will result in in- or out- migration respectively such that all households in any city of the system display the same utility level at any given time: ; whereas the population , becomes endogenous. When flood prone areas are distributed proportionally to the distance from the city center (configurations uu= uu� NN a) to d) in Figure 1), it is possible to show analytically, using Cobb-Douglas functions and building on Avner and Hallegatte (2019), how household utility ( ), aggregate land values ( ) and city populations ( ) are affected by floods compared to a no-flood situation in all four polar city configurations (CCA, CCP, OCA uu AALLAA NN and OCP).

||**Closed Cities**<br> <br>|**Open Cities**<br>|
|---|---|---|
||CCA<br> <br><br><br>|OCA<br> <br>|
|**Absentee Landowners**|⎩<br>⎨<br>⎧ufCC= Cf<br>1<br>γ<br>∗uNCC<br>ALAfCC= ALANCC<br>NfCC= NNCC= N<sup>�</sup><br> <br>|�<br>ufOC= uNOC= u�<br>ALAfCC= Cf∗ALANCC<br>NfOC= Cf∗NNOC<br>|
||<br> <br>CCP<br> <br><br><br>|<br> <br>OCP<br> <br>|
|**Public Ownership of Land**<br>_Table 1:Household utility (_u_), Aggregate lan_<br>_flood situation in the four classic urban con_|<br>⎩<br>⎨<br>⎧ufCC= Cf<br>1<br>γ<br>∗uNCC<br>ALAfCC= ALANCC<br>NfCC= NNCC= N<sup>�</sup><br>_d values (_ALA_), and city population (_N_)_<br>_figurations (open or closed cities with a_|<br>�<br>ufOC= uNOC= u�<br>ALAfOC= Cf∗ALANOC<br>NfOC= Cf∗NNOC<br>_when floods affect a city compared to a no-_<br>_bsentee landowners or public ownership of_|

<u>⎩ fC C NC C</u> _Table 1:Household utility ( ), Aggregate land values ( ), and city population ( ) when floods affect a city compared to a no-_ OOC OOC _flood situation in the four classic urban configurations (open or closed cities with absentee landowners or public ownership of_<sup>�</sup> _land) when flood-prone areas are proportional to the distance to the city center, corresponding to cases a), b), c and d) in Figure_ uu AALLAA NN _1._ <u>ρ</u> and in Table 1 are parameters that intervene in calculations with bb , with ~~�~~ aa 1 ρ+ii CCff being identical over all flood-prone areas (there is only one flood type) and γγ CCff = (1 −θθ) + θθ� ~~.~~ ρρ+ii+ρρff ff In closed cities (CCA and CCP), two results, already reported in Avner and Hallegatte (2019), are ρρ γγ= aaββ 1 noteworthy. First, household utility is decreased by a factor <1, compared to a ‘no-flood’ situation, γγ translating the fact that reduced housing supply will increase unitary housing rents and lower housing ff CC consumption, all else equal. Second, Aggregate land values (ALV) remain at their ‘no-flood’ level as the decrease in housing supply and increase in housing construction costs are exactly compensated by higher housing rents. This result, documented in section 2.2, is true in the CCA situation also.

In open cities (OCA and OCP), household utility is exogenously given such that it is equal in flooded or nonflooded settings. However, city population will adjust through in- or out-migration in order for household utility to remain constant. Two results here are also noteworthy. First, both population and aggregate land values are decreased by a factor < 1 , with homogenously distributed flood-prone areas, 1 compared to a no-flood situation in both OCA and OCP settings. It should be noted that ff , so that CC γγ the decrease in population and aggregate land values is higher in open cities than the decrease in utility ff ff CC > CC in closed cities. Second, aggregate land values per household remain at their ‘no-flood’ level, such that C C . ffOOC NNOOC NN CC ff CC LL When floods are central or peripheral, analytic calculations are more difficult to derive. We use<sup>�</sup> = LL<sup>�</sup> = LL<sup>�</sup> = LL<sup>�</sup> When floods are central or peripheral, analytic calculations are more difficult to derive. We use simulations informed by the same set of parameters as in section 2.2 and described in section 6 and the 10