Proposition 2: There exist two formalization cost thresholds, _<u>k</u>_ = _Ra_ <u>(1</u> _π−_ (1 _<u>q−</u>_ <u>)(</u> _q_ 1)+ _−πq_ <u>)</u> and _k_<sup> ̄</sup> = (1 _− q_ )(1 _− π_ )( _y − u_ ) , such that:<sup>18</sup>

- If _k_ _<u>≤ k</u>_ <u>(Case</u> 1): The city is fully formal (all owners register and sell their plots). It extends ˆ

- until _xb_ = _x_ ( _k_ ) =<sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −k −u_].</sup>

- If _<u>k < k ≤</u> k_<sup> ̄</sup> <u>(Case 2):</u> The city extends until _xb_ = _xa_ and is organized in two zones, one formal on [0 _,_ ˇ _x_ ( _k_ )] (Zone 1) and one informal on ]ˇ _x_ ( _k_ ) _, xa_ ] (Zone 2), where _x_ ˇ( _k_ ) =<sup><u>1</u></sup> _t_ � _y −_ (1 _−q_ )(1 _<u>k</u> −π_ )<sup>_−u_</sup> � , with the border _xa_ being the same as in the benchmark case (see Section 2 above). The central residential zone [0 _,_ ˇ _x_ ( _k_ )] is fully formal, with all owners registering their plot and participating in the market. The price on that segment is _Rf_ ( _x_ ) = _y − xt − u_ . The peripheral residential zone ]ˇ _x_ ( _k_ ) _, xa_ ] is fully informal, with all owners participating in the market without registering their plot. The informal price is _R_ ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − xt − u_ ) .

- If _k_ _<u>></u> k_<sup> ̄</sup> <u>(Case 3):</u> The city extends until _xa_ and is fully informal (all owners participate in the land market without registering their plot).

# Proof: See Appendix B.

Case 1 can be viewed as a developed country case where an aordable registration system is available and all land is registered. Case 2 is representative of a developing country context where registration is costly and where formal and informal land uses coexist. There is perfect spatial sorting between registered and non-registered plots, as landowners decide to register their land in the vicinity of the city center (zone [0 _,_ ˇ _x_ ( _k_ )] ), but not further away. Case 3 corresponds to the unlikely case where the registration cost is so high that no one registers and the equilibrium conguration is exactly the same as in the benchmark model presented in the previous section (just as if registration were not an option). Graphs in Figure 2 represent the equilibrium city structure and land prices in Cases 1 and 2.<sup>19</sup> Note that the urban fringe in Case 1 ( _x_ ˆ( _k_ ) ) is greater than in Case 2 ( _xa_ ).<sup>20</sup> This property is consistent with the observation we previously made that tenure insecurity and information asymmetry

> 18Throughout the rest of the paper, we assume that _y − u >_ max( (1 _−q_ )(1 _<u>k</u> −π_ )<sup>_,_</sup> _π_ + _qR_ (1 _<u>a−π</u>_ )<sup>)toensuretheexistenceof</sup> the city and the possibility of having unregistered plots. This implies that _<u>k</u> < k_<sup> ̄</sup> . It is a stronger assumption than the one we had to justify the existence of the city under the benchmark model.

> 19In Appendix B, we also present a gure that plots the payos of sellers underlying their participation and registration decisions (see Figure 7).

> 20Indeed, observing that _x_ ˆ( _k_ ) is only dened for _k ≤_ _<u>k</u>_ <u>,</u> we see that _x_ ˆ( _k_ ) _− xa_ =<sup><u>1</u></sup> _t_<sup>[</sup><sup>_<u>k</u>_</sup> _− k_ ] _>_ 0 .

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