Richard Grimal

Is there a limit to car traffic growth ? Potential demand and convergence paths towards saturation

Keywords: car equipment, traffic growth, potential demand, saturation

Car traffic growth has temporarily stopped during the 2000's, before coming back to growth after 2010, in relation with economic recovery and decreasing fuel prices. However, there are strong reasons to believe that car traffic growth is potentially limited, among which close completion of the diffusion process for car ownership, limited travel time budgets in relation with stable or declining travel speeds, decreasing marginal returns of additional car travel and infrastructure capacity restrictions. Representing car ownership as the result of an equilibrium between potential demand and economic constraints, and assuming additional car travel to be of decreasing marginal utility, one can implement a model to estimate saturation thresholds and describe the convergence path towards saturation. The model is disaggregated by household type and vehicle rank to account for heterogeneous choice sets and structural change in demographics and activity rates. It highlights the existence of an incomplete diffusion process for some groups and allows to break down potential demand between negotiable and non-negotiable needs. Projection results for France prove the existence of residual potential growth for car equipment among non-working adults which is nonetheless limited, while population ageing and changing cohabitation patterns will have a downwards effect on demand. By combining these results with car mileage projections, we find average car traffic per adult to increase gently until 2040, and remain almost stable afterwards. From this date, traffic trends would be essentially determined by demographic factors.

Introduction

There is a long-admitted relation between car traffic and economic growth, which is well documented through the literature and widely supported by empirical evidence (Gilbert and Nadeau, 2001;Banister and Stead, 2002). This correlation has resulted in expectations about car traffic to increase indefinitely in line with economic growth. From 1990 to 2003, the total amount of car traffic generated by personal vehicles in France still rose by 30 %. Likewise, the number of personal cars increased from 24 to 32 million between 1990and 2016[START_REF] Cgdd | Les comptes des transports en 2016, Tome 1, 54 ème rapport de la Commission des comptes des transports de la nation[END_REF]. Though car traffic growth was partially caused by demographic expansion, it also resulted from increasing car ownership and car mileage per adult, doubling between 1974 and the early 2000's [START_REF] Grimal | Is the stagnation of individual car travel a general phenomenon in France ? An analysis by zone of residence and standard of living[END_REF]. However, after decades of continuous growth, the average car mileage per adult levelled off, in France as in many other developed countries (CGDD, 2012a ;[START_REF] Millard-Ball | Are we reaching peak travel ? Trends in Passenger Transport in Eight Industrialized Countries[END_REF], though for different levels and at different periods : for instance, the average car mileage per person started levelling off from as long as the mid 90's in Germany and Japan, respectively by 8 000 and 6 000 km ; in the early 2000's in the United Kingdom and France, respectively by 8 000 and 10 000 km; in the half 2000's in Australia and the United States, respectively by 10 000 and 16 000 km (ITF, 2011;BITRE, 2012). Differences in traffic trends between developed countries might result from various motives such as land use patterns -with for instance an opposition between Japan and the United States in terms of urban density -transport supply and infrastructures, fuel taxation, public policies, mobility cultures, etc. Nothing similar is yet noticeable in emerging economies where auto-mobility is still expanding rapidly, driven by demographic growth, galloping urbanization and lower diffusion rates for car ownership [START_REF] Madre | Are we heading towards a reversal for the trend towards ever-greater mobility ?[END_REF]ITF, 2011;Qian and Soopramanien, 2014). In some countries, the transition towards auto-mobility is occurring through an intermediary stage based on the access to motorized two-wheelers (Hansen, 2017). However, they are already signs of car traffic slowing down in some megacities from emerging countries (Gao and Newman, 2018).

Whatever, recent changes in traffic trends raise the issue of future traffic growth in developed countries in a situation of increasing uncertainty. In particular, transport demand forecasts tend to be frequently overestimated, such inaccuracies being partly caused by an optimistic bias and political orientations, and challenging current traffic models (Mackinder and Evans, 1981;World Bank, 1986;Thomson, 1993;Flybjerg et al., 2003;2007). Facing the stagnation of car traffic, several authors conjectured that this plateau could be the prelude to a future decline, caused by multiple convergent transitions in the way of life and the comeback of alternatives to the car (Le Vine et al., 2009;[START_REF] Metz | Saturation of demand for daily travel[END_REF]ITF, 2011;[START_REF] Millard-Ball | Are we reaching peak travel ? Trends in Passenger Transport in Eight Industrialized Countries[END_REF][START_REF] Newman | Peak car use : understanding the demise of automobile dependence[END_REF]Goodwin, 2011). However, this assumption is only one among two other scenarios, that were equally suggested by Goodwin (2010-11) as possible readings of the events : the « business-as-usual » scenario, assuming the current stagnation to be a temporary break caused by difficult economic circumstances (rising fuel prices, economic crisis after 2008…), within a long-run trajectory of growth ; the theory that car traffic could have reached saturation, as main travel needs likely to be realized by car would now be satisfied.

Since 2010 though, car traffic gradually came back to growth in France, even accelerating after 2013 in relation with economic recovery and decreasing fuel prices, following the peak of 2012 [START_REF] Maurin | L'usage de la voiture de nouveau en progression. Futuribles[END_REF]. Traffic comeback was also noticed in other countries such as Germany and the USA, highlighting the key contribution of economic drivers (fuel prices, GDP…) to recent traffic trends, including the 2000's plateau (BITRE, 2012 ;Bastian et al., 2016). This recovery does not imply, however, that the very concept of saturation is irrelevant and that it is not to be expected in a more or less distant future. More, prospects of coming saturation are made very credible by a number of trends, such as decreasing levels of car ownership and car mileage from the early 1990's in the largest French metropolitan areas such as Ile-de-France [START_REF] Cornut | Diffusion of car ownership and use in Paris metropolitan area since the mid 70's[END_REF], national vehicle fleet growth slowing down from the early 2000's [START_REF] Cgdd | Les comptes des transports en 2016, Tome 1, 54 ème rapport de la Commission des comptes des transports de la nation[END_REF], and the gradual decoupling between traffic growth and the economy [START_REF] Madre | Can a decoupling of Traffic and Economic growth be envisaged ?[END_REF]OCDE/ITF, 2011), which is also manifested through decreasing income elasticities of car traffic [START_REF] Pendyala | A repeated cross-sectional evaluation of car ownership[END_REF][START_REF] Goodwin | Elasticities of Road Traffic and Fuel Consumption with respect to price and income: a review[END_REF]. In addition, traffic recovery was much less pronounced for personal cars than for the traffic generated by lightduty vehicles, which is more directly related to economic activity [START_REF] Cgdd | Les comptes des transports en 2016, Tome 1, 54 ème rapport de la Commission des comptes des transports de la nation[END_REF]. Though hardly accounted for in official models -for instance through decreasing income elasticities -ideas of saturation appear as an interesting subject to explore, notably in relation with issues of forecasting accuracy and traffic externalities.

In section 2, we present different arguments supporting the theory of saturation for car ownership and car traffic. They can be synthesized within a conceptual framework based on the idea of potential demand, where the gap between realized and potential demand is assumed to come from a set of technical, economic, environmental and regulatory constraints. The notion of pseudo-saturation is also introduced for car mileage. This paper will essentially focus on the role of economic drivers by estimating saturation thresholds and modelling the convergence path towards saturation with the relaxation of financial constraints. In section 3, the relation between car ownership, car mileage and income is analyzed by household categories and vehicle rank. Model specification is also exposed. Estimation results are displayed in section 4, while section 5 is devoted to forecasting. Finally, in section 6, the main results of this study are reminded, before we draw up research tracks for the future, underlining some uncertainties and possible model improvements.

Discussion on the idea of saturation

The idea that car traffic growth would sooner or later come to an end is not entirely new: as soon as from the 1950-60's, potential limits to traffic growth were conjectured in the early works from Tanner at the UK Road Research Laboratory (Goodwin and Van Dender, 2013). In particular, it was believed then that car traffic growth would come to an end once everyone who wants and needs a car would have one, so that the diffusion process of car ownership was conceived as a major cause for potential saturation from the origin of traffic forecasts [START_REF] Goodwin | Peak travel, peak car and the future of mobility -Evidence, unresolved issues, policy implications and a research agenda[END_REF]. This research has been going on in the 1970's [START_REF] Tulpule | Forecasts of vehicles and traffic in Great Britain 1972 revision[END_REF], by chance anticipating saturation to come during the first decade of the 2000's, where car traffic indeed levelled off, though probably because of economic circumstances (cf. supra). During the 1980's, these works were abandoned in the context of the oil counter-shock resulting in lower fuel prices and the comeback of traffic growth caused by the widespread diffusion of a second vehicle within households, itself supported by the increasing activity rate of women and the expansion of outer suburbs (Madre, 1992). In the 1990's, a number of works underlined the contribution of population ageing and cohort effects to long-run car ownership trends, possibly resulting in saturation with behaviours of successive cohorts becoming closer in later cohorts (Lambert et Madre, 1989;Gallez, 1994;[START_REF] Bussière | Vers la saturation ? Une approche démographique de l'équipement des ménages en automobile dans trois régions urbaines[END_REF]Dargay et al., 2000). During the 2000's, the theory of a gradual decoupling between transport activity and the economy emerged (Banister and Berechmann, 2001;OECD, 2001;[START_REF] Madre | Can a decoupling of Traffic and Economic growth be envisaged ?[END_REF][START_REF] Madre | Can a decoupling of Traffic and Economic growth be envisaged ?[END_REF]Tapio, 2005), sometimes also presented as the decreasing transport intensity of the economy (Peak, 1994;Banister and Stead, 2002;EEA, 2006). Finally, the stagnation of car traffic during the 2000's led to the conjecture about "peak car" in the early 2010's.

Going further, a number of works attempted to forecast car ownership and/or car use at different horizons in France and other developed countries and to estimate empirical saturation thresholds, such as works by Glaude and Moutardier (1978), Latapie and Lefol (1987), Madre and Gallez (1993), [START_REF] Bussière | Vers la saturation ? Une approche démographique de l'équipement des ménages en automobile dans trois régions urbaines[END_REF], [START_REF] Schafer | The future mobility of the world's population[END_REF], BITRE (2012), [START_REF] Madre | Are we heading towards a reversal for the trend towards ever-greater mobility ?[END_REF], [START_REF] Collet | Vers quels plafonds pour la motorisation et l'usage de la voiture ?[END_REF], [START_REF] Cornut | Diffusion of car ownership and use in Paris metropolitan area since the mid 70's[END_REF], based on different types of econometric and demographic models. Estimated saturation levels and temporalities may differ from a country to another according to observed car ownership and traffic patterns, themselves dependent on land use frameworks, transport policies, environmental regulations, etc. At the international level, BITRE et al. ( 2012) estimated saturation levels for VKT/person for twenty-five countries, as well as for aggregate traffic using demographic forecasts, as a function of fluctuations of fuel prices, changes in the economy (unemployment and income statistics), and a quadratic time trend leading to saturation. Though traffic growth consistently slowed down in many countries, levels of traffic per capita differ much from a country to another, from very high levels in United States, followed by Australia and Canada, to a much lower average car mileage per adult in Japan and Korea. While the average VKT/person should exceed 18 000 km in the United States in 2030, it should level off by 7 000 VKT/person in Japan from as soon as 2015. The situation is intermediary in European countries, with an estimated saturation mileage of 10 000 VKT/person in France, which would also be reached by 2015. [START_REF] Schafer | The future mobility of the world's population[END_REF] uses a heuristic model to project future levels of car use for different world areas, assuming the stability of time and money budgets, and people to move to faster modes with increasing income, thus slowing car traffic growth. According to these forecasts, the average car traffic volume per capita would start decreasing in North America from 2010, after a peak at 22 000 PKT/capita. Saturation thresholds may also change from a study to another according to the variety of employed data and methods. For instance, the saturation threshold for the proportion of motorized households in France was estimated at 96 % according to Glaude and Moutardier (1978), and 88 % according to Latapie and Lefol (1987), using data from the French National Transport Survey from 1981-82. According to forecasts by Madre and Gallez (1993) using a demographic model and French Household Continuous Surveys, the French vehicle fleet was estimated at 33 million vehicles in 2010 and the proportion of multi-motorized households to 40 %, consistently with results from existing data, either from the National Transport Survey 2008 or French Car Fleet Surveys. The saturation threshold was expected to be reached by the middle of the 21 st century. [START_REF] Collet | Vers quels plafonds pour la motorisation et l'usage de la voiture ?[END_REF], using an econometric model with a flexible Chapman-Richards specification, estimated saturation levels for car ownership and car use in France, respectively to 0,76 cars/adult, and 16 200 km/year/household, assuming fuel prices to remain at their 2010 level. In [START_REF] Grimal | Is the stagnation of individual car travel a general phenomenon in France ? An analysis by zone of residence and standard of living[END_REF], the saturation threshold for the average car mileage per adult at the 2010 fuel price level was estimated at 8 000 km in core cities, 9 300 km for inner suburbs, and 13 200 km for outer suburbs. [START_REF] Cornut | Diffusion of car ownership and use in Paris metropolitan area since the mid 70's[END_REF], using logistic curves as a function of time and income, estimated saturation thresholds specific to the Paris metropolitan area (Ile-de-France), where car use has already began to level off from the 1990's: these are much lower than for the rest of France, due to high density and a massive recourse to transit, which is still expected to increase with the future "Grand Paris" transit network at the metropolitan scale.

By contrast, demographic expansion and economic development should result in the pursuit of car traffic growth in the next few decades in emerging economies. According to a study by Bussière (1991), the demographic factor alone should result in the growth of car mobility until 2035 in the city of Puebla in Mexico. However in practice, the expansion of auto-mobility in emerging economies should also depend much on the trajectory of economic development, along with the average standard of living and the distribution of income, as car ownership diffusion only takes off from a certain level of income (Chamon et al., 2008;Pongthanaisawan and Sorapipatana, 2010;ITF, 2011). The contribution of economic development to car ownership diffusion and traffic increases is more uncertain and dependent on national economic patterns. In Mexico for instance, the persistent poverty did not allow a significant improvement in the diffusion of individual car ownership until now, which even fell down between 1994 and 2011 [START_REF] Madre | Are we heading towards a reversal for the trend towards ever-greater mobility ?[END_REF], while in China, the market is expanding very rapidly due to economic growth and has even become the first world market in 2010 (Qian and Soopramanien, 2014). On the whole, car ownership and car traffic should grow much more rapidly in emerging economies than in developed countries: for instance, car traffic growth is expected to increase by a factor 5 to 6.5 between 2000 and 2050 in emerging economies -vs. only 30 to 40 % for the OECD zone (ITF, 2011). According to Chamon et al. (2008), the number of cars could increase from 1,9 billion between 2005 and 2050 in emerging markets and developing economies (vs. only 400 million in the OECD zone), notably due to the economic expansion of India and China, where important reserves of potential growth still remain, given much lower car ownership rates than in countries from the OECD zone (Qian and Soopramanien, 2014). In addition, aspirations to own a car remain strong among the youth of emerging countries (Belgiawan et al., 2014).

From a more theoretical point of view, the idea of saturation was justified by different arguments through the literature. Following [START_REF] Goodwin | Peak travel, peak car and the future of mobility -Evidence, unresolved issues, policy implications and a research agenda[END_REF], we expose some of these :

(a) One of them is based on the diffusion process of durable goods, which can be incorporated within the general framework for the diffusion of innovations [START_REF] Rogers | Diffusion of innovations[END_REF]. The diffusion of durable goods can usually be modelled through an S-shaped sigmoid-curve tending towards an asymptote that will represent the saturation level for this type of asset. The diffusion process can be complete, if every agent is destined to be equipped with the gradual relaxation of constraints, or incomplete, as definitive limitations to diffusion might persist, related either to the lack of need or to remaining constraints, for instance the permanence of disabilities or inequalities resulting in insufficient financial resources for some population groups, or even to individual preferences (Madre and Gallez, 1993). In particular, the diffusion of car ownership could attain its limits, and therefore represent one of the major reasons for traffic stagnation (NSTPRC, 2008 ;[START_REF] Metz | Saturation of demand for daily travel[END_REF][START_REF] Metz | Peak car and beyond: the Fourth Era of Travel[END_REF]. However, this observation implies to determine the exact nature of the relation between car ownership and traffic growth. One possible statement is to consider that the increasing number of cars available would result in induced traffic growth through the increasing number of individuals driving their personal car (BITRE, 2012). The effect of increasing car ownership on the average car traffic per capita can be demonstrated through the observation of traffic trends over a long-run period, with the average car traffic per adult in France increasing in line with the average number of cars per adult from the early 1970's, while the average car use per vehicle has remained roughly stationary during the same period, though with mid-term fluctuations related to fuel price volatility [START_REF] Grimal | Is the stagnation of individual car travel a general phenomenon in France ? An analysis by zone of residence and standard of living[END_REF]. Therefore, one could expect average car traffic per adult to reach saturation once every one who needs it would be equipped with a car, or before if the diffusion process remains incomplete for the above-mentioned reasons. For instance, car-related costs, along with driving and parking difficulties, may be perceived as exceeding advantages of the car for households living in dense urban areas well-provided with alternative transport supply. However in turns, one could assume that increasing the number of household vehicles could just result in lowering the average car mileage per vehicle, for an unchanged total distance travelled, by changing the distribution of household car mileage between household's vehicles. For instance, purchase of a second vehicle in a family may result in lowering the first vehicle mileage. Both mechanisms may be true within their own scope, requiring an appropriate modelling framework to disentangle them.

(b) Another argument is based on the willingness to devote time to travel. As daily travel time has remained remarkably stable over a long-run period [START_REF] Metz | Peak car and beyond: the Fourth Era of Travel[END_REF], travel growth was mainly caused by higher travel speeds, allowing travel distances to increase [START_REF] Zahavi | Traveltime Budgets and Mobility in Urban Areas[END_REF]. Here again, this dynamic being caused by the broader availability of a personal car, one can expect car travel growth to come to an end once every adult would be equipped. Also, car travel speeds have themselves increased, contributing to traffic growth by means of improved road infrastructures, both quantitatively and qualitatively. However, real car travel speeds may now be declining, both because of increasing congestion (IAURIF, 2013), infrastructure investment cuts, and due to policy sets going towards decreasing the efficiency of the car, through reducing authorized speeds, dedicated transit lanes, and parking restrictions [START_REF] Jarrige | Politiques de stationnement et mobilité locale : approche empirique[END_REF][START_REF] Cairns | Traffic impact of highway capacity reductions : assessment of the evidence[END_REF][START_REF] Fraser | Peak travel in a Megacity : Exploring the role of Infrastructure saturation on the suppression of automobile use[END_REF]. One may assume these measures to moderate car travel distances in relation with the limited consent for additional travel time.

(c) The average car travel by agent -either the individual or the household -may also be limited by decreasing marginal returns of additional gains in terms of accessibility (trip distances) and/or mobility (trip frequencies) as these correspond to ever-less essential travel needs [START_REF] Schafer | The future mobility of the world's population[END_REF][START_REF] Metz | Saturation of demand for daily travel[END_REF][START_REF] Metz | Peak car and beyond: the Fourth Era of Travel[END_REF]. In addition, these potential benefits are themselves weakening as car travel speed is coming closer to its upper limit, or is even decreasing. More, the increasing value of time leads to transferring new trips to faster modes such as high speed train and plane [START_REF] Schafer | The future mobility of the world's population[END_REF], allowing either substantial time cuts or accessibility benefits, while presenting a greater intrinsec utility, increasing the value of travel by allowing to realize different activities (Papon et al., 2008).

(d) Finally, road infrastructure capacity constraints may also hinder car travel growth, along with road investment cuts caused by the increasing scarcity of budgetary resources and the reorientation of public funds towards alternatives to the road [START_REF] Papon | Existe-t-il des seuils de saturation de la mobilité des personnes ? Réalités Industrielles, une série des Annales des Mines[END_REF].

However, these theoretical arguments are partially undermined by a number of behavioural uncertainties, regarding either limits to the diffusion of car ownership, the real stability of travel time budgets on the long-run, or potential thresholds for long-distance travel which is a priori unconstrained by nature, making the empirical estimation of saturation thresholds uneasy. For instant, the relevant level of analysis for the diffusion process of the car is ambivalent. While some durable goods have always been considered individual, such as the mobile phone or the personal computer, others always remained household equipments, such as most domestic electrical assets (fridge, washing-up machine…). However, the status of the car is transitional between an individual and a collective good, and has moved over time with the growing diffusion of a second vehicle among households, in relation with the expansion of outer suburbs and the increasing integration of women into the job market (Barjonet et al., 1987;Madre, 1992;[START_REF] Papon | Adulterous behavior within the car-owner couple: some analyses from French panel data on car rental and car sharing within households[END_REF]. As a result, the estimation of saturation thresholds for car ownership was frequently revised upwards over time [START_REF] Ingram | Determinants of motorization and road provision[END_REF]. In addition, the relevant level for modelling the diffusion process of an equipment is directly related to its function which, in the case of the car, appears to be hybrid between common, shared and individual usage. Especially, the main vehicle household is rather multipurpose, being simultaneously used for the commuting needs of the household head, and common leisure/holiday trips of the family, for instance. Within mono-motorized households, the single vehicle is also to be shared between its adult members for their daily travel needs. In contrast, the second vehicle is rather of individual use as being devoted principally to the partner's daily travel needs. The status of the third vehicle is more uncertain as it may correspond, either to an individual equipment devoted to a third adult household member -for instance an adult child still living at the parental home -or to a specialized vehicle -for instance devoted to long-distance trips. By the way, vehicle specialization may also have changed over time: with the increasing availability of a second vehicle, the first vehicle may have become less universal and more focused on the household head's travel needs along with a lesser need of shared usage between adult household members. For a given household composition, every vehicle use may therefore be dependent on the number of household vehicles. Given the hybrid status of the car between an individual and collective equipment, either individual or household-based models could be implemented, though most of them are household-based, in part for reasons of survey design and data availability (De Jong et al., 2004), but also for theoretical reasons, in particular the amount of common and shared vehicle use making difficult to utterly assign a car to a given individual. Not only being halfway between an individual and a collective equipment, the car is also a good of intermediary status between an essential and a luxury item. While being simply motorized has become an essential for most households, upmarket models consist in luxury goods and remain symbols of social status [START_REF] Coulangeon | L'équipement automobile entre contrainte et distinction sociale[END_REF]. In addition, while the main vehicle is often considered mandatory, ownership of a second vehicle is more subject to trade-offs depending on individual travel needs and financial constraints, though this status is moving over time along with the diffusion stage of the second vehicle. As a result, while the first vehicle market may already have attained maturity, reserves of potential growth may still exist for the second vehicle as its social diffusion remains incomplete. Therefore, either the collective or individual function of the car, its degree of necessity or social status depend much on vehicle rank.

In addition, despite limited consent for additional travel time, people could be forced to adopt more timeconsuming travel habits under the pressure of opposite constraints. For instance, increasing tensions on metropolitan housing markets induced by metropolization reduce the financial accessibility of their central areas [START_REF] Rougerie | Prix des logements anciens -Dans les années 2000, ils ont augmenté beaucoup plus vite que les loyers et les revenus[END_REF][START_REF] Vanco | L'accès à la ville : vers une contrainte financière de plus en plus forte ? in La mobilité urbaine des années[END_REF], forcing middle-class households to handle increasing commuting distances by orientating residential choices away from employment areas [START_REF] Breton | Domicile-travail : les salariés à bout de souffle[END_REF]. Greater commuting distances might also partially be a consequence of the growing conflict between professional and family obligations [START_REF] Vignal | Logiques professionnelles et logiques familiales : une articulation contrainte par la délocalisation de l'emploi[END_REF][START_REF] Vincent | Carrières académiques : comment concilier mobilités spatiales et vie de famille ? Synergies Pays Riverains de la Baltique[END_REF], with on one side the increasing injunction to job mobility, and one another side a rather low residential flexibility, resulting from the subscription of residential loans and the permanency of relational, educational and family links locally. Finally, the phenomenon of "metropolization", along with the systematic job rationalization associated with digitalization, might lead to the increasing concentration of wealth and activities in the largest cities connected to globalization, correlated with a decline of small cities and rural areas -that were designed through the socio-geographical concept of "periferical France" by (Guilluy, 2014) -potentially and paradoxically resulting in increasing travel distances for a number of activities. Higher commuting distances associated with speed moderation policies would then result in higher travel times and could increase rather than reduce car dependence, in contradiction with the expectations of planners and local authorities. While until now, individuals have adapted through reducing discretionary trips in order to control daily travel budgets, this strategy might attain its limits as some trips are mandatory, pushing individuals either to relocate, shift to transit when alternatives are attractive enough, or consent to increase daily travel time. Depending on opportunities, opposite constraints may therefore result in different tradeoffs, leading either to reduced or increased travel distances and times, and therefore to travel growth or decline. Indeed, though [START_REF] Marchetti | Anthropological Invariants in Travel Behavior[END_REF] hypothesizes average daily travel time to be some kind of « anthropological invariant », nothing guarantees, however, that this apparent stability would not be broken under the pressure of contradictory injunctions generated by the economy, public policies, urban trends and spatial patterns of activities. Rather than assuming a constant travel time, a more general argument supporting saturation -though it doesn't give an operational criterion to estimate saturation thresholds -is that willingness to devote time to travel is necessarily bounded by other activities, within the framework of a trade-off between concurrent allocations of time budgets.

Finally, it seems quite difficult to establish saturation thresholds at the household level for vehicle use, especially when it comes to long-distance travel, which is only limited by available income and leisure time. The current social distribution of long-distance travel suggests that huge reserves of potential growth still remain, if the distribution of income was to be modified, or the prices of long-distance travel to be cut through the generalization of low-cost trips.

Though arguments in favour of saturation may be different in nature, they can all be subsumed through the notion that some mechanisms are rather finite than infinite, and may have an upper limit which is determined by capacity constraints or potential needs, or at least may slow down from a certain level. When it comes to car ownership or any durable good, its diffusion rate can be represented as the result from an equilibrium between perceived needs and a set of constraints or limiting factors, hindering out their realization. Then, the saturation level can also be described as the potential demand or market size for this type of good, being an indicator of demand in the absence of constraints. The notion of saturation, however, is imperfect when dealing with car mileage as, unlike the diffusion rate of a given equipment, there is no a priori reason to believe that its intensity of usage is limited by nature (cf. supra). However, because of decreasing marginal returns of additional car travel, one can assume car travel growth to slow down with the level of car mileage already reached -or with increasing income -in such a way that from a certain stage, growth would become unsubstantial, a notion which can be designed as « pseudo-saturation ».

Travel needs notably derive from household composition, as they increase with the number of adults and children, from the activity programs of household members -in particular whether they are working or not -from the built environment and the spatial distribution of activities. However, the notion of needs may also be understood dynamically, as these might not be entirely predetermined by household characteristics from the beginning of the diffusion process, but could evolve over time under the influence of exogenous and unpredictable changes in social norms and the way of life. For instance, increasing woman activity and external city growth have contributed to increase car ownership needs, therefore uprising the level of potential demand over time for a given level of income. In addition, needs are not purely determined by objective constraints such as activity programs, household composition or land-use patterns, but also derive from subjective determinants like individual preferences, travel experience and mental habits. Given the role of transport experience, the characteristics of individual travel are partly induced by the increasing availability and diversity of transport supply. In particular, the widespread diffusion of the car, itself permitted through an increasing average income, has considerably widened the horizon of choice in terms of potential destinations over time, while simultaneously contributing to change the way of life.

Constraints explaining the gap between realized and potential demand may be diverse in nature, either technical, economical, regulatory or related to the physical environment, for instance. Technical constraints are notably related to travel speeds of available transport modes. With the broader availability of a personal car, bounds to mobility are lowering, while the average travel speed is approaching to its potential limit, which is related to car equipment rates at saturation. Car travel speed growth is also related to road infrastructure improvements which might have ceased. In addition to technical factors, economical constraints also contribute to explain the gap between realized and potential demand. For instance, limitations to the diffusion of personal car availability are weakening with an increasing average income, while decreasing fuel prices may contribute to a higher car mileage. However, upper limits to car use are also determined within the scope of trade-off mechanisms between different expenditures (food, housing, energy, etc.). Apart from financial limitations, regulatory measures may also hinder car use (authorized speed limits, parking restrictions, road pricing, alternate driving schemes, traffic restrictions for pollutant vehicles in some areas…). Finally, additional constraints may come from the physical environment. For instance, traffic-calming oriented urbanism might result in greater traffic and parking difficulties, thus providing an incentive to reduce car use, especially in dense metropolitan areas well-provided with transit, dedicated bike lanes and bike-sharing schemes.

Yet, both travel needs and constraints are changing over time. During several decades, increasing travel needs associated with the relaxation of constraints favoured the expansion of auto-mobility. Travel needs have been increasing with the diffusion of professional activity among women -thus generating additional home-to-work car trips -changes in the built-environment and the increasing spatial mismatch between working and residential areasresulting in higher travel distances and increasing car dependence (Newman and Kenworthy, 1989) -and the increasing demand for leisure travel, resulting from a higher average income. At the same time, the expression of potential demand was stimulated by an increasing license diffusion among women through generational renewal, decreasing vehicle costs, an increasing average income, contained inequalities and moderate fuel prices, leading to a relative decline of car travel inequalities depending on income (Collet et al., 2010). However from the 2000's, constraints on car use have been reinforced. Indeed, driving license diffusion tends to level off with the exhaustion of cohort effects, while household financial constraints have reinforced with rising fuel prices, resulting from increases in both petrol prices and fuel taxation. Other financial constraints have been increasing during this period, especially the weight of household mandatory expenditures such as housing, contributing to the increased pressure on transport budgets (Hivert et Madre, 2012), as household's trade-offs between different expenditures are dependent on relative prices (Banks et al., 1997). In addition, transport and planning policies have become less favourable to the car.

In theory, one could expect car travel needs to level off in the future while constraints limiting car use should be reinforced, consecutive to higher fuel taxation, changes in the built environment resulting from increasing urban densities, traffic-calming policies and anti-pollution measures taken by local authorities, and the phenomenon of metropolization resulting in the increasing concentration of population and activities in the largest metropolitan cities. Constraints coming from the built environment should become deterrent in many cities and therefore contribute to reduce facilities for car ownership and car use. In addition, metropolization should contribute to mitigate the utility of additional travel by providing people local accessibility to a large number of opportunities for jobs and activities. Finally, new cohorts tend to be less licensed and to drive less than their elders, at least in the early stages of their life course (Kuhnimof et al., 2012;Delbosc and Currie, 2013), a trend that could reflect less car-focused travel preferences, thus resulting in the decline of car travel in the future through generational renewal (Polzin et al., 2014).

However, as already mentioned, the effects of metropolization and digitalization are potentially ambivalent as they are associated with increasing housing tensions in the largest cities, along with a relative decline of small/medium cities and rural areas, both possibly resulting in higher daily travel distances. De facto, higher commuting distances are seen in the largest urban areas, especially in Ile- de-France (ref). They are also favoured by the growing conflict between injunctions towards professional mobility and anthropological aspirations towards settlement. Joint effects of metropolization and job market liberalization could therefore paradoxically reduce rather than increase proximity benefits, thus reinforcing car dependence. In addition, transport financing for rural areas and small cities could be challenged by the increasing scarcity of public funds and the lack of cost-effectiveness of new investments, thus potentially limiting future alternatives to the car. Even within metropolitan areas, outer suburbs might remain largely car-dependent in the future. Finally, generation effects are also ambivalent, as found in a number of empirical research works: in particular, rather than a general disinterest for the car, the millennials could manifest a new pattern of transition towards adulthood, characterized by longer studies, a delayed entrance into the worforce and later family foundation, all of which also become more uncertain (Billari and Liefbroer, 2010). In this case, access to auto-mobility would be delayed rather than abandoned (Kim, 2014;Garikapati et al., 2016), so that long-term impacts of generational renewal on car traffic would be mitigated.

The effects of reinforced constraints on the mobility system is therefore uncertain because of potentially contradictory injunctions from the economy, social change, public policies and spatial patterns towards the individual. Their potential impacts on car travel are difficult to model and probably require prospective thought and a better understanding of the complex network of relationships auto-mobility maintains with changes in the economy, the spatial distribution of jobs and opportunities, public policies and residential choices. Therefore by now, we will focus on the long-term impacts on car ownership and the average car mileage per adult/household from structural sociodemographic change and the relaxation of economic constraints, in order to determine whether saturation of automobility shall be expected, at which level and according to which temporality, assuming other constraints to be stable. In order to do so, we will design a model to estimate saturation thresholds and model growth paths towards saturation with an increasing average income over time.

Data analysis and modelling framework

The French Car Fleet Surveys

Analysis relies on the French car fleet survey waves from 1993 to 2010, based on a panel sample containing six to seven thousand households living in France surveyed on their vehicle fleet and driven mileage, being renewed by a third every year. Realized by the polling institute TNS-SOFRES from 1976, the survey is financed by a bunch of public and private stakeholders and analyzed by the French Institute of Sciences and Technologies for Transports, Planning and Networks (IFSTTAR). The dataset contains information on the household vehicles and their characteristics, up to three vehicles per household, including personal cars but also light-duty vehicles. This information can be used to design models of car availability and car mileage by vehicle rank, by dissociating the first, second and third household vehicle. Given the existence of missing values for some variables, in particular for income and annual mileage, the "hot-deck" method 1 was used to complete missing values and prevent the loss of information that would lead to less precise and possibly biased estimates. The income per consumption unit is used rather than the total income, in order to compare households from different size and composition 2 .

Data analysis

Following the arguments exposed in section 2, the analysis is performed by vehicle rank, and for every vehicle 1 After having identified the determinants of the dependent variable with missing values, observations are then classified according to the values of these determinants. An observation with missing value is finally given the value of the observation immediately preceding or following. 2 The number of consumption units per household is calculated using the OECD scale of equivalence, where the household head is weighted for 1, other household adult members for 0.5, and children aged less than 14 for 0.3. rank we disentangle car ownership rates from the annual car mileage per vehicle among equipped households. In addition, the analysis is split by household category in order to account for heterogeneous choice sets, corresponding to different classes of needs. In particular, the choice set for the number of vehicles per household is strongly related to the number of adult members. For instance, for households containing a single adult member, the choice set is usually restricted to being non-motorized or owning one single vehicle, while the case for owning two vehicles or more is extremely marginal, as additional vehicles would be superfluous. In the same way, households containing two adult members will tend to choose between holding zero, one or two vehicles, while the case for three or more vehicles will be very scarce, given the lack of necessity. A consequence is that in the vast majority of cases, the number of adults represents an upper limit for the number of household vehicles. Using a unified model without separating different household types would therefore inaccurately describe choice sets, leading to overestimate some marginal choices by ignoring the notion of superfluous equipment. In addition to the number of adults, we also include segmentation variables for professional activity and the presence of children, as indicators of additional travel needs notably related to commuting and accompanying trips. Distinguishing different classes of needs or household categories also allows accounting for different levels of potential demand and income sensitivities. Sixteen segments were used for data analysis and model estimation. The last segment IAE was made up of three initial categories, some of which were statistically too small for data analysis and model estimation, including either one, two or three and more non-working adults with children.

The calculation of equipment rates confirms the heterogeneity of choice sets between household types depending on the number of adults. For single-adult households, alternatives are usually restricted to being de-motorized or owning one single vehicle, the alternative of owning two vehicles or more remaining marginal. In the same way, households with two adults are confirmed to choose between owning zero, one or two vehicles, while the alternative of holding a third vehicle is in minority. The alternative of holding three vehicles or more becomes substantial only for households containing at least three adults. This is why households with three vehicles remain in strong minority as few of them contain more than two adults, apart from the temporary situation of adult children still living at home during their college studies. In addition, in most household categories, equipment growth is found to slow down from a certain level of income and converge towards an equilibrium, which may also be considered as a saturation threshold.

It is also confirmed that saturation thresholds differ much according to the number of adults and by vehicle rank : in particular, they testify that the diffusion process of a given equipment may remain incomplete. For instance, the maximal equipment rate for the first vehicle reaches roughly 80 % among single working adults without children in the highest income groups, while it is close to 99 % among dual workers with children. Within the same group, the maximal equipment rate for the second vehicle reaches roughly 75 %, while it remains marginal among single working adults. In addition, in every household category, saturation thresholds also differ much by vehicle rank : for instance, the status of the first vehicle as an equipment shifts from an essential to something vital between single-adult households and households with two adults, with equipment rates at almost completion in the second case ; between the same groups, the status of the second vehicle comes from a superfluous equipment to an essential. On the contrary, the maximal equipment rate doesn't depend much on the activity status of adult household members. For instance, the maximal diffusion rate for the first vehicle is similar for working and non-working adults without children. In the same way, the maximum diffusion rate of the second vehicle is roughly 60 % among households with two adults and no children, no matter how much is the proportion of adults working. Finally, the existence of children is also a case for higher levels of potential demand, though less influential than the number of adults. For instance, the maximum equipment rate goes from 90 % to 80 % between single working adults with or without children. The distribution of vehicle rates by income group also highlights the existence of minimum diffusion rates in lower income groups, which may be interpreted as non-negotiable equipment needs. By differentiation between maximum and minimal equipment rates, one can estimate the amount of negotiable equipment needs, which can be abandoned in case of insufficient financial resources. For the first vehicle for instance, this amount is quite negligible for dual workers with children, for which the equipment rate reaches nearly 100 % even in lower income groups, indicating that the first vehicle is considered vital and merely non-negotiable. On the other hand, the equipment rate shifts from 50 to 80 % between lower and higher income groups for single working adults without children, while the equipment rate for the second vehicle increases from 50 to 75 % with an increasing income among dual workers with children. The amount of negotiable needs, altogether with minimum and maximum equipment rates, are therefore similar to those observed for the first vehicle among single working adults. Professional activity and the existence of children also strongly influence the minimal equipment rate and therefore the amount of negotiable needs. While the minimum equipment rate for the first vehicle approaches 40 % for single non-working adults without children, it reaches 52 % for working adults without children, and 66 % for working adults with children.

Finally, the level of potential demand by vehicle rank can be empirically estimated for every household type from the observation of equipment rates by income groups, giving an indication of saturation thresholds to be reached with the complete relaxation of financial constraints. Potential demand consists of both non-negotiable and negotiable needs, to be removed in case of insufficient resources. For the main vehicle, the saturation threshold is usually very close to completion in most household groups, indicating that it is considered a first necessity good. The influence of household characteristics (the number of adults, activity status, the existence of children…) on the level of potential demand exists but is rather limited. Contrary to the main household vehicle, the potential demand for the second vehicle is much differentiated by household type : the status of the second vehicle varies from a medium-range to a first necessity good with travel needs increasing. However, the market is made both of negotiable and non-negotiable needs. While the potential market size for the first vehicle is roughly independent from household travel needs, household characteristics are more influent on the amount of negotiable needs. For instance, the first vehicle is partly negotiable among singles while it is essentially non-negotiable among couples. The amount of negotiable needs according to the household resources also varies in relation with travel needs for the second vehicle. This is a fortiori the case for the third vehicle, which can be considered an up-range good until now, with limited diffusion. Then we consider the average car mileage per vehicle 3 , depending on vehicle rank, in relation with household income group. The average car mileage for the first vehicle is found to be sensitive to income, but ever-less with income increasing, which is a consistent result with the prior assumption of pseudo-saturation. In addition, the average car mileage tends to level off from a certain income group for many household types, or is even decreasing in the highest income groups. However, the dependence of car mileage on income is usually weak, the minimum mileage, corresponding to non-negotiable needs, being close to the maximum mileage. For instance, among single working adults without children, the average car mileage is increasing from 11 000 to 14 000 km between lower and higher income groups. The mileage is more dependent on household type, varying from 15 200 to 18 000 km depending on income for households with two working adults and children, vs. 8 500 to 11 000 km for single non-working adults without children. The number of adults, the existence of professional activity and children are all factors of increasing car mileage in relation with higher travel needs. However, the relation between these factors and the amount of negotiable needs in relation to income is less obvious than for equipment rates. A reason might be the existence of a selection bias resulting from vehicle ownership decisions already endogenizing travel plans, which would also explain the low income sensitivity of vehicle mileage for equipped households. Car mileage for the second vehicle appears to be almost independent from income. The difference with the main vehicle can be interpreted in terms of their respective functions. Given its multipurpose usage, the main vehicle is also used for discretionary motives such as leisure and holiday trips, which are more sensitive to income as they can more easily be removed in case of insufficient resources, while the second vehicle is rather dedicated to daily trips, by nature less sensitive to income. The average car mileage is also very stable dependent on household type and approximately equates 10 000 km in every group, apart from single adults holding a second vehicle and households containing no working adult. Again, this apparent homogeneity can be interpreted assuming the proportion of negotiable needs to become negligible among equipped households as car ownership decisions are already endogenizing travel plans. Finally, car mileage for the third vehicle is negatively correlated to income. This apparently paradoxical result might express some endogenous bias, third vehicle ownership being even less correlated to travel needs as income is increasing or in other words, the mileage threshold triggering purchasing decisions would be decreasing with income.

Modelling framework

Given behavioural uncertainties affecting the diffusion process of car ownership and the identification of limits to travel by car, empirical methods should be preferred for the estimation of saturation thresholds, rather than theoretical models relying on heuristic assumptions, that would require much more research to be fully confirmed through appropriate experimental designs.

These considerations led us to model car ownership and mileage by vehicle rank, separating the first, second and third vehicle by considering them as distinct goods at different stages of their diffusion process.

Different household groups can indeed be interpreted as different classes of travel needs, being characterized by heterogeneous choice sets for the number of vehicles, while vehicle ranks can be understood as goods of different status and in more or less advanced diffusion stage.

In order to account for the dynamic nature of perceived travel needs, some models figure potential or desired demand as a function of income, either for car ownership or car use, and represent the adjustment process between potential demand and real demand as a function of time, like in partial adjustment models (e.g. Dargay and Vythoulkas, 1999;Dargay, 2002). However, if assuming the existence of limits to the demand for auto-mobility, long-term car ownership and travel needs may then be assimilated to saturation thresholds, which are essentially a function of household composition, employment, the built environment and spatial activity patterns. In particular, structural change in household composition and participation to the workforce well explain changing allocations of household expenditures and time over the life course and between cohorts (Blundell et al., 1994). In particular, increasing car ownership across cohorts is well accounted through the increasing women's participation to the workforce, while changes in household composition and employment rates also well account for the bell-shaped curve of car ownership over the life course which is usually observed. Limited financial resources then represent, within each household category and in every time period, the main reason explaining the gap between desired and observed travel patterns. Insofar as we are mainly interested in long-term forecasting rather than in the estimation of short-run and long-run elasticities, we can then ignore the temporal adjustment process between real and potential demand, and adopt a static model where structural change in socio-demographics, employment rates and income distribution are sufficient to account for long-term change in car ownership and car use. In order to account for the potential effect of the number of cars on the distribution of car mileage between household vehicles, we introduce dummy variables for existence of a 2 nd and 3 rd vehicle in the car mileage model of the 1 st vehicle, and a dummy variable for the 3 rd vehicle in the car mileage model of the 2 nd vehicle. In such manner, we may be able, for a given household composition, to account for both effects of travel induction and mileage transfer between vehicles related to the number of household cars.

In order to estimate saturation thresholds for car ownership rates and average car mileage per person, and later on to calculate their growth path towards saturation -or pseudo-saturation -over time, we design the following model :

= ( ) + (1)
Diffusion rate of vehicle rank j, for household i belonging to category k Income per consumption unit of household i, belonging to category k , , Parameters ( < 1, > 0)

Disturbance term

In this formula :

stands for the saturation threshold -or alternatively, the potential demand -for vehicle j in household category k, which is also an indicator of market size. This is the equipment rate that would be reached with the complete relaxation of financial constraints, i.e, when the income becomes infinite. As already mentioned, < 1 shall express the lack of need for vehicle j for some households or the persistence of other constraints at equilibrium -for instance the existence of a handicap, or the lack of parking supply -resulting in an incomplete diffusion process.

is an income parameter that should be positive, as the diffusion rate of equipment j is assumed to increase with the relaxation of financial constraints. A greater value of is an indicator of the concavity of the diffusion curb, also meaning that the equipment rate of vehicle j is converging more rapidly towards saturation. With greater values of , an increasing proportion of households are closer to saturation, only the lower income groups being underequipped. A greater value of can therefore also be considered an indicator of lower inequalities.

is an intercept, which can be positive or negative, of which depends the minimum level of diffusion corresponding to a zero income. As we assume the diffusion rate to be described by a logistic curve, > 0 implies that (0) > /2 or said otherwise, that the diffusion process always stands in its second phase of convergence towards saturation. It also means that the majority of equipment needs are non-negotiable due to low financial resources and therefore gives an indication that equipment j is considered an essential. On the contrary, < 0 means that (0) < /2, implying that a majority of equipment needs are negotiable due to lower income, and therefore that equipment j is rather a mid-range or even luxury good. therefore defines market size or the amount of potential demand for equipment j, -modulo the logit c.d.f function -the proportion of non-negotiable needs, while the difference -Λ% & stands for the amount of negotiable needs due to insufficient resources, and increases the concavity and therefore the velocity of the convergence path towards saturation.

The following model is then implemented for the annual car mileage expectation of vehicle j for individual i in household category k, conditional on vehicle j ownership.

%' () = 1& = * + + ,-.( + 1) + / (2)
' Annual mileage of vehicle rank j for household i in category k, for households holding vehicle j )

Indicator for vehicle j ownership Income per consumption unit of household i in category k + , * Parameters / Disturbance term According to this formula, the car mileage for a given vehicle rank, conditional on vehicle holding, is increasing with income -provided + > 0 -however ever-more slowly, traducing the notion of pseudo-saturation. Though is theoretically non-limited, an empirical threshold could be estimated that would fulfil the needs of 99 % households.

Here again, a minimum car mileage * is estimated for households with zero income, corresponding to nonnegotiable needs. As in ParcAuto, up to three vehicles are described for every household, the expectation for the total car mileage of a given household can then be expressed as :

0(' ) = ∑ 0 2 3 %' () = 1& 4() = 1) (3)

Model estimation

Car equipment models by household category and vehicle rank are then estimated by maximum-likelihood. Initial values are deducted from empirical thresholds being emphasized by data analysis. Then, the algorithm is proceeding through successive iterations to perform maximum likelihood estimation. Log-linear models for car mileage are estimated through regular regression procedures. Estimation results are synthetized in tables 4 and 5, respectively for car ownership and car mileage models. In most cases, saturation thresholds are consistent with empirical observations. They confirm the diffusion process of the first vehicle to be often almost complete, with a level of potential demand varying between 80 and 100 % depending on household type. Potential demand is slightly lower for singles, where it establishes between 80 and 90 % according to household type, while it reaches 95 to 99 % among households containing two adults. Results are clearly different for the second vehicle, for which the diffusion process is always incomplete. As expected, the potential demand for a second vehicle is very limited among singles for which it can be assimilated to a superfluous equipment, reaching only 5 to 10 %. For households containing two adults, its status can be considered as intermediary with potential diffusion rates ranging from 60 to 75 %, essentially depending on the existence of children, representing an additional case for a second vehicle. Finally, potential demand for a second vehicle ranges from 75 to 90 % of households containing three adults or more, making of it an essential in these groups, especially for households where at least two adults are working. Coming to the third vehicle, potential demand is almost non-significant for singles, still remains marginal among households containing only two adults, with a potential market size ranging from 5 to 10 %, but becomes more substantial in households with three or more adults. However, apart from households with two working adults and no children, the third vehicle still remains an up-range equipment, with a level of potential demand comprised between 20 and 35 %.

Values of are almost always positive, suggesting that the majority of potential demand for the first vehicle is non-negotiable. Only for singles and households with no working adult is potential demand for the first vehicle mostly negotiable. The highest amounts of non-negotiable needs are found for households with two working adults or even more. On the contrary, values of 5 are in majority negative, indicating a higher amount of negotiable needs due to insufficient resources for the second vehicle. This is obviously the case for singles, for which a second vehicle is generally superfluous, but also for households including only one working adult, and even more for households with no working adult at all. The number of professionals therefore increases the degree of necessity of a second vehicle among households with two adults or more. The proportion of negotiable needs is even higher for the third vehicle, which has been shown to represent an up-range equipment, even for households with three adults or more.

Finally, values of are found to increase with the number of adults, the proportion of working adults and the existence of children, the most important factors being the number of adults and the existence of children. It means that income gaps in vehicle rates are decreasing with the intensity of travel needs, being a factor of homogeneity in equipment behaviors. However for a given household category, values of are decreasing with vehicle rank j, showing the existence of increasing income differentiation with a higher vehicle rank, as the corresponding equipments are becoming less of a necessity. The degree of necessity for a given vehicle is therefore a function of household category k and vehicle rank j, which have an influence on both the market size or potential demand for this type of good, the amount of negotiable needs and the rapidity of convergence towards saturation or income-related inequalities in equipment rates. This result is important and should be kept in mind when interpreting incomedifferentiated behaviors which do not necessarily express pure inequalities, but result from an interaction between inequalities and necessity. Therefore, sharper income distributions may in some cases illustrate a higher degree of necessity rather than pure inequalities. Car mileage models per household category and vehicle rank are then estimated. The *'s are representing the estimated amount of non-negotiable travel among equipped households, while the 's stand for the income sensitivity of car mileage. For every vehicle rank, the *'s are higher for households with at least two adults, and generally increase with children and the number of professionals, though some results may appear counter-intuitive possibly due to biased parameters for some segments which might not benefit from sufficient sample size. Estimates for the +'s confirm that the first vehicle mileage is in general more sensitive to income than for the second and third vehicle, which we explained by its type of usage, including some proportion of discretionary trips.

Equipment and traffic forecasts

Methodology

Starting from this model, we realize equipment and traffic projections up to 2050, also estimating levels of potential demand for car equipment, in the absence of financial constraints. All projections account for structural demographic change, which is notably related to population ageing and the evolution of family models. Indeed, as models are defined by household category, the results must be weighted by predictable household structure to obtain aggregate forecasts, which implies making projections of the household structure at the forecasting horizon4 . The methodology to obtain projections of the household structure was the following. We started from official projections from the Department of Observation and Statistics of the French General Council of Sustainable Development (CGDD), distinguishing between single adults with or without children, couples and other household types5 . Projections include intermediary points every five years from 2015 to 2050, allowing to figure convergence paths towards saturation or pseudo-saturation. However, to further distribute households between the different groups, one also has to make assumptions about activity rates of adults by cohabitation mode. In order to do this, we first estimated the weight of the age group 15-69 by aggregate household type -singles with or without children, couples -using projections of cohabitation patterns by age group (CGDD, 2012c)6 . Then we used projections of activity rates for the age group 15-69 from the National Institute of Statistics and Economic Studies [START_REF] Filatriau | Projections à l'horizon 2060 : des actifs plus nombreux et plus âgés[END_REF]. These calculations already allow to estimate the distribution of adults living alone or within couples -either with or without children -between working and non-working adults, assuming that adults aged more than 69 are non-working. However, an additional assumption is required to determine the distribution of working adults living with a partner between bi-active and mono-active households, depending on behaviors of social endogamy. The assumption was made that behaviors of social endogamy were essentially unchanged, i.e that the probability for one working adult to be associated with another working adult would be the same than in 2010. The household typology which we used to forecast future levels of car equipment and traffic finally distinguishes between single adults, either working or non-working, with or without children, and couples sorted by the number of working adults (dual working, single working and no working). Future household distribution is summarized in figure 1. Predictably, population ageing will result in decreasing activity rates and related commuting travel needs, while combined with changing family models, it will lead to a greater proportion of single adults. These structural changes must be kept in mind when analyzing projections of equipment and traffic growth. These calculations allow the estimation of levels of potential demand for car equipment by vehicle rank at different dates by weighting disaggregate saturation thresholds by projected household structure. However, in order to estimate real equipment rates -differing from potential equipment rates because of persistent financial constraints -and car mileages, one also has to make projections of the income distribution at intermediary points, relying on assumptions about income growth. The assumption retained here was of an homogeneous and constant income growth of 1 % on average.

Projection results

For every household type and vehicle rank, we calculated growth paths of equipment rates towards saturation levels estimated in section 4, which allows to evaluate both market maturity and correlatively, reserves of potential growth. For the first vehicle, reserves of potential growth are weak for couples as diffusion is already very close to completion, while they appear to be greater for singles and especially for non-working singles. For instance for non-working singles with children, the equipment rate for the first vehicle will increase from 31, 2 to 53,2 % between 2010 and 2050 in relation with an increasing income. For non-working singles without children, it will raise from 60,6 to 70,4 %. By comparison, the diffusion rate for dual workers will stay almost stable -from 96,6 to 97,3 %. For the second vehicle, the saturation threshold is almost already attained for dual workers, but reserves of potential growth still remain for couples with at least one unemployed adult. For instance, for single workers, the diffusion rate will increase from 43,9 to 52,1 %, and from 25,5 to 34,9 % for couples with no workers. Finally, the diffusion of the third vehicle will remain almost stationary among dual workers at roughly 11,4 %, while low reserves of potential growth still remain for couples with one or no working adult -with a corresponding equipment rate raising from 5,5 to 6,8 % in the first case, and from 2,0 to 3,0 % in the second. As a whole, reserves of potential growth for car equipment still remain among households with at least one non-working adult. However, potential equipment growth is globally limited and convergence towards saturation appears to be quite slow with equipment growth slowing down from 2025. These results are then combined with household structure projections to calculate global indicators of real and potential demand by vehicle rank. In relation with the increasing proportion of singles and retired resulting from population ageing and changing cohabitation patterns, potential demand for the first and the third vehicle will sligthly decrease -respectively from 92,2 to 90,3 %, and from 3,4 to 3,3 %, while demand for the second vehicle will fall down more substantially from 40,0 to 32,2 %. In addition, real equipment rates will slowly approach levels of potential demand by increasing softly for the first vehicle and decreasing for the second vehicle, as real car equipment will decrease more gently than potential demand for the second vehicle due to an increasing income. The equipment rate for the first vehicle will increase from 81,9 to 83,4 %, while it will decrease from 27,6 to 24,6 % for the second vehicle. Aggregating all vehicle ranks, the potential number of cars per household would decrease substantially along with the narrowing market for the second vehicle -from 1,36 to 1,26 vehicle per household -while the potential number of cars per individual would slightly increase, from 0,59 to 0,61. The real number of cars per household will slightly decrease while the average number of cars per individual would softly increase, approaching the estimated saturation threshold of 0,6 car per person. We finally estimate car mileages. In most categories -apart from single non-working adults with children -the main vehicle mileage is slightly increasing with income but at an ever-slower pace, seemingly attaining pseudosaturation by 2040, apart from single working adults where growth would maintain afterwards. By contrast, the second vehicle mileage remains relatively stable on average, apart from single non-working adults with children. By aggregating all vehicle ranks, one obtains estimations of the average car mileage per household and individual. The average car mileage per household is softly increasing up to 2050 for every household type -for instance, for single workers, it is increasing from 19 100 to 20 750 km -but remains relatively stable on average because of household structural change already mentioned. By contrast, the average car mileage per individual keeps on increasing, from 5 140 to 5 780 km, but at a decreasing pace, especially from 2040, remaining almost stable afterwards.

Conclusion

After decades of continued increase, car traffic has temporarily levelled off during the 2000's, before coming back to growth once the economic crisis was passed, along with decreasing fuel prices. Though economic conditions may have decisively contributed to recent trends, there are strong reasons to believe, however, that traffic growth will not be infinite : among these, the diffusion process of car ownership as an equipment good will sooner or later be completed, daily travel time is rather stable on average and potentially limited by competing allocations of time budgets, real car travel speeds are getting stable or even decreasing, individuals benefit from decreasing marginal returns of additional car travel, and infrastructure supply is limited by credit cuts and re-orientation towards transport alternatives for motives of sustainability. In addition, car traffic levels are increasing proportionally to the number of vehicles, so that one may assume that the average car traffic per adult will come to saturation once every one who needs it will be equipped with a car. Also, empirical observation of car equipment rates by household type and vehicle rank provides evidence of convergence towards saturation from a given level of income. While it seems more arbitrary to assume the existence of an absolute limit to car use, car mileage growth is nonetheless decreasing with income, providing evidence for decreasing marginal returns of additional travel leading to pseudo-saturation. Assuming potential demand to be limited, one convenient way to describe realized demand is to model it as an equilibrium between potential demand and constraints, which may be of different natures, either technical, economical, regulatory or related to the physical environment. Assuming other constraints to be stable, one can study which effect has the relaxation of economic constraints on the level of demand. The estimation of an adapted logit model confirms the existence of saturation thresholds, which may sometimes express an incomplete diffusion process, also allowing to break down potential demand between negotiable and non-negotiable needs, in relation with insufficient financial resources, and to evaluate the rapidity of convergence towards saturation. Forecasting results obtained with these models confirm that while reserves of potential growth for car equipment still remain, they are nonetheless limited, and thwarted by population ageing and changing cohabitation patterns. In addition, mileage projections lead to anticipate a limited growth of average car traffic per adult until 2040, and quasi-stability afterwards. From this date, traffic growth would be essentially determined by demographic growth. In the light of these results, pollutant and GHG emissions abatement goals might be facilitated. Finally, we draw out some research tracks for the future. First on the side of model improvements : joint choices of car equipment and car mileage could be tested through tobit models, to account for possible internalization of travel plans into car ownership decisions, which would explain the low income sensitivity of car mileage, or even the counterintuitive relation between car mileage and income for the third vehicle. A theoretical basis for car mileage should also be defined, leading to offer alternative specifications and empirically decide which is the best. Another field of improvement consists in the formulation of alternate scenarios. For instance, traffic forecasts may also depend on energy prices and energy efficiency, which were not accounted for at this stage as our goal was to obtain a central scenario based on economic growth, assuming everything else to be equal. Yet, energy prices might keep on increasing in relation with heavier taxation -in particular with the implementation of a carbon tax -while this effect will be partially countered by technological progress resulting in increasing energy efficiency. In order to model their effect on traffic growth, income could be replaced by an indicator of energetic purchasing power, for instance. Purchasing prices for cars should also be introduced, in relation with different scenarios in terms of price dynamics -for instance, the development of low-cost vehicles, or quite the opposite, heavier vehicle taxation resulting in higher vehicle prices. Nonetheless, future levels of car use will also depend on regulatory constraints or related to the physical environment, which are likely to be reinforced in the future, along with the multiplication of restricted traffic areas, more severe regulations on pollutant vehicles, restrictive speed limits, etc. Finally, beyond the prospect of saturation, some factors could lead to a declining car use. Some of them might be economical, for instance the possible expansion of mass poverty, or the lack of efficiency gains combined with heavier environmental taxation -with the implementation of a carbon tax, for instance -while others might be technological and organisational, such as the autonomous vehicle, a wider deployment of telecommuting or car-based devices such as car-sharing and carpooling. , A., Börjesson, M., Eliasson, J., 2016. Explaining 'peak car' with economic variables. Transportation Research Part A 88, 236-250.

Bastian

Figure 1 :

 1 Figure 1 : Projected distribution by household type, 2010-2050

  Figure 2a,b : Estimated growth paths and saturation thresholds by household type for the 1st and 2 nd vehicles

Figure

  Figure 3a,b : Real and potential estimated car equipment levels 2010-2050

Figure

  Figure 4a,b : Main and second vehicle mileage per household type

Figure

  Figure 5a,b : Generated car traffic per household and individual, generally and by household type.

Table 1

 1

	. Segment names and descriptions
	Segment name	Description
	PSISE	Non-working adult without children
	PSASE	Working adult without children
	PSAAE	Working adult with children
	1A1ISE	One working adult, one non-working adult, without children
	2ASE	Two working adults without children
	2ISE	Two non-working adults without children
	1A1IAE	One working adult, one non-working adult, with children
	2AAE	Two working adults with children
	3A+SE	Three and more working adults without children
	3A+AE	Three and more working adults with children
	2A1I+SE	Two working adults, one non-working adult, without children
	2A1I+AE	Two working adults, one non-working adult, with children
	1A2I+SE	One working adult, two non-working adults, without children
	1A2I+AE	One working adult, two non-working adults, with children
	3I+SE	Three and more non-working adults without children
	IAE	Only non-working adults with children

Table 2 .

 2 Equipment rates by household category, income group and vehicle rank Source: French Car FleetSurveys, 1993Surveys, -2010

			PSISE			PSASE			PSAAE			1A1ISE
		V1	V2	V3	V1	V2	V3	V1	V2	V3	V1	V2	V3
	< 500 €	38,4	2,0	0,2	52,9	3,0	0,4	62,8	1,0	0,0	76,2	19,7	2,2
	500-1000 €	47,1	1,7	0,2	63,7	2,1	0,4	82,5	2,7	0,3	86,4	24,9	2,1
	1000-1500 €	61,2	3,1	0,3	77,4	3,8	0,2	87,5	4,2	0,2	92,6	38,5	2,6
	1500-2000 €	72,8	4,1	0,2	79,9	4,4	0,3	88,8	7,0	0,0	95,6	45,1	2,9
	2000-2500 €	78,9	4,0	0,3	79,1	4,5	0,4	90,7	7,0	0,0	97,2	51,3	4,3
	2500-3000 €	80,5	6,6	0,4	74,9	3,0	0,3	92,9	14,3	0,0	96,9	64,3	5,1
	> 3000 €	82,7	8,0	0,9	74,3	5,3	0,2	83,3	5,6	0,0	97,1	63,4	8,5
			2ASE			2ISE			1A1IAE			2AAE
		V1	V2	V3	V1	V2	V3	V1	V2	V3	V1	V2	V3
	< 500 €	81,5	36,4	3,1	77,6	13,4	0,7	84,2	25,6	1,5	91,9	47,9	2,7
	500-1000 €	89,7	36,7	2,4	90,5	14,0	1,2	96,2	41,4	1,9	98,4	58,4	3,0
	1000-1500 €	95,2	52,2	3,7	95,3	20,3	1,2	98,2	55,0	3,0	98,9	72,3	4,8
	1500-2000 €	96,5	58,5	4,8	96,9	30,0	2,1	98,0	61,7	4,9	98,5	72,9	4,7
	2000-2500 €	96,3	60,2	5,3	98,2	40,4	3,4	97,0	66,4	1,5	97,8	74,7	5,0
	2500-3000 €	92,3	59,2	5,5	97,6	51,5	6,1	96,5	77,2	3,5	97,5	70,1	4,9
	> 3000 €	93,5	60,9	5,0	97,3	54,5	5,4	100,0	64,2	4,5	96,3	64,6	4,9
			3A+SE			3A+AE			2A1I+SE			2A1I+AE
		V1	V2	V3	V1	V2	V3	V1	V2	V3	V1	V2	V3
	< 500 €	87,1	64,5	45,6	86,7	58,1	25,7	89,4	49,3	16,5	91,6	47,6	10,8
	500-1000 €	95,5	77,2	43,0	95,9	70,7	31,4	97,1	64,3	22,7	99,1	65,3	12,8
	1000-1500 €	97,0	81,5	48,8	97,2	80,0	30,6	98,8	75,4	24,2	99,1	74,4	12,5
	1500-2000 €	97,6	86,9	53,9	98,7	81,8	28,6	99,1	81,1	26,3	100,0	80,9	16,1
	2000-2500 €	97,0	87,8	56,1	100,0	87,9	30,3	100,0	85,0	33,1	100,0	83,7	24,8
	2500-3000 €	100,0	90,9	56,4	94,7	57,9	0,0	98,6	84,1	31,2	95,7	85,5	27,5
	> 3000 €	93,7	76,2	41,3	93,3	80,0	0,0	97,6	84,6	34,2	100,0	68,4	10,5
			1A2I+SE			1A2I+AE			3I+SE			IAE
		V1	V2	V3	V1	V2	V3	V1	V2	V3	V1	V2	V3
	< 500 €	86,2	33,5	5,0	84,2	27,0	4,2	82,2	33,5	8,9	35,8	3,3	0,0
	500-1000 €	95,6	48,9	11,9	97,1	47,9	6,7	96,1	50,0	16,4	65,5	16,1	0,0
	1000-1500 €	98,0	64,0	18,4	99,3	70,8	13,4	99,3	70,8	26,6	89,5	36,8	2,6
	1500-2000 €	98,8	73,9	20,5	98,8	79,8	21,4	97,8	72,3	22,8	100,0	28,6	28,6
	2000-2500 €	98,9	73,9	24,4	100,0	78,3	17,4	100,0	73,8	29,5	100,0	50,0	16,7
	2500-3000 €	95,8	70,4	16,9	94,7	57,9	5,3	97,6	64,3	21,4	0,0	0,0	0,0
	> 3000 €	98,2	65,5	34,6	100,0	75,0	0,0	90,5	52,4	19,1	100,0	100,0	0,0

Table 3 :

 3 Average car mileage (in km) by household category, income group and vehicle rank Source : French Car FleetSurveys, 1993Surveys, -2010

Table 4 :

 4 Estimated saturation thresholds of diffusion rates by household category and vehicle rank

	Source : French Car Fleet Surveys, 1993-2010
		V1	V2	V3	V1	V2	V3	V1	V2	V3
	PSISE	0,863	0,095	0,010	1,12 e -4	7,2 e -5	2,7 e -5	-0,767	-1,902	-6,367
	PSASE	0,795	0,056	0,003	1,9 e -4	6,6 e -5	3.5 e -3	-0,028	-0,454	0,558
	PSAAE	0,899	0,099	-	2,58 e -4	1,4 e -4	-	0,039	-2,332	-
	1A1ISE	0,976	0,660	0,213	1,55 e -4	1,07 e -4	4,5 e -5	0,671	-1,372	-2,637
	2ASE	0,958	0,624	0,058	2,3 e -4	1,46 e -4	9,9 e -5	0,876	-0,664	-0,802
	2ISE	0,978	0,623	0,082	2,13 e -4	1,04 e -4	9,1 e -5	0,522	-2,227	-2,891
	1A1IAE	0,985	0,706	0,046	3,54 e -4	1,76 e -4	1,4 e -4	0,548	-1,246	-1,465
	2AAE	0,986	0,747	0,052	3,78 e -4	2 e -4	1,91 e -4	1,710	-0,338	-1,173
	3A+SE	0,976	0,876	0,564	2,6 e -4	1,48 e -4	7,8 e -5	1,330	0,563	0,721
	3A+AE	0,974	0,806	0,287	4,74 e -4	2,32 e -4	3,2 e -3	0,555	0,098	-2,748
	2A1I+SE	0,991	0,860	0,329	3,39 e -4	1,54 e -4	1,05 e -4	1,049	-0,319	-0,373
	2A1I+AE	0,994	0,851	0,280	4,46 e -4	1,54 e -4	2,7 e -5	1,075	-0,266	-2,214
	1A2I+SE	0,985	0,764	0,241	3,23 e -4	1,74 e -4	2,06 e -4	0,745	-0,947	-1,923
	1A2I+AE	0,993	0,808	0,204	4,43 e -4	2,41 e -4	1,57 e -4	0,180	-1,644	-1,935
	3I+SE	0,982	0,647	0,134	4,03 e -4	2,49 e -4	2,75 e -4	-0,730	-2,305	-3,733
	IAE	0,942	0,437	-	2,63 e -4	3,11 e -4	-	-1,335	-3,312	-

Table 5 :

 5 Parameter estimates of car mileage models by household category and vehicle rank Source : French Car FleetSurveys, 1993Surveys, -2010

			*			+
		V1	V2	V3	V1	V2	V3
	PSISE	3145	2739	1750	592,4	247,8	309,8
	PSASE	3250	7970	5010	1064	119,9	-256,0
	PSAAE	7588	5077	-	703,5	550,4	-
	1A1ISE	4894	8648	69	1006,1	73,7	611,4
	2ASE	5718	4454	7218	1113,1	745,9	231,5
	2ISE	10210	3761	1912	131,7	279,0	286,9
	1A1IAE	5314	5628	16000	1186,7	602,7	-585,9
	2AAE	7375	4632	8610	997,5	713,6	50,6
	3A+SE	3957	18313	8585	1242,7	-620,3	439,3
	3A+AE	9971	11054	8691	679,2	48,4	475,0
	2A1I+SE	6753	10371	13877	978,6	141,4	-217,0
	2A1I+AE	1165	5869	6970	1667,0	568,8	333,4
	1A2I+SE	3582	11712	12361	1135,9	-93,8	-201,4
	1A2I+AE	18443	1125	28444	-275,7	1084,3	-2111,2
	3I+SE	8713	17190	12437	422,4	-909,7	-586,0
	IAE	724	8394	-	1464,6	65,4	-

However, we implemented a simplified segmentation which was consistent with household categories used in official demographic projections, which also implied to re-estimate corresponding models.

These projections were realized to estimate the potential housing demand in France, relying on regular assumptions about fertility, net migration, mortality and cohabitation behaviors (CGDD, 2012b).

As projections were available only up to 2030, trends were prolongated up to 2050, in particular the decreasing proportion of adults living with a partner and correlatively, the increasing proportion of singles.

14248 10865 10518 14847 10886 10560 500-1000 € 14964 12455 11980 16780 11950 12597 15287 11521 11308 15872 10490 8227 1000-1500 € 15618 11915 13421 14082 11096 12758 16054 11370 11592 17274 11531 10187 1500-2000 € 16062 12177 11687 15697 12453 13432 16602 11941 11168 17268 10698 10175 2000-2500 € 15972 11818 13027 19154 9267 9680 16497 11271 12049 19074 13433 12254 2500-3000 € 16953 12560 12677 17883 10401