According to the classical urban ecomomics model of Fujita and Ogawa [23], individuals choose job and dwelling places that maximize their net income after deduction of rent and commuting costs. More precisely, an agent will choose to live in x and work at location y such that the quantity Z(x,y)=W(y)−CR(x)−G(x,y) (1) is maximum. The quantity W(y) is the typical wage earned at location y, C_R(x) is the rent cost at x, and G(x, y) is the generalized transportation cost to go from x to y. There is also a similar equation for the profit of companies (that they want to maximize) and that we do not need here: we assume that employment is located at a unique center y = 0 and that wages and rent costs are of the same order for all individuals. Most large cities are polycentric [24, 25] with the existence of many different activity centers but this first approach assumes that polycentricity does not change the order of magnitude of commuting trip distances (see also [25] on this point). We also assume that the residence location x is given and random—residence choice is obviously a complex problem and replacing a complex quantity by a random one is a typical assumption made in the statistical physics of complex systems. Within these assumptions, we obtain a simplified Fujita-Ogawa model where we focus on the mode choice: individuals have already a home and work at the CBD, and the problem is about choosing a transport mode. Within these assumptions, the maximization of Z(x, 0) implies the minimization of the transport cost G(x, 0): max Z(x, 0) ⇒ min G(x, 0). Thus, individuals simply tend to choose the mode that minimizes their commuting costs to go the office located at 0. In this perspective, we assume that individuals change more frequently their jobs than their homes. More generally, the Fujita Ogawa model gives a density that results of the interplay between residential and commuting costs and due to our simplifications, the density is now an exogenous quantity.

In order to discuss commuting costs, we assume that a proportion p of the population has access (meaning having to walk less than 1km) to mass rapid transit such as the subway or elevated rail (we neglect buses or tramways here) whereas a share 1 − p of the population has no choice but to commute by car (we assume that all individuals can drive a car if needed). In other words, p is the probability to have access to the MRT (see Fig 1), but it doesn’t imply that it is the mode chosen: if the agent has access to the subway, he needs to compare the costs G_car and G_MRT in order to choose the less costly transportation mode.

The existence of a single central business district, the location of homes, and the density of MRT are considered as exogenous variables. The important endogenous variable is here the share of car users and the time spent in traffic jams (allowing then to estimate CO2 emissions). All the assumptions used in this model are of course approximations to the reality but we claim here that our model captures the essence of the traffic in large urban areas phenomenon. Starting with a model containing all these various parameters would actually be not tractable and would hide the critical ingredients.

In order to define the model completely we have to specify the expressions for the generalized costs. We will omit all other forms of commuting (walking, cycling, etc.), and we neglect spatial correlations between the densities of public transport and residence or population, which is an important assumption that certainly needs to be refined in future studies. The fraction p of individuals that have a choice between car and MRT will choose the transport mode with the lowest generalized cost which takes into account both monetary costs and trip duration (see for example [26, 27]).

For cars, we include congestion described by the Bureau of Public Roads function (see [28] and Methods) which captures the main effect: increasing the traffic on a road will decrease the effective speed on it. For the MRT, we neglect its monetary cost which is small in comparison with the one for cars. We then obtain the corresponding generalized costs for cars and the MRT under the form Gcar(x)=Cc+rvcV(1+(Tc)μ) (2) GMRT(x)=V(f+rvm) (3) where C_c is the daily cost of a car, v_c and v_m are respectively the car and MRT velocities, c is the road capacity of the city, f the walking plus waiting time for transit, r the distance between home and the central business district, T the total car traffic, and μ the exponent that characterizes the sensitivity to traffic. The quantity V is the value of time defined in transport economics as the money amount that a traveler is willing to pay in order to save one hour of time. It is an increasing function of income and is bounded by the hourly wage. We assume here that driving is faster than riding public transport (v_c > v_m) but is more expensive. Once an individual has chosen a mode he sticks to it and will not reconsider his choice even if the traffic evolves. In other words, we assume here that individual habits have a longer time scale than traffic dynamics.

Individual mobility is then governed by comparing these costs G_car and G_MRT and will depend on exogenous parameters such as car and subway velocities, car costs, etc. In the general mode choice theory (see for example [29]), given the values of the costs G_car and G_MRT there is a probability P_C = F(G_car − G_MRT) to choose the car. The function F is in general smooth and satisfies F(−∞) = 1 and F(+∞) = 0 and we consider here the simplest case where F(x > 0) = 0 and F(x < 0) = 1. An individual located at x with access to the MRT will then choose to use the car if G_car < G_MRT which implies a condition on the value of time of the form V < V_m(r) where V_m(r) depends on the parameters of the system and on the distance to the center r and reads Vm(r)=Ccf+r(1vm−1vc(1+(Tc)μ)) (4) Also, writing the equality G_car = G_MRT between generalized costs of car and MRT leads to the critical distance given by d(V,T)=min(L,CcV−f1vm−1vc(1+(Tc)μ)) (5) where L∼A (A is the area size of the city) is the largest extent of the city. This critical distance evolves as traffic increases (Fig 2), translating the fact that driving is less advantageous when the traffic is large. This expression also shows that individuals with a small value of time are more likely to use public transport, since they are more apt to spend time than money. Distance to the center is pivotal in this decision process: too far from the center—further than a critical distance d(V, T)—individuals favor driving to avoid lengthy journeys, and the richer they are, the smallest this distance.

Writing d(V, T = T*) = L, gives the critical maximal traffic T* for which driving is less beneficial in the whole agglomeration: T*=c[vc(1vm−1vc−CcV−fL)]1/μ (6)

When population increases, car traffic has essentially two sources: first, individuals may not have access to the MRT and second, if they have access to it, they might be too far and will prefer to take the car. This can be summarized by the following differential equation for the variation of car traffic T when P varies dTdP=1−p+p[1−η(d(V,T))] (7) where the first term of the r.h.s. corresponds to the 1 − p share of individuals far from MRT stations, and the second term to the fraction 1 − η of individuals that have access to the MRT and are living further from the center than d(V, T). This equation is valid for T < T* and for T > T* we have η = 1. By considering statistically isotropic cities with population density ρ(r) (r is the distance to the CDB), we obtain η(d(V,T))=∫0d(V,T)drrρ(r)∫0Ldrrρ(r) (8)

We have to plug in this expression together with Eq 5 of d(V, T) and to solve the differential equation Eq 7 which in general is too difficult. The density is endogenous in the original Fujita-Ogawa model but due to our simplifications is now an exogenous quantity. In general the density is decreasing with r and a simple choice is ρ(r) = ρ₀exp(−r/r₀) and we then obtain at dominant order in T/C dTdP≈1−pe−r/r0(1+ar0)+O((TC)μ) (9) where a = (C_c/V − f)/(1/v_m − 1/v_c). We note here that in the uniform density case ρ(r) = ρ₀, we obtain η = πd(V, T)²/A and T≃(1−pAb2)P+O((Tc)μ) where b is a function of the exogenous parameters b=(1/vm−1/vc)/π(Cc/V−f).

This result Eq 9 is valid For P<P*=T*/(1−pe−a/r0(1+a/r0)) since for d(V, T) > L the only source of car traffic comes from individuals who do not have access to the mass rapid transit, which leads to dT/dP = 1 − p implying T=(1−p)(P−P*)+T* (10) We note that even if this result seems somewhat simple, it derives from non-trivial considerations such as the comparison of the critical distance and the area size. The important fact to retain here is that cost considerations in the case where a mode choice is available usually lead for large urban areas to leave the car and take the MRT. Also, we note that the effect of the density is essentially on the first regime where T < T* and therefore does not impact our conclusions.

Compiling data from 25 megacities in the world (see Material and methods) for which we found an estimate of the population having access to the MRT (and therefore the quantity p), we compute the critical levels T* and P*. We note that for all these cities the definition of p is the same: it is the share of individuals living within 1km from a MRT station. For reasonable values of time [26, 30], most of cities have T* = 0 and all have T* ≪ P (T* ranges from 0 to 50% of P). Cases where T* = 0 mean that even at zero traffic the critical distance d(V, T = 0) is larger than the typical size of the city L. This depends a bit on the value of time but in most cases we do observe small values of T* indicating that cities are mostly in the saturated regime where the MRT is always more advantageous than the car. Public transports are so economical (compared to cars) that people living near rapid transit stations are highly likely to ride them. Thus, traffic does not appear as a determinant parameter in individual mobility choices as it concerns mostly individuals who have no choice but to drive and who suffer from onerous commuting costs and unavoidable time-consuming trips as traffic increases. We note here that correlations between the neighborhoods with low MRT density (small p) where individuals have expensive commuting costs and their revenue is an interesting field of study and appears as an important source of urban segregation [27]. Since we have in general T*, P* ≪ P, we obtain the simple prediction that TP≃1−p, a non-trivial consequence of rapid transit cheapness and individual choices of mobility: we recall that p is the probability to leave in the vicinity of a MRT station but that doesn’t imply the use of this transport mode. We compare the empirical car modal shares TP of these cities (see data description in Material and methods) to our prediction on Fig 3 and observe—considering the simplicity of the model—a very good agreement and a relevant linear trend highlighting the efficiency of public transportation in reducing traffic. In particular, most of the European cities are well described by our prediction and we observe a few deviations. These discrepancies can probably find their origin in the existence of other modes of commuting, lower car ownership rates (e.g. in Buenos Aires [31]), lower road capacities, higher cost of MRT, or a high degree of polycentrism.

Our model also provides a prediction for the transport-related gas emissions and we will focus on the CO₂ case for which we obtained data. We make the simplest assumption where these emissions are proportional to the total time spent on roads. The quantity of CO₂ emitted for a driver residing at x is then given by QCO2(r)∝r[1+(T/c)μ] which leads to a total QCO2∝∑driversiri[1+(Tc)μ] (11) ∝gA(1−p)P[1+(Tc)μ] (12) where we used our result for the total traffic T = (1 − p)P. We assumed that the sum ∑_i r_i is of the form gAT where A sets the scale of displacement and where the prefactor g encodes the geometrical aspect of car mobility in the city, including the spatial distribution of residences and activities, and the transport infrastructure. Its estimation probably requires a more detailed, specific calculation but the important aspect here is the scaling with A. We thus obtain that the annual CO₂ emitted by car and per capita is given by QCO2P∝A(1−p)(1+τ) (13) where τ=(Tc)μ is the average delay due to congestion and is empirically accessible from the TomTom database [32] (while the capacity c is usually not empirically accessible). It is interesting to note that QCO2 is then the product of three main terms: size of the city × fraction of car drivers × congestion effects, which correspond indeed to the intuitive expectation about the main ingredients governing car traffic. Also, we note here that the dominant term of QCO2 is proportional to the population indicating a simple linear scaling, and that nonlinear term could possibly appear as corrections, in contrast with previous results [25, 33], and which could explain the difficulties to reach a consensus about the behavior of this type of quantities (see for example [34]). We compare our prediction Eq 13 to disaggregated values of urban CO₂ emissions on Fig 4 and observe a good agreement. We observe some outliers like Buenos Aires which has a very small car ownership rate and thus lower than expected CO₂ emissions and areas such as New York which appears to be one of the largest transport CO₂ emitter in the world [35]. This result illustrates the role played by public transport and traffic in modulating transport-related CO₂ emissions. Most importantly, we identify urban sprawl (A) as a major criterion for transport emissions. We note that if we introduce the average population density ρ=PA, we can rewrite our result as QCO2P∝Pρ(1−p)(1+τ) (14) i.e. QCO2P∝ρ−12 since P is a slowly-varying function within the scope of large urban areas. We understand here how Newman and Kenworthy [7] could have obtained their result by assuming the density to be the control parameter. However, even if fitting data with a function of ρ is possible, our analysis shows that it is qualitatively wrong: the area size A and the public transport density p seem to be the true parameters controlling car-related quantities such as CO₂ emissions. Mitigating the traffic is therefore not obtained by increasing the density but by reducing the area size and improving the public transport density. Increasing the population at fixed area would increase the emission of CO₂ (due to an increase of traffic congestion leading to an increase of τ) in contrast with the naive Newman-Kenworthy assumption where increasing the density leads to a decrease of CO₂ emissions.

We also note that ∂logQCO2∂logp=−p1−p (15) can be relatively large (in absolute value) while ∂logQCO2∂logA=12 (16) is small which suggests that the increase of transport density is in general more efficient than an area decrease (which is also less feasible). We also understand that if mass rapid transit is efficient in reducing transport-related greenhouse emissions, on the other hand reducing the road capacity c (to deter individuals from driving) may not be a good remedy, as we identified drivers with individuals with no other solution. Worse, by making their trips longer, diminishing c indirectly contributes to higher emissions and pollution as well as potentially socially segregating situations.

Finally, we can also estime the average commuting time (details are given in Material and methods) and we obtain for the one-trip commuting time τ_c averaged over the population the following expression τc¯=p(f+gAvm)+(1−p)gAvc(1+τ) (17) where g is a geographical factor that encodes the spatial complexity of trips. The comparison with the data displays relatively large fluctuations, but our analysis seems to capture the main trend and the one-parameter fit over g using our expression Eq 17 leads to the average value g ≈ 0.203. We note that for a uniform distribution of residences, a simple calculation leads to g=2/3π≈0.376 and in the simple isotropic case where the density decreases with the distance r to the center as ρ(r) = ρ₀(1 − r/L), we obtain g=1/3π≈0.188. The average value g obtained by the fit is in between these two theoretical estimates. However, polycentrism and more generally the spatial organization of the city and its infrastructure certainly play an important role here and our model can only provide a first approximation to the average commuting time.

We studied 25 metropolitan areas from Europe, America, Asia and Australia. The number of cities was limited by the availability of data on MRT accessibility and reliable modal share estimates. All the data is freely available and we list here their sources. We also provide a file together with this submission with all the data used in this study (see S1 Dataset).

The definition of a metropolitan area varies from one country to another but we aimed at assembling a statistically coherent set of agglomerations. Populations and areas were collated from Wikipedia data on metropolitan statistical areas in accordance with national definitions.

The commuting time for MRT users is given by f+gA/vc while for car users it is gA(1+τ)/vc where g is a geographical factor. We therefore obtain for the one-trip commuting time τ_c averaged over the population the following expression τc¯=p(f+gAvm)+(1−p)gAvc(1+τ) (18) The comparison of this result with empirical data is shown on Fig 5. Even if we observe relatively large fluctuations, our analysis seems to capture the main trend and the one-parameter fit using our expression Eq 18 leads to the average value g ≈ 0.203. We note that for a uniform distribution of residences, a simple calculation leads to g=2/3π≈0.376 and in the simple isotropic case where the density decreases with the distance r to the center as ρ(r) = ρ₀(1 − r/L), we obtain g=1/3π≈0.188. The average value g obtained by the fit is in between these two theoretical estimates. From Fig 5 we can compute the ‘effective g_eff’ for each city and compare it to the average value g. This quantity g_eff (and the ratio η = g_eff/g) encodes both the complexity of the population and activity densities, and of the transportation infrastructure. For our set of cities, the average ratio is of order 1.37 and we observe outliers with small values such as Barcelona (η = 0.04), Seoul (η = 0.62), and large ones (Buenos Aires 1.92, Rotterdam 2.32, Singapore 3.38). Most of the cities (76%) have however here a ratio η larger than 1 which implies that our model underestimates in general the commuting distance. Many local effects can explain the variations observed in the commuting distance: the existence of polycentricity could reduce the commuting distance (for example, it seems that we overestimate the commuting time for New York which might be a consequence of polycentrism), but other factors such as poor transportation infrastructures (or traffic bottlenecks due to geographical constraints) could have the opposite effect of increasing this quantity. In general we expect that the transport network structure will have an important impact, especially if it is very anisotropic in space. The determination of the commuting distance for each city probably implies to take into account a large number of specific details, but our analysis shows that it is consistent to assume that the order of magnitude of this quantity is A.