_Cost of time._ In the model, rents (per surface unit) decrease when moving away from the center of Paris because households have to pay a generalized transportation cost, which is the sum of a perceived monetary cost (interpreted here as the cost of fuel) and of the cost associated with transport time, assuming that households do a round-trip per day towards the center of Paris. In the simulation, cost associated with transport time represents generally the bigger part of generalized cost, and the way we assess this cost has an important role in our results.

Numerous studies have dealt with this issue, but no conclusive result exists on this complex subject. In Ile-de-France, French Government’s Strategic Analysis Center proposed to use net hourly wage as an estimate for commuting time cost, but explained that the value of actual commuting time cost depends greatly on several factors such as households characteristics or modal choice. (Boiteux and Baumstark, 2001) Due to the importance of time cost choice in the simulation, we calibrated time cost instead of using an a priori fixed value. We computed this cost using our data on rent spatial distribution: out of these data, assuming our model perfectly exact, it is indeed possible to estimate a theoretical generalized transportation cost. Assuming that this generalized cost reflects the sum of the direct cost of transport and of the cost associated with transport time, and assuming that households do a round-trip per day towards the center of Paris, the transport time cost was estimated as a function of journey time.

Marginal time cost seems to decrease with travel time, and we chose to model simply this decrease using a piecewise affine function. This representation leads us to use a cost time worth 105% of the net hourly wage when the travel time is less than 25 min (or, equivalently, when the distance to the center of Paris is less than 15 km), then a very low cost (6.6% of the net hourly wage) for portions of journey in excess of this limit. The value of time for journeys during less than 25 min is therefore very close to commuting time cost in Ile de France according to French Government’s Strategic Analysis Center.

This observed decrease in marginal time cost can be attributed to the limits of our approach, in particular to the monocentric framework and to the hypothesis that households do a round-trip per day towards the center of Paris. In the real world, in places where travel time exceeds 25 minutes, a large fraction of households do not commute to the center of Paris. This leads to a shorter average trip length than in the mono-centric case, and using actual average trip length would enable to use more realistic time cost values and smaller total fuel costs for locations far from Paris city center. In absence of needed data, we did not take into account explicitly this variation in trip length, and modeled it with a non-linear time cost.

# _Appendix D.3. Validation: urbanized surface evolution_

As can be seen on Fig. D.11a, the model reproduces well the current Paris urban area. The main mismatch is in the west (Mantes la Jolie) and in the south of the urban area (Melun), where the model does not capture observed urbanized areas. These two zones correspond to cities which were built long before being included in Paris urban area, and are important employment centers on their own, whereas the model only represents built areas due to Paris urban area sprawl.

It is possible to use this model to simulate city evolution from 1900 to 2006. For instance, Fig. D.11d, Fig. D.11c and Fig. D.11b compare simulated urban area with actual urbanized area, in 1900, 1960 and 1982, respectively. Because of the lack of data, we used the same transport 37