated by the arrows, a deviation from ne would induce migration flows which restore the equilibrium. Therefore, there is a continuum of equilibria with ne>n^ where n^=n∗ maximises v(ni). We summarise this as: Proposition 1 If cities are symmetric and pollution is purely local, cities are too large in equilibrium. The economic intuition for this result is that in a stable equilibrium, all cities are on the decreasing part of the indirect utility curve so that negative externalities dominate at the margin. Since migrants fail to internalise the consequences of their location choices, cities are too large in equilibrium. 4.2. Global pollution Now, let δ = 1 so that pollution is global from the viewpoint of the economy. Since pollution is global, we can drop the index i from pollution Ei and write the utility difference of living in city i versus j as v(ni)−v(nj)=E−β(v^(ni)−v^(nj)). (10) For E > 0, the individual migration decision is determined by the difference v^(ni)−v^(nj), so global pollution does not affect migration decisions. Let n^ denote the city size which solves max nv^(n). Setting v^′(n)=0 and solving gives n^=γrA(α−γ)t. (11) Then, by the same argument as in Henderson (1974), there is a continuum of stable equilibria with city sizes ne>n^. Figure 3 shows possible equilibrium city sizes. As before, any equilibrium with ne>n^ is stable. Figure 3 View largeDownload slide Equilibrium and optimum city size with global pollution. Notes: The figure shows optimal ( n*) and possible equilibrium (some n>n^) city size with global pollution. If equilibrium city size is on the thick red portion of the v(n) curve, it is smaller than optimal. Figure 3 View largeDownload slide Equilibrium and optimum city size with global pollution. Notes: The figure shows optimal ( n*) and possible equilibrium (some n>n^) city size with global pollution. If equilibrium city size is on the thick red portion of the v(n) curve, it is smaller than optimal. The optimum city size n∗ is found by maximising v(n)=v^(n)E(n)−β. The first-order condition can be written v^′(n)nv^(n)=βE′(n)nE(n). (12) At the optimum, the elasticity of (private utility) with respect to population size should equal the elasticity of total emissions, multiplied by the marginal damage of emissions. We know that ne≥n^ and that n^ maximises v^(n). Since β>0,E(n)>0 and v^(n) is quasi-concave, evaluating Equation (12) at n^ implies that n*<n^ if E′(n^)>0. Since E(n)=m·e(n)=Nne(n), we find cities are definitely too large if per-capita pollution is increasing in city size. Intuitively, in this case making cities larger increases pollution, which increases the disutility from pollution. This reinforces the argument in Henderson-style models which make cities too large. However, if per-capita emissions are decreasing in city size, we find n∗>n^. This opens up the possibility that in equilibrium, cities may be too small. However, since there is a continuum of equilibria with ne>n^, cities may also be too large. Summarising this discussion, we have: Proposition 2 Suppose that pollution is global, i.e., δ = 1. If per-capita emissions increase with n, cities are too large in equilibrium. However, if per-capita emissions decrease with n, cities may be either too small or too large in equilibrium.Figure 3 illustrates the case where pollution is global and per-capita emissions are decreasing with city size. The equilibrium city size is some ne>n^, where n^ is the maximum of the function v^(n). The optimum city size n* is the maximum of the v(n) curve.10 The thick (red) part of the v(n) curve shows the part where the possible equilibrium city size (with ne>n^) is smaller than the optimum size. However, the equilibrium city size may also be larger than n*. As Proposition 2 makes clear, in the case of global emissions whether cities are over- or undersized depends on how per-capita emissions change with city population. However, as already stated in Section 3, not much is known about this relationship. Therefore, we estimate this relationship in Section 6, where we use numerical simulation to gauge whether cities will be over- or undersized in equilibrium. 5. Asymmetric cities 5.1. Equilibrium and social optimum with asymmetric city sizes We now introduce asymmetric cities into the model. To do so, we assume that an individual living in city i obtains utility vi(ni)=Ainiγ(rA+tni)−αEi−β. (13) The variable Ai is a city-level amenity, which could be a consumption amenity such as good weather or a production amenity such as good infrastructure or a favourable geographic location. Without loss of generality, we assume A1=1 and Ai>Ai+1 for i=1,2,…,m−1. As before, pollution is given by Ei(n)=e(ni)+δ∑j≠ie(nj), with e=nθ, θ>0. We assume the number of cities m is fixed and then ask how the optimum allocation of population among these cities differs from the equilibrium one.11 Let v^ be the equilibrium utility level that is attained under free migration and let the equilibrium population vector be n^={n^1,…,n^m}}. The equilibrium city size distribution satisfies vi(n^)=v1(n^)=v for all i=1,…,m. Using Equation (13) and setting A1=1, we can then solve for the amenity levels that are compatible with a free migration equilibrium: Ai=(n^1n^i)γ(rA+tn^irA+tn^1)α(Ei(n^)E1(n^))β. (14) Note that our formulation implies that the amenity levels are uniquely identified by the equilibrium distribution of population up to the normalisation that A1=1. We want to compare the equilibrium city size distribution to the optimal distribution. To characterise the latter, we assume the social planner maximises the sum of utilities max n∑i=1mnivi(n) subject to the population constraint ∑i=1mni=N. Letting λ be the Lagrangean multiplier on the population constraint, the first-order conditions are given by12 vi+ni∂vi∂ni+∑j≠inj∂vj∂ni=λ, i=1,…,m. (15) The last term on the left-hand side of Equation (15) shows the pollution spillovers between cities. The sign of vi−vi+1 is important in the following analysis. While vi=vi+1 holds at the equilibrium, suppose vi>vi+1 holds for all i at the social optimum. Because ∂vi/∂ni is negative in the neighbourhood of a stable equilibrium for almost all i,13 it must be that the optimum ni is smaller than the equilibrium ni in large cities, whereas the optimum ni is larger than the equilibrium ni in small cities. The opposite is true when vi<vi+1. Therefore, we can now show the following: Proposition 3 Assume that θ<1. Then (i) if pollution is close to local, the optimal utility is higher in larger cities. Large cities are too large and small cities are too small at the equilibrium; (ii) if pollution is close to global and the marginal damage of pollution is sufficiently large, the optimal utility is lower in larger cities. Large cities are too small and small cities are too large at the equilibrium. Proof See Appendix A.   ▪ The intuition is as follows. Suppose that pollution is local, as might be the case, say, for NOX. Then, the indirect utility vi is a function of its city size ni only. As shown by Henderson (1974), the indirect utility is decreasing in ni at a stable equilibrium. Start from the equilibrium vi=vi+1 with ni>ni+1 and consider the effect of moving one person from the larger city i to the smaller city i + 1. The utility vi rises to vi+Δi whereas the utility vi+1 falls to vi+1−Δi+1 because vi decreases with ni. The rise Δi and fall Δi+1 are similar in magnitude when the one person is sufficiently small relative to total city size. Since there are more people in city i, however, the sum of niΔi exceeds the sum of ni+1Δi+1. Therefore, it is optimal to reduce the size of larger cities and raise that of smaller cities. As a result, the utility levels in larger cities are higher than those in smaller cities at the optimum. By contrast, if pollution is global, such as in the case of CO2, concentrating population in bigger cities decreases total emissions if θ<1, which benefits residents in all cities. When moving one person from a smaller city i + 1 to a larger city i, utility of city i residents falls while that of i + 1 residents rises. However, due to the global externality, utility of the residents of all other cities also rises. Therefore, as long as the marginal damage of pollution is large enough, social welfare rises. Examining Equation (A.3) in Appendix A, we can further say the following. Given sufficiently large δ (i.e., close to global pollution), large cities are more likely to be too small if the housing expenditure share α, the agricultural land rent rA and the commuting cost t are small. In this case, the crowding effects induced by commuting and tight housing markets in larger cities are outweighed by the beneficial effect of reduced pollution for all other cities. In order to correct the discrepancy between the equilibrium and optimal distributions of city sizes, the national government may impose location taxes and subsidies according to city size. In the case of global pollution with θ<1, in our setup, living in large cities should be subsidized to make them more attractive.14 We compute the optimal tax/subsidy scheme that achieves the socially optimal city size distribution under free mobility numerically below in Section 6.2. 6. Numerical simulation 6.1. Parameter values We now try to assess to what extent optimum and equilibrium city size may diverge, using numerical simulation. We present here the results for the asymmetric city case. The symmetric city case can be found in Borck and Tabuchi (2016). We use the following parameter values. We set the expenditure share of housing to α=0.24 following Davis and Ortalo-Magné (2011), and the agglomeration elasticity to γ=0.05 [see Combes and Gobillon (2015) for an overview].15 From Borck and Brueckner (2018), we set rA=$58,800, the annual land rent of agricultural land in the USA, and t=$503, the annual (monetary plus time) commuting cost per mile in the USA. As described in Appendix C, we calibrate β, using central estimates of the social cost of carbon (SCC) from the literature. Using the central estimate of USD 40.54 per metric ton CO2 for 2015 (assuming 3% discounting, value updated to 2015 USD) from the recent study by Interagency Working Group on SCC, US Government (2015), we find a value of β=0.022.16,17 The estimates of the SCC are surrounded by a lot of uncertainty and some controversy. Therefore, we also use a higher estimate for the SCC. In particular, we use the 95th percentile estimate of USD 118 for 2015 from Interagency Working Group on SCC, US Government (2015) (again at 3% discounting, in 2015 USD), which gives a value of β=0.064. Since much less is known about the emissions intensity θ than about the other parameters (see our discussion in Section 3), we estimate this parameter using US city data. Suppose that total emissions in city i in year t are eit=Bnitθ. Then, per-capita emissions decrease with population size if and only if θ<1. We can then estimate a linear regression of the form log eit=c+θ log nit+εit, (16) where c≡ log B is a constant and ε is the error term. We use data from Fragkias et al. (2013) to estimate CO2 emissions in US core-based statistical areas (CBSAs) (metropolitan statistical areas and micropolitan areas) from 1999 to 2008. The dataset contains CO2 emissions and population for 933 CBSAs. Emissions are based on data from the Vulcan Project, which quantifies US fossil fuel carbon dioxide emissions at 10 km × 10 km grid cells and at the scale of individual factories, power plants, roadways and neighbourhoods on an hourly basis. These are aggregated by Fragkias et al. (2013) to annual observations by CBSA. Table 1 shows the summary statistics. Figure 4 displays a binned scatter plot of log emissions against log population, where all data are pooled. Population varies from 12,340 in the smallest city to 18.7 million in the New York metro area. Per-capita emissions vary by a factor of over 200. Table 1 Summary statistics Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 aMillion metric tons. Source: Fragkias et al. (2013). Table 1 Summary statistics Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 Variable Observation Mean Standard deviation Min Max Emissions (MMT)a 9330 1.542 3.924 0.036 71.06 Per-capita emissions (MMT)a 9330 8.4×10−6 0.000017 1.1×10−6 0.0002556 Population 9330 290,029 994,482.1 12,340 1.87×107 aMillion metric tons. Source: Fragkias et al. (2013). Figure 4 View largeDownload slide CO2 emissions and city size. Notes: The figure shows a scatterplot of CO2 emissions by city size as well as a linear regression (solid line) and local polynomial smoothing (dashed). Source: Own calculations based on data from Fragkias et al. (2013). Figure 4 View largeDownload slide CO2 emissions and city size. Notes: The figure shows a scatterplot of CO2 emissions by city size as well as a linear regression (solid line) and local polynomial smoothing (dashed). Source: Own calculations based on data from Fragkias et al. (2013). We start by estimating Equation (16) by pooled OLS. Results are shown in Column (1) of Table 2. Standard errors are clustered at the CBSA level. The coefficient on population is 0.938 and it is significantly smaller than 1.18 According to this estimate, if population doubles, per-capita emissions would fall by −(2θ−1−1)×100=4.2%.19 Table 2 CO2 emissions and city size Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). Table 2 CO2 emissions and city size Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 0.938*** 0.938*** 0.834*** 0.802*** (0.0168) (0.0168) (0.0978) (0.120) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9330 9330 9330 9330 R-squared (within) 0.681 0.682 0.128 0.147 Number of CBSAs 933 933 933 933 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). This estimate may be biased due to omitted variables or reverse causality. If pollution were local, then our model would predict that individual migration decisions are based on city emissions, so population would be endogenous and OLS estimation would consequently be biased. Given that CO2 is a global pollutant, however, this is not a concern in the present setup, since migration should be independent of local emissions. Therefore, reverse causality may not be a big concern in the current setup. However, cities may still differ in unobserved factors that affect population size and emissions. To mitigate potential biases, we will add various fixed effects to our baseline regression. First, in Column (2), we include time-fixed effects to allow for any time varying factors that are common across CBSAs and affect emissions, such as national business cycles. If these cycles were correlated with population size (say because some cities grow more than others when the economy grows) and also affect CO2 emissions, the OLS coefficient would be biased. The coefficient in Column (2), however, is the same as in the model without time effects. In Column (3), we include CBSA-fixed effects. Some cities may have disproportionately many power plants that service larger geographic areas. Also, cities may differ in some unobserved dimension such as industry structure, climate or other factors that may affect population size and emissions at the same time. As long as this heterogeneity is time invariant, we can control for it by estimating a model with CBSA-fixed effects. As shown in Column (3), the coefficient on population drops to 0.83 once we control for CBSA- and time-fixed effects. Thus, the result that θ<1 does not seem to be driven by unobserved heterogeneity among CBSAs. Finally, there may be macroeconomic effects that affect regions differentially and may be correlated with city size and emissions. For instance, cities in industrial regions and those in regions with a large service sector will be differentially affected by business cycles, and so will their CO2 emissions. To deal with time varying regional differences of this sort, in Column (4), we include interaction effects between year and US census divisions (there are four census regions and nine divisions). As can be seen from the table, the coefficient on population slightly drops to 0.8 and it remains significantly smaller than one.20 Hence, doubling population would reduce per-capita CO2 emissions by −(20.8−1−1)×100=12.8%. This is our preferred estimate, since it controls extensively for time-varying regional heterogeneity, and we will use this value of θ for the numerical simulation. However, we will also use the higher value of 0.94 as a robustness check.21 6.2. Simulation results We now simulate numerically the equilibrium and optimal city size distribution with asymmetric cities. We assume a given number of cities, m, and given total population N. So we exclude the formation of new cities. For our simulation, we use the 180 largest US CBSAs. The total population is the sum of the population sizes of these 180 cities, n = 225,678,243. In Appendix D, we present an alternative simulation where we assume 180 cities with the same total population and a city size distribution which follows Zipf’s law. We then compare the equilibrium number of cities to the social optimum. 6.2.1. Results Out of all CBSAs in the year 2008, we keep the largest 180 cities (see Table 3 for the 10 largest and smallest CBSAs).22 These cities comprise 90% of the total population living in MSAs (252 million) and 80% of the population in CBSAs (283 million). Table 3 The 10 largest and smallest MSAs in the simulation Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Notes: The table displays population and emissions levels by MSA for 2008 from Fragkias et al. (2013). Amenity levels are computed from Equation (14); Δ refers to simulated population and emissions changes at optimum as described in the text. Table 3 The 10 largest and smallest MSAs in the simulation Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Rank MSA Amenity Population Δ Population (%) Emissions Δ Emissions (%) 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 1 18,672,355 4.18 40.7 3.33 2 Los Angeles–Long Beach–Santa Ana, CA 0.929 12,692,740 3.55 24.5 2.83 3 Chicago–Joliet–Naperville, IL–IN–WI 0.877 9,384,555 3.02 42.6 2.41 4 Dallas–Fort Worth–Arlington, TX 0.81 6,158,022 2.24 17.5 1.79 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 0.804 5,906,917 2.16 15.9 1.72 6 Houston–Sugar Land–Baytown, TX 0.798 5,702,270 2.09 24. 1.67 7 Miami–Fort Lauderdale–Pompano Beach, FL 0.792 5,454,633 2.0 12.7 1.6 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 0.79 5,391,607 1.98 19.3 1.58 9 Atlanta–Sandy Springs–Marietta, GA 0.783 5,152,141 1.89 25.4 1.51 10 Boston–Cambridge–Quincy, MA–NH 0.763 4,483,141 1.61 13.7 1.28 171 Clarksville, TN–KY 0.444 261,530 −5.94 4.9 −4.78 172 Myrtle Beach–North Myrtle Beach–Conway, SC 0.444 260,609 −5.95 1.01 −4.79 173 Santa Cruz–Watsonville, CA 0.443 256,520 −6.0 0.626 −4.83 174 Cedar Rapids, IA 0.443 255,503 −6.02 0.92 −4.84 175 Binghamton, NY 0.442 252,527 −6.06 0.869 −4.88 176 Merced, CA 0.441 250,538 −6.09 0.747 −4.9 177 Lynchburg, VA 0.44 249,299 −6.1 0.972 −4.91 178 Bremerton–Silverdale, WA 0.44 246,912 −6.13 0.4 −4.94 179 Amarillo, TX 0.439 244,454 −6.17 3.3 −4.96 180 Olympia, WA 0.439 244,332 −6.17 0.467 −4.97 Notes: The table displays population and emissions levels by MSA for 2008 from Fragkias et al. (2013). Amenity levels are computed from Equation (14); Δ refers to simulated population and emissions changes at optimum as described in the text. We assume that the current distribution is an equilibrium. We can then compute the level of amenities that rationalise the equilibrium from Equation (14) (the values are in Table 3 for the 10 largest and 10 smallest cities, and in Table OA.1 in the Online Appendix for all cities). Figure 5 shows the equilibrium (light blue/gray) and optimal (dark red/gray) city size distribution. For better visibility, the figure plots the equilibrium and optimal distributions assuming the larger β value of 0.064. Zipf’s law holds fairly well except for the upper tail of the distribution. Note that the largest city in the sample, New York, has 18.7 million whereas according to Zipf’s law the largest city has more than twice that many inhabitants. Table 3 shows the equilibrium population level, emission level as well as the percentage change rates of the optimum relative to equilibrium levels (denoted by Δ). Figure 5 View largeDownload slide Optimal (dark red/gray) and equilibrium city size distributions (light blue/gray). Source: Own calculations based on data from Fragkias et al. (2013). Figure 5 View largeDownload slide Optimal (dark red/gray) and equilibrium city size distributions (light blue/gray). Source: Own calculations based on data from Fragkias et al. (2013). As shown in Figure 5 and Table 3, the three largest cities are undersized by 3–4%. At the social optimum, the largest city in the sample, New York, is undersized by 779,843 of its 18.7 million inhabitants, while the second largest, Los Angeles, is undersized by 450,180. The smallest city in the sample is oversized by about 6.2%. Note that out of the 180 cities, 23 are undersized and the other 157 are oversized at the equilibrium. In total, moving from the equilibrium to the optimal allocation would require moving 5.35 million people or 2.4% of the total population. The table also shows the changes in emissions, which amount to 80% of the changes in population levels. Total emissions fall from 3.74×108 MMT to 3.72×108, or by 0.6%. Mirroring the population changes, Table 4 shows changes in wages and housing rents.23 At the social optimum, the largest cities command higher wages and rents, but as shown in Proposition 3, at the optimum, the large city residents receive lower utility than small city residents. Table 4 The 10 largest and smallest MSAs in the simulation [Equation (2)] Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Note: Δ refers to simulated wage and rent changes at optimum as described in the text. Table 4 The 10 largest and smallest MSAs in the simulation [Equation (2)] Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Rank MSA Population Wage Δ Wage (%) Housing rent Δ Housing rent 1 New York–Northern New Jersey–Long Island, NY–NJ–PA 18,672,355 61,165.1 0.20 23.3157 4.18 2 Los Angeles–Long Beach–Santa Ana, CA 12,692,740 59,995.9 0.17 15.8491 3.55 3 Chicago–Joliet–Naperville, IL–IN–WI 9,384,555 59,096.8 0.15 11.7183 3.02 4 Dallas–Fort Worth–Arlington, TX 6,158,022 57,865. 0.11 7.68946 2.24 5 Philadelphia–Camden–Wilmington, PA–NJ–DE–MD 5,906,917 57,744.6 0.11 7.37592 2.16 6 Houston–Sugar Land–Baytown, TX 5,702,270 57,642.9 0.10 7.12038 2.09 7 Miami–Fort Lauderdale–Pompano Beach, FL 5,454,633 57,515.1 0.10 6.81117 2.00 8 Washington–Arlington–Alexandria, DC–VA–MD–WV 5,391,607 57,481.7 0.10 6.73247 1.98 9 Atlanta–Sandy Springs–Marietta, GA 5,152,141 57,351.3 0.09 6.43345 1.89 10 Boston–Cambridge–Quincy, MA–NH 4,483,141 56,953.8 0.08 5.5981 1.61 171 Clarksville, TN–KY 261,530 49,410.5 −0.31 0.32671 −5.94 172 Myrtle Beach–North Myrtle Beach–Conway, SC 260,609 49,401.8 −0.31 0.32556 −5.94 173 Santa Cruz–Watsonville, CA 256,520 49,362.8 −0.31 0.320454 −6.00 174 Cedar Rapids, IA 255,503 49,353. −0.31 0.319184 −6.02 175 Binghamton, NY 252,527 49,324.1 −0.31 0.315468 −6.06 176 Merced, CA 250,538 49,304.6 −0.31 0.312984 −6.08 177 Lynchburg, VA 249,299 49,292.4 −0.31 0.311437 −6.10 178 Bremerton–Silverdale, WA 246,912 49,268.7 −0.32 0.308457 −6.13 179 Amarillo, TX 244,454 49,244. −0.32 −0.305388 −6.17 180 Olympia, WA 244,332 49,242.8 −0.32 0.305235 −6.17 Note: Δ refers to simulated wage and rent changes at optimum as described in the text. The welfare gain from moving to the optimal city size distribution is small, less than 0.1% of income.24 Small welfare gains from optimal policies are also found in similar models, e.g., by Eeckhout and Guner (2017) and Albouy et al. (2016). We can also compute the level of taxes that could be used by a social planner to support the social optimum. To do so, let the utility level in city i be v(ni,τi), where τi is a city-specific tax or subsidy. Denoting optimal population levels by ni*, the Pigouvian taxes/subsidies τi solve the linear system of equations v(ni*,τi)=v(nj*,τj) (17) ∑j=1mτj=0,  for i,j=1,…,m, i≠j. (18) When the optimal tax/subsidy scheme is applied, free migration will lead to the city size distribution that replicates the social optimum. The computation reveals that the optimum is supported by a subsidy on the 72 largest cities and taxes on the rest of the cities. Living in the largest city, New York, would have to be supported by a subsidy corresponding to 1.4% of the local wage. The tax levels for the small cities are all below 1% of the local wage. Finally, note that we have assumed that individuals are freely mobile. One could study a model with imperfect mobility, for instance by assuming heterogeneous location preferences. Depending on the distribution of these preferences, a social planner would want to move fewer individuals than in the absence of moving costs. 6.2.2. Sensitivity We now briefly describe how the results change when we vary some of the model parameters. First, we increase θ to 0.94 to reflect a potentially higher pollution elasticity. We find that the largest city is undersized by 1.6% and the smallest oversized by 2%. Conversely, when θ decreases to 0.75, the biggest city is undersized by 4.7% and the smallest city is oversized by 7.5%. Since the benefit of concentration is increased the more per-capita pollution decreases with city size, this finding is intuitive. Next, suppose that in line with a high estimate of the SCC, the emissions damage, β, increases from 0.022 to 0.064 (see Section 6). Now, the largest cities are undersized by 9–13% and the smallest oversized by 18%.25 When the agglomeration elasticity, γ, increases from 0.05 to 0.08, agglomeration becomes more efficient, and again, concentration increases at the optimum: the largest city is undersized by 4.8% at the equilibrium and the smallest oversized by 7%.26 Conversely, when we decrease γ to a lower value of 0.02, the largest city is undersized by 3.7% and the smallest one oversized by 5.5%. Lastly, we change the spillover parameter δ. As a first variation, we use completely local emissions, that is, δ = 0. As shown in Proposition 3, the biggest cities will then be oversized and the smallest cities undersized. However, our computations reveal small overall differences between optimum and equilibrium city sizes. Therefore, we conclude that when pollution is completely local, big cities are only slightly too big and the equilibrium city size distribution is close to optimal. A natural question to ask is, what is the actual degree of spillovers from local emissions? To approach this question, we borrow from Borck and Brueckner (2018) who consider optimal energy taxation with local and global emissions. Their results imply that local emissions make up 60.7% of emissions from commuting and 53% of emissions from residential energy use.27 Let us take the average of these values, 57%, so we set δ=1−0.57=0.43. We then find that the largest city is undersized by 3.5% and the smallest is oversized by 5.8%. In the past two decades local pollution has decreased relative to global pollution and this trend is likely to continue (see, e.g., Amann et al., 2013). Moreover, the damage from global warming is projected to increase over time, since GHGs accumulate in the atmosphere and warming is caused by the stock of pollution. Therefore, we tentatively conclude that using realistic parameters, the case for large cities being undersized and small ones oversized persists and will get stronger over time. On the other hand, increasing agricultural land rent rA to $100,000 per year increases the costs of agglomeration. The effect on the optimum size of cities is rather small, however. A very similar result obtains when the per mile commuting cost increases to $750 per year. 7. Conclusion The paper has analyzed the optimum size of cities in an urban model with environmental pollution. When pollution is purely local and cities are symmetric, we find that equilibrium cities are too large, mirroring the finding of Henderson (1974) and others. With asymmetric cities, this translates into the result that big cities are oversized and small cities undersized. However, when pollution is global and per-capita pollution decreases in population size, we find that in a symmetric city model, cities might be inefficiently small, contrary to the standard model. When cities are asymmetric, big cities are undersized and small cities oversized. Over the last decades, global pollution has increased relative to local pollution, and the damage from global warming increases over time. Hence, we conclude that for the future, a policy that favours big cities might actually be warranted. Some possibilities for future research suggest themselves. First, our analysis was based on one estimate of the population elasticity of pollution, which is a central parameter in the analysis. More robust evidence on this parameter clearly seems important. Second, we think it would be interesting to redo the quantitative analysis with data from different countries. For instance, there is a growing number of papers on Chinese cities (e.g., Au and Henderson, 2006). Since the properties of the equilibrium city system and pollution patterns in China and other developing economies are undoubtedly different from developed countries, studying equilibrium and optimum city systems in this context would seem to be relevant. Finally, we could include city governments that maximise residents’ utility. We conjecture that a city government would want a city size even smaller than the free mobility case. The reason is that in the symmetric case, the free mobility outcome is at least the n which maximises v^(n), the utility without pollution, whereas a city government would maximise roughly the utility v˜(n) which would correspond to v^(n) plus the disutility from local pollution, since the local government cares only for the locally produced pollution affecting its own citizens.28 This could magnify the welfare gain from imposing optimal city sizes. Supplementary material Supplementary data for this paper are available at Journal of Economic Geography online. Footnotes 1 Au and Henderson (2006) show that many Chinese are too small due to the migration restrictions of the hukou system. 2 See also Hsieh and Moretti (2017), who show that local housing supply inefficiently restricts migration of workers to high productivity cities such as San Francisco and San Jose. 3 See Section 3 for more on this literature. 4 Redding and Rossi-Hansberg (2017) survey related quantitative spatial economics models. 5 In contrast to Albouy et al. (2016), we don’t consider a rural sector in the economy. 6 Duranton and Puga (2004) show that several different mechanisms lead to the same functional form, such as gains from specialisation, matching, sharing intermediate inputs or learning. 7 Albouy et al. (2016) study cross-city externalities arising from landownership requirements on migrants. 8 Some papers not reviewed here estimate other functional forms, where, for instance, pollution is assumed to be a quadratic function of population. 9 Note that there is a continuum of equilibria, so all that can be said in general is that ne>n^, but the exact location of the equilibrium is indeterminate. 10 The functions have been rescaled so that v(n*)=v^(n*) for better visibility. 11 This differs from, e.g., Albouy et al. (2016) who study a city system with an endogenous number of asymmetric cities. However, looking at a varying number of cities in our context may not be reasonable. When we find that big cities are too small, we might want to close some small cities. Conversely, when big cities are too large, we might want to add more small cities. However, both exercises are not possible to implement in our numerical model, since we do not have data on the universe of all cities, so there is no “smallest” city. 12 We assume an interior solution where 0<ni*<N for all i and that the second-order conditions hold. 13 Tabuchi and Zeng (2004) show that a stable equilibrium requires ∂vi/∂ni<0 for at least m – 1 cities. 14 See also Eeckhout and Guner (2017) for an analysis of taxes related to city size. 15 On the one hand, recent papers have found values of γ lower than 0.05 (Combes and Gobillon, 2015), on the other hand, accounting for dynamic externalities (De La Roca and Puga, 2017) or consumption benefits (Tabuchi and Yoshida, 2000; Glaeser et al., 2001), a value of 0.05 or higher may seem reasonable. We use both higher and lower values of γ later in our sensitivity checks. 16 The study reports averages over three different integrated assessment models. We use their average value across the three models for 2015 at 3% discounting, USD 36 as our central value. 17 In Appendix C, we calibrate β to match the lower SCC value for the USA instead of the SCC for the world used by Interagency Working Group on SCC, US Government (2015). Interestingly, since US population is smaller and per-capita income higher, the resulting value of β=0.0266 does not differ much from this baseline. 18 Using Japanese data for 105 metropolitan employment areas in 2005, we obtain a similar result with a coefficient of 0.902, which is significantly smaller than one. 19 To check for a possibly nonlinear relation between the log of emissions and log population, we also include the square of log population. When we do so, the coefficient on log population drops slightly to 0.9, but the coefficient on the square term is insignificant. Figure 4 compares a linear regression to local polynomial smoothing. The latter shows that the relation between log emissions and log population may be slightly nonlinear, but this nonlinearity seems restricted to the upper and lower tail of the city size distsribution. 20 The p-values for the test for θ<1 are: 0.0001 in Column (1), 0.0001 in Column (2), 0.045 in Column (3) and 0.0499 in Column (4). In all cases, the hypothesis that θ<1 cannot be rejected at the 5% significance level. 21 An interesting question is whether the effect we estimate is due to population size or population density, since the two may affect emissions differently. In Appendix Table A1, we add the log of population density as a regressor. In the OLS regressions (Columns 1 and 2 of Table A1), the coefficient on population increases to 1.04, while the coefficient on density is −0.16. On this count, one could argue that population density, rather than population size, drives the result that population increases emissions less than proportionately. In the fixed effects regressions (Columns 3 and 4 of Table A1), however, the coefficient on population drops to 0.6–0.65 once we add density, while the coefficient on density is positive, although not significant. On this count, one could argue that density does not influence emissions, while population size increases emissions less than proportionately. However, since our model predicts that density is itself a function of population, the results should be interpreted with caution. 22 The full sample of cities is displayed in Table OA.1 in the Online Appendix. 23 The wage rate in city i is assumed to be Bniγ, where we set B to target the average wage in the USA in 2008, $52,029. In order to compute rents, we choose units of housing to match average dwelling size in the USA, 2196 sq ft. [see Borck and Brueckner (2018) for details]. 24 When population is efficiently allocated, as noted, total emissions fall by 0.6%, relative to the equilibrium. Note, however, that the welfare gain from efficiently allocating population in the absence of pollution would also be small with our parameters. 25 For completeness, we also report results for a low SCC value. When β is 0.012 (corresponding to a low SCC value of USD 20), the largest cities are undersized by around 2% and the smallest oversized by 3.3%. 26 In fact, Tabuchi and Yoshida (2000) show that agglomeration externalities from consumption in Japanese cities are about the same size as productive externalities. 27 It is assumed that local pollution is measured in units such that the same marginal damage value can be applied to global and local pollution. 28 This assumes global pollution. With local pollution, a city government maximises v(n) so the allocation would be efficient in the symmetric case. See, e.g., Abdel-Rahman and Anas (2004) and the literature therein. 29 We note that this is partly due to the assumption that the value of β is the same for local and global pollution, but local pollution is much smaller than global pollution. Therefore, for a more realistic simulation, the marginal damage of local emissions relative to global emissions should probably be increased. Acknowledgements We thank two referees, Stefan Bauernschuster, Jan Brueckner, M. Morikawa, T. Morita, Michael Pflüger and M. Yano as well as participants at MCC Berlin, Tinbergen Institute, University Duisburg-Essen, Free University Berlin, in Minneapolis (UEA), Lisbon (EMUEA), Münster and Dresden (VFS), Tokyo (RIETI), Osaka (spatial economics conference) and Pau (Location choices and environmental economics workshop) for comments and suggestions. The first author thanks the German Science Foundation (DFG) and the second author thanks RIETI for financial support. References Abdel-Rahman H. M. ( 1988 ) Product differentiation, monopolistic competition and city size . Regional Science and Urban Economics , 18 : 69 – 86 . Google Scholar CrossRef Search ADS Abdel-Rahman H. M. , Anas A. ( 2004 ) Theories of systems of cities. In Henderson J. V. , Thisse J.-F. (eds) Handbook of Regional and Urban Economics , vol. 4 , pp. 2293 – 2339 . Amsterdam : Elsevier . Albouy D. , Behrens K. , Robert-Nicoud F. , Seegert N. ( 2016 ) The optimal distribution of population across cities. NBER Working Paper No. 22823. Amann M. , Klimont Z. , Wagner F. ( 2013 ) Regional and global emissions of air pollutants: recent trends and future scenarios . Annual Review of Environment and Resources , 38 : 31 – 55 . Google Scholar CrossRef Search ADS Arnott R. ( 1979 ) Optimal city size in a spatial economy . Journal of Urban Economics , 6 : 65 – 89 . Google Scholar CrossRef Search ADS Au C. C. , Henderson J. V. ( 2006 ) Are Chinese cities too small? Review of Economic Studies , 73 : 549 – 576 . Google Scholar CrossRef Search ADS Blaudin de Thé C. , Lafourcade M. ( 2016 ) The carbon footprint of suburbanization: Evidence from French household data. Mimeographed. Borck R. ( 2016 ) Will skyscrapers save the planet? Building height limits and urban greenhouse gas emissions . Regional Science and Urban Economics , 58 : 13 – 25 . Google Scholar CrossRef Search ADS Borck R. , Brueckner J. K. ( 2018 ) Optimal energy taxation in cities . Journal of the Association of Environmental and Resource Economists , 5 : 481 – 516 . Google Scholar CrossRef Search ADS Borck R. , Pflüger M. ( 2015 ) Green cities? Urbanization, trade and the environment. IZA discussion paper 9104. Borck R. , Tabuchi T. ( 2016 ) Pollution and city size: can cities be too small? CESifo Working Paper No. 6152. Combes P.-P. , Gobillon L. ( 2015 ) The empirics of agglomeration economies. In Duranton G. , Henderson J. V. , Strange W. C. (eds) Handbook of Regional and Urban Economics , vol. 5 , pp. 247 – 348 . Amsterdam : Elsevier . Dascher K. ( 2014 ) City Silhouette, World Climate. Available at http://ssrn.com/abstract=2250673. Davis M. A. , Ortalo-Magné F. ( 2011 ) Household expenditures, wages, rents . Review of Economic Dynamics , 14 : 248 – 261 . Google Scholar CrossRef Search ADS De la Roca J. , Puga D. ( 2017 ) Learning by working in big cities . Review of Economic Studies , 84 : 106 – 142 . Google Scholar CrossRef Search ADS Diamond R. ( 2016 ) The determinants and welfare implications of US workers’ diverging location choices by skill: 1980–2000 . American Economic Review , 106 : 479 – 524 . Google Scholar CrossRef Search ADS Duranton G. , Puga D. ( 2004 ) Micro-foundations of urban agglomeration economies. In Henderson J. V. , Thisse J.-F. (eds), Handbook of Regional and Urban Economics , vol. 4 , pp. 2063 – 2117 . Amsterdam : Elsevier . Eeckhout J. , Guner N. ( 2017 ) Optimal spatial taxation: Are big cities too small? CEMFI Working Paper No. 1705. Fragkias M. , Lobo J. , Strumsky D. , Seto K. C. ( 2013 ) Does size matter? Scaling of CO2 emissions and U.S. urban areas . PLoS One 8 ( 6 ): e64727 . Google Scholar CrossRef Search ADS PubMed Gabaix X. ( 2016 ) Power laws in economics: an introduction . Journal of Economic Perspectives , 30 : 185 – 206 . Google Scholar CrossRef Search ADS Gaigné C. , Riou S. , Thisse J.-F. ( 2012 ) Are compact cities environmentally friendly? Journal of Urban Economics , 72 : 123 – 136 . Google Scholar CrossRef Search ADS Glaeser E. ( 2011 ) Triumph of the City: How Our Greatest Invention Makes Us Richer, Smarter, Greener, Healthier, and Happier . New York : Penguin Press . Glaeser E. L. , Kahn M. E. ( 2010 ) The greenness of cities: carbon dioxide emissions and urban development . Journal of Urban Economics , 67 : 404 – 418 . Google Scholar CrossRef Search ADS Glaeser E. L. , Kolko J. , Saiz A. ( 2001 ) Consumer city . Journal of Economic Geography , 1 : 27 – 50 . Google Scholar CrossRef Search ADS Gudipudi R. , Fluschnik T. , Garcìa Cantú Ros A. , Walther C. , Kropp J. P. ( 2016 ) City density and CO2 efficiency . Energy Policy , 91 : 352 – 361 . Google Scholar CrossRef Search ADS Henderson J. V. ( 1974 ) The sizes and types of cities . American Economic Review , 64 : 640 – 656 . Hsieh C.-T. , Moretti E. ( 2017 ) Housing constraints and spatial misallocation. NBER Working Paper No. 21154. Interagency Working Group on Social Cost of Carbon [SCC], US Government . ( 2015 ) Technical Support Document: Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis under Executive Order 12866 . Washington, DC : US Government Printing Office . Lamsal L. N. , Martin R. V. , Parrish D. D. , Krotkov N. A. ( 2013 ) Scaling relationship for NO2 pollution and urban population size: a satellite perspective . Environmental Science & Technology , 47 : 7855 – 7861 . Google Scholar CrossRef Search ADS PubMed Larson W. , Yezer A. M. ( 2015 ) The energy implications of city size and density . Journal of Urban Economics , 90 : 35 – 49 . Google Scholar CrossRef Search ADS Larson W. , Liu F. , Yezer A. ( 2012 ) Energy footprint of the city: effects of urban land use and transportation policies . Journal of Urban Economics , 72 : 147 – 159 . Google Scholar CrossRef Search ADS Morikawa M. ( 2013 ) Population density and efficiency in energy consumption: an empirical analysis of service establishments . Energy Economics , 34 : 1617 – 1622 . Google Scholar CrossRef Search ADS Nordhaus W. D. ( 2017 ) Revisiting the social cost of carbon . Proceedings of the National Academy of Sciences of the United States of America , 114 : 1518 – 1523 . Google Scholar CrossRef Search ADS PubMed Redding S. J. , Rossi-Hansberg E. A. ( 2017 ) Quantitative spatial economics . Annual Review of Economics , 9 : 21 – 58 . Google Scholar CrossRef Search ADS Rybski D. , Reusser D. E. , Winz A.-L. , Fichtner C. , Sterzel T. , Kropp J. P. ( 2016 ) Cities as nuclei of sustainability? Environment and Planning B , 44 : 425 – 440 . Sarzynski A. ( 2012 ) Bigger is not always better: a comparative analysis of cities and their air pollution impact . Urban Studies , 49 : 3121 – 3138 . Google Scholar CrossRef Search ADS Seto K. C. , Güneralp B. , Hutyra L. R. ( 2012 ) Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools . Proceedings of the National Academy of Sciences of the United States of America , 109 : 16083 – 16088 . Google Scholar CrossRef Search ADS PubMed Tabuchi T. , Yoshida A. ( 2000 ) Separating urban agglomeration economies in consumption and production . Journal of Urban Economics , 48 : 70 – 84 . Google Scholar CrossRef Search ADS Tabuchi T. , Zeng D.-Z. ( 2004 ) Stability of spatial equilibrium . Journal of Regional Science , 44 : 641 – 660 . Google Scholar CrossRef Search ADS Tolley G. S. ( 1974 ) The welfare economics of city bigness . Journal of Urban Economics , 1 : 324 – 345 . Google Scholar CrossRef Search ADS Tscharaktschiew S. , Hirte G. ( 2010 ) The drawbacks and opportunities of carbon charges in metropolitan areas—a spatial general equilibrium approach . Ecological Economics , 70 : 339 – 357 . Google Scholar CrossRef Search ADS Appendix A. Proof of Proposition 3 Since ni∂vi∂ni=γvi−αtnirA+tnivi−βθniθviEi−1 ∑j≠inj∂vj∂ni=−βθδniθ−1∑j≠injvjEj−1, we have vi+ni∂vi∂ni+∑j≠inj∂vj∂ni−λ=(1+γ−αtnirA+tni)vi−β(1−δ)θviniθEi−1−βδθZniθ−1−λ=Xivi−βδθZniθ−1−λ=0, (A.1) where Xi≡1+γ−αtnirA+tni−β(1−δ)θniθEi−1 and Z≡∑jnjvjEj−1 is constant across cities. Since Expression (A.1) is the same for i and for i + 1, we can eliminate λ as follows: Xivi−βδθZniθ−1=Xi+1vi+1−βδθZni+1θ−1, which can be rewritten as vi=1Xi(Xi+1vi+1+βδθZniθ−1−βδθZni+1θ−1). Thus, the utility differential is Δv≡vi−vi+1=1Xi[(Xi+1−Xi)vi+1+βδθZ(niθ−1−ni+1θ−1)]=ΔVa+ΔVb, where ΔVa≡αtrAvi+1(ni−ni+1)(rA+tni)(rA+tni+1)ΔVb≡βθ[(1−δ)vi+1(niθEi−1−ni+1θEi+1−1)+δZ(niθ−1−ni+1θ−1)]. While ΔVa>0, the sign of ΔVb is indeterminate. However, the first term of ΔVb is positive whereas the second term of ΔVb is negative because niθEi−1−ni+1θEi+1−1=1EiEi+1(niθEi+1−ni+1θEi)=δEiEi+1(niθ−ni+1θ)∑jnjθ>0 and niθ−1−ni+1θ−1<0,∀θ∈(0,1). Let δ = 0. Then, ΔVb>0 and thus Δv>0. By continuity, this also holds for δ positive but close to zero. Let δ = 1. Solving Δv<0 for β, we have Δv<0⇔β>β˜≡αtrAvi+1(ni+1−ni)θZ(rA+tni)(rA+tni+1)(niθ−1−ni+1θ−1)>0. (A.2) By continuity, Δv<0 holds for sufficiently large β when δ is close to but smaller than one.   ▪ B. Local landownership Suppose that all land in a city is owned by residents, so the total differential land rent is distributed equally to all residents. Let income be given by y=w+R/n, where R=∫0x¯(r(x,v)−rA)dx (A.3) is the total differential land rent. Rewriting Equations (8) and (7) gives x¯=(w+R/n)[1−rAα(rA+tn)−α]t. (A.4) v=(w+R/n)(rA+tn)−αE−β. (A.5)  Substituting from Equation (A.5) into r(x,v)=(w+R/n−tx)1/αE−β/αv−1/α with w=nγ gives r(x,v)=(rA+tn)(nγ+R/n)−1/α(nγ+R/n−tx)1a. Using this in Equation (A.3) and solving gives R=n1+γ[rA1+α−(rA+tn)α(rA−αtn)](rA+tn)1+α−rA1+α. (A.6) Finally, substituting in Equation (A.5) gives v=(1+α)tE−βn1+γ(rA+tn)1+α−rA1+α, (A.7) which is also inverted U-shaped in n. We then redo the simulation exercise from Section 6.2. For the city size distribution described by Zipf’s law with δ = 1, we find the largest city is undersized by 3.6% and the smallest is oversized by 7.3%, so results are very close to the baseline simulation. Varying δ shows that this also holds for local pollution and for the intermediate case δ=0.43. C. Calibration of β We now calibrate β using central estimates of the SCC from Interagency Working Group on SCC, US Government (2015). The MRS between pollution and (non-housing) consumption is MRS=−∂u/∂E∂u/∂z=βz(1−α)E. (A.8) Substituting optimal consumption, z(y)=(1−α)(w−tx) gives MRS=β(w−tx)/E, and integrating over the city gives citywide MRS MRS¯=∫0x¯β(w−tx)E1s(x)dx (A.9) =βw[tn+rA−rA1+α(tn+rA)−α](1+α)tE, (A.10) where we have substituted the optimal s(x) and used Equations (7) and (8). Finally, letting M be world population and n be city population, we get the SCC SCC=βMw[tn+rA−rA1+α(tn+rA)−α](1+α)tnE. (A.11) We use the following parameters: world population in 2015 was M = 7.35 billion (source: UN World Population Prospects, http://esa.un.org/unpd/wpp/Download/Standard/Population/), world per-capita income in 2015 was w=$10,743 (source: UN National Accounts Main Aggregates Database, http://unstats.un.org/unsd/snaama/dnllist.asp) and total CO2 emissions were E = 34,649 million metric tons CO2 in 2011 (source: World Bank, World Development Indicators, http://data.worldbank.org/indicator/EN.ATM.CO2E.KT/countries). We set n = 750,000 and from Borck and Brueckner (2018), we use t=$503.53,α=0.24 and rA=$58,800. The target value for the SCC is USD 40.54 per ton CO2, using the central value from Interagency Working Group on SCC, US Government (2015) (converted from 2007 to 2015 USD). Using the stated parameters, setting Equation (A.11) equal to 40.54 and solving gives β=0.022. The other values in the text are solved likewise using different values for the SCC. C.1. Calibration using SCC for USA An alternative approach would be to calibrate β to target the SCC for the USA, on the assumption that national policy makers care about national welfare, not the welfare of world citizens. Therefore, we recalibrate β to target the SCC value for the USA computed by Nordhaus (2017). His central US estimate is $4.78, much lower than the $31.21 value for the world. We therefore now use a target value of $5 instead of the world value of $40. Setting income per capita for the USA at $56,516 and population at 324 million and solving Equation (A.11) gives a value of β=0.0266. Interestingly, therefore, the results do not differ much when we use an SCC value for the USA instead of the world SCC. Table A1 CO2 emissions and city size (including population density) Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). Table A1 CO2 emissions and city size (including population density) Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Dependent variable: log CO2 emissions (1) (2) (3) (4) Log population 1.040*** 1.040*** 0.648*** 0.599*** (0.0281) (0.0281) (0.176) (0.186) Log density −0.158*** −0.158*** 0.181 0.200 (0.0369) (0.0369) (0.147) (0.142) Constant 2.335*** 2.343*** 3.533*** 3.896*** (0.202) (0.201) (1.117) (1.369) Observations 9244 9244 9244 9244 R-squared (within) 0.689 0.691 0.128 0.147 Number of CBSAs 928 928 928 928 Year-fixed effects No Yes Yes Yes CBSA-fixed effects No No Yes Yes Division×Year-fixed effects No No No Yes Notes: Standard errors are clustered at the CBSA level. ***p < 0.01, **p < 0.05, *p < 0.1. Source: Own calculations based on data from Fragkias et al. (2013). D. Simulation assuming Zipf’s law In this exercise, we assume that the city size distribution follows Zipf’s law. As is well known, this is a good approximation for city systems in most countries, except at the very top and bottom of the distribution (Gabaix, 2016). We continue to assume n = 225,678,243 and distribute this population to 180 cities so the city size distribution exactly follows Zipf’s law. The largest city has 39 million inhabitants (more than twice the New York City metro area), the second largest 19.5 million and the smallest city has 217,180 inhabitants. We compute the amenity levels from Equation (14) for these given population sizes. We first assume δ = 1 so pollution is global. We find that the biggest city is undersized by 4.8% (1.9 million) while the smallest one is oversized by 7.8%. When pollution is purely local, δ = 0, we find that the divergence between optimal and equilibrium city sizes is small, as in the real cities sample.29 The largest city is oversized by 0.01% and the smallest city is undersized by 0.06%. Finally, for an intermediate value, δ=0.43, the largest city is undersized 3.6% and the smallest oversized by 7%. © The Author(s) (2018). Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

Journal

Journal of Economic GeographyOxford University Press

Published: Apr 10, 2018

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off