metadata: authors: 0: author: Letrouit,Lucie Michele Maya
1: author: Selod,Harris
count: Western Africa
docna: 0: docna: Trust or Property Rights ? Can Trusted
            Relationships Substitute for Costly Land Registration in
            West African Cities ?
docty: Policy Research Working Paper
subtopic: Agricultural Economics,Regulatory Regimes,Judicial System Reform,Legal Reform,Legal Products,Legislation,Social Policy,Common Property Resource Development,Urban Housing and Land Settlements,Urban Housing,Municipal Management and Reform,Urban Governance and Management,Transport Services
teratopic: Agriculture,Rural Development,Public Sector Development,Transport,Law and Development,Governance,Social Development,Communities and Human Settlements,Urban Development
abstracts: cdata!: The paper studies the market failures
            associated with land tenure insecurity and information
            asymmetry in an urban land use model, and analyzes
            households' responses to mitigate tenure insecurity.
            When buyers and sellers of land plots can pair along trusted
            kinship lines whereby deception (the non-disclosure of
            competing claims on a land plot to a buyer) is socially
            penalized, information asymmetry is attenuated, but overall
            participation in the land market is reduced. Alternatively,
            when owners can make land plots secure by paying to register
            them in a cadaster, both information asymmetry and tenure
            insecurity are reduced, but the registration cost limits
            land market participation at the periphery of the city. The
            paper then compares the overall surpluses under these trust
            and registration models and under a hybrid version of the
            model that reflects the context of today's West African
            cities where both registration and trusted relationships are
            simultaneously available to residents. The analysis
            highlights the substitutability of trusted relationships to
            costly registration and predicts the gradual evolution of
            economies towards the socially preferable registration
            system if registration costs can be sufficiently reduced.
repnme: repnme: Trust or Property Rights ? Can Trusted
            Relationships Substitute for Costly Land Registration in
            West African Cities ?
display_title: Trust or Property Rights Can Trusted
            Relationships Substitute for Costly Land Registration in
            West African Cities
listing_relative_url: /research/2020/06/32202200/trust-or-property-rights-can-trusted-relationships-substitute-costly-land-registration-west-african-cities
topicv3: Personal Property Rights m1326571 1907,Cadastre and Land Registration m1326617 890,Land Tenure m1326631 888,Transport m1326661 135,Land and Housing m1327648 817
publishtoextweb_dt: 2020-06-30T00:00:00Z
historic_topic: Agriculture,Rural Development,Public Sector Development,Transport,Law and Development,Governance,Social Development,Communities and Human Settlements,Urban Development
origu: DEC-Sustainability & Infrastruct (DECSI)
no_of_pages: 76
author_keys: 0: author_key: 1542992
1: author_key: 750638
model_extractions: Policy Research Working Paper 9310 Trust or Property Rights? Can Trusted Relationships Substitute for Costly Land Registration in West African Cities?

_Lucie Letrouit Harris Selod_ Development Economics Development Research Group June 2020
Policy Research Working Paper 9310

# **Abstract**

The paper studies the market failures associated with land tenure insecurity and information asymmetry in an urban land use model, and analyzes households’ responses to mitigate tenure insecurity. When buyers and sellers of land plots can pair along trusted kinship lines whereby deception (the non-disclosure of competing claims on a land plot to a buyer) is socially penalized, information asymmetry is attenuated, but overall participation in the land market is reduced. Alternatively, when owners can make land plots secure by paying to register them in a cadaster, both information asymmetry and tenure insecurity are reduced, but the registration cost limits land market participation at the periphery of the city. The paper then compares the overall surpluses under these trust and registration models and under a hybrid version of the model that reflects the context of today’s West African cities where both registration and trusted relationships are simultaneously available to residents. The analysis highlights the substitutability of trusted relationships to costly registration and predicts the gradual evolution of economies towards the socially preferable registration system if registration costs can be sufficiently reduced.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at lucie.letrouit@neuf.fr and hselod@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Trust or property rights?

Can trusted relationships substitute for costly land registration in West African cities?

Lucie Letrouit<sup>_∗_</sup> Harris Selod<sup>_†_</sup> Keywords: Land markets, property rights, information asymmetry, informal land use, land registration, ethnic kinship JEL classication: P14, Q15, R14<sup>1</sup>

> _∗_ Paris School of Economics, Ecole Nationale des Ponts et Chaussées, Paris, France. Email: lucie.letrouit@neuf.fr

> _†_ The World Bank, 1818 H Street NW, Washington, DC 20433. Email: hselod@worldbank.org.

> 1 We are grateful to Amadou Cissé for the many discussions about social structures in West Africa that initially motivated this paper, and to Pierre M. Picard, Jan Brueckner and Tony Yezer for suggestions and technical comments on the model. We are also grateful to Eliana la Ferrara and Karen Macours for useful insights on kinship and land markets in developing countries, as well as to Thierry Verdier and Miren Lafourcade and to the participants to the Labor and Public Economics Seminar at the Paris School of Economics far various comments and suggestions. We gratefully acknowledge the funding of the Paris School of Economics Mobility Grant, the ANR SOCOCITY project, the Labex OSE and the Labor and Public Economics Group at the Paris School of Economics.
# 1 Intro

In developing countries, informally holding land is more often the norm than the exception. In subSaharan African cities, in particular, a large fraction of landownersin some cases up to 80 percentdo not hold a formal property right on their land. This high level of informality mirrors the deciencies of land registration systems which, in sub-Saharan Africa, remain prohibitively costly and unaordable to most households. It is all the more problematic as informal tenure can have large private and socioeconomic costs. A key reason is the risk of eviction associated with informal land, which reduces investment in land (Besley, 1995) or may reduce labor market participation due to the necessity of spending time guarding one's land plot (Field, 2007). Informal land tenure also hinders the tradability of land, possibly leading to land misallocation. Eventually, households residing on informal plots are exposed to a wide range of social ills, including crime, poor health from low housing quality, and negative human capital externalities (Galiani and Schargrodsky, 2010, Galiani et al., 2017, Nakamura, 2017).

To our knowledge,the theoretical model presented in this paper is the rst to focus on the interaction between social norms and land markets, and how land transactions among trusted parties can address the information asymmetry that has come to characterize today's urban land markets in sub-Saharan Africa (Durand-Lasserve et al., 2015, World Bank, 2019). In our model, purchasing informal land is risky for buyers, as plot ownership might be contested in the future. In an ideal world, competing land ownership claims could be extinguished through adjudication and registration of land ownership in a cadaster leading to the issuance of a property title. In practice, it can be very costly to do so and this solution is only chosen by a fraction of the population. The objective of our paper is thus to investigate how urban households who do not pay for land registration may rely on an alternative way to address tenure insecurity when transacting informal land. More specically, we study a mechanism whereby buyers and sellers match in the informal market according to a trusted ethnic relationship that reduces the information asymmetry and the likelihood of purchasing an insecure plot. We formalize this idea in an urban land use model with tenure insecurity and information asymmetry, where we study equilibrium land market transactions and associated ineciencies. In our framework, plots are of two types: risky plots, which ownership may be contested in the future, and risk-free plots, which ownership cannot be contested. In addition, buyers and sellers of land plots may have reciprocal duties based on trusted ethnic kinship. If a risky plot is exchanged between individuals linked by ethnic kinship without disclosure of the risk to the buyer, the seller will be considered to 2
have violated his duty and a social penalty will be imposed on him. In that context, a buyer expects that a seller he is ethnically related with will be more likely to sell him a secure plot and the buyer will consequently be ready to pay a premium. Knowing this, sellers may decide whether to transact with kin or non-kin members, depending on the intrinsic risk on their plot, the social penalty and the ethnic premium. An important prediction of the model is that although matching along ethnic lines reduces information asymmetry, it also lowers overall market participation. Alternatively, when owners are oered the possibility to make plots secure by paying to register them in a cadaster, both information asymmetry and tenure insecurity are reduced, but the cost of registration limits transactions at the periphery of the city. We compare the overall surplus under these two polar cases and under a hybrid version of the model, where both registration and trusted relationships are available options, as is the case in many sub-Saharan-African cities.

Our approach relates to two existing trends of literature from anthropology and economics. First, a well-established anthropological literature describes links among groups in a wide range of societies. Individuals from such groups are referred to as allies, kins or cousins and exhibit codied reciprocal duties along those links (Mauss,1923). These duties may take various forms, including the requirement to treat one another fairly or to exchange gifts such as food or shelter. Such links are very commonly found in sub-Saharan Africa. We found published analyses in the context of Burkina Faso, Burundi, the Gambia, Guinea, Mali, Rwanda, Senegal, Tanzania and Zambia (see Freedman, 1977, Ndiaye, 1992, Fouéré, 2004, Smith, 2004 and 2006, Diallo, 2006, Dunning and Harrison, 2010). In West Africa and central Africa, the social institution underpinning those links is referred to under the generic French term of cousinage (referring to the social links between groups of so-called cousins), a term that we will use throughout this paper.<sup>2</sup> These cousinage relationships often correspond to alliances between pairs of social groups dened by ethnicity, patronyms and/or the professions traditionally exerted by members of these groups.<sup>3</sup> Although cousinage relationships come from a very old tradition,<sup>4</sup> they are still widely used nowadays.In Senegal, it was found in a survey that 46 percent of Senegalese practice cousinage everyday and an additional 30 percent practice it sometimes (Smith, 2004). Although the

> 2An alternative term for cousinage is joking relationships (in French, cousinage à plaisanterie), which refers to the codied jokes that individuals exchange upon their rst encounter. Joking according to ritualized mocking allows to identify the nature and intensity of bilateral relationships and stress reciprocal duties before engaging in social interactions.

> 3For example, the Sérère and Poular are two allied ethnic groups in Senegal and the Gambia and the Ba and Diallo are two allied family names in Senegal. In Mali, groups that traditionally exerted the profession of blacksmith are linked with all other castes.

> 4In Mali, for instance, cousinage is believed to have been ordained by the 13th century ruler Sundiata Keita, as part of the oral constitution of the Mali Empire.

3
anthropological literature has mostly focused on reciprocal social relationships, several authors mention the role of cousinage in markets, as revealed by price bargaining along ethnic lines (see Hagberg 2006 for Burkina Faso, and Birkeland 2007 and Jones 2007 for Mali). In the economics literature, however, only a small number of papers have focused on kin relationships and land markets, with no specic focus on cousinage. Relevant studies include the work of Marx et al. (2019), who show how ethnicity aects the bargaining power of slum dwellers in Kenya, and Macours et al. (2010) and Macours (2014), who study social and ethnic matching on informal rental markets in the Dominican Republic and Guatemala in response to insecure property rights.

The second trend of literature that we build on involves an emerging urban economics literature on land tenure insecurity in developing countries and its implications for land markets. This literature trend began with Jimenez's (1985) seminal model of squatting in which informal dwellers coordinate land invasions to protect themselves from evictions. Brueckner and Selod (2009) further studied the emergence of a city's squatter settlements in a general equilibrium with inelastic land supply. In their model, squatting squeezes the formal land sector, explaining the high price of formal land in an equilibrium conguration where formal and informal settlements coexist.<sup>5</sup> Our paper, however, does not involve squatting, whereby land is occupied without being purchased or rented out from its rightful owner, but focuses on the larger context of tenure informality and insecurity, whereby the occupant of a land plot may be its legitimate owner but does not have a fully-edged property right, leading to the possibility of competing claims and conicts. A small number of recent models have extended the standard monocentric land use model of urban economics initially developed by Alonso (1964), Mills (1967) and Muth (1969), in order to account for these issues. In Selod and Tobin (2018), urban households compete for land and simultaneously decide the type of property right to purchase from a land administration among a menu of rights that provides various degrees of tenure security. The model leads to an equilibrium with formal and more secure property rights at the proximity of the city center, a prediction that also holds in our model. Cai et al. (2018) extend the Selod and Tobin model to a calibrated dynamic stochastic model with internal migration that allows them study the long term trajectory of formal and informal land uses and the persistence of informal settlements over time. Picard and Selod (2020) extend the canonical model of urban economics to study the conversion of agricultural land into urban residences and the associated changes in land tenure. They introduce

> 5Brueckner (2013) further extended the model with the introduction of a rent-seeking organizer. Shah (2014) modied the model to account for squatting on public land. Turnbull (2008) proposed a non-spatial but dynamic model of the landowner-squatter relationship that focuses on the timing of evictions.

4
information asymmetry between buyers and sellers of risky plotsa feature that is also present in our modeland nd that information asymmetry deters land market participation and hinders the land use conversion process at the periphery of the city. Other spatial papers model specic types of informal housing. This is the case of Brueckner et al. (2019), who develop a theory explaining the emergence of a rental market for backyard structures in South African cities, and Pfeier et al. (2019), who propose a dynamic land-use model with formal and informal housing, including traditional informal settlements as well as backyarding . Other recent studies have focused on the determinants of informal housing and urban slums, stressing the role of migration and the relative elasticities of formal and informal housing supply in determining the amount of informal housing (Alves, 2017, Henderson et al., 2019, Cavalcanti et al., 2019).

Our model builds on the above urban economics literature, especially on Selod and Tobin (2018) and Picard and Selod (2020), as we introduce tenure insecurity and information asymmetry in a monocentric land use model. To our knowledge, however, our model is the rst land use model with interpersonal transactions, an important feature that was largely missing in the theoretical literature on land markets in developing countries, in spite of its likely high prevalence.<sup>6</sup> The introduction of ethnic matching allows us to assess and compare the respective advantages of transactions sanctioned by property rights registration and of transactions under trusted relationships. It also allows us to study the coexistence of the two practices within a single city. Our framework generates novel predictions regarding the prevalence of ethnic matching in land markets in the presence of costly registration, and highlights the substitutability of trusted relationships to costly registration as well, as the gradual evolution of economies towards full cadastral coverage as registration costs are reduced.

The paper is organized as follows. We start by presenting a benchmark urban economics model with tenure insecurity in section 2. In section 3, we sequentially present two polar cases: a rst extension where households have the possibility to establish a registered property right (see sub-section 3.1) and a second extension where land cannot be registered but where buyers and sellers may pair according to a trusted relationship (see sub-section 3.2). In section 4, we then present a hybrid modelclosest to capturing the reality of developing country citieswhere both registration and reliance on trusted relationships for transactions of informal land are available options. The nal section concludes.

> 6For a model of impersonal transactions and a theory of cadasters, see Arruñada, 2012. The idea of interpersonal transactions regarding land sales were only previously explored in a theoretical model by Lanjouw and Levy (2002) who contrasted land transaction conditions between family members and outside parties in a non-spatial setting.

5
# 2 An urban land-use model with tenure insecurity (benchmark

# model)

We rst present a benchmark model based on the standard land use model of urban economics, in which we introduce tenure insecurity. This allows us to derive city structure and surplus in the absence of a property right system that makes land tenure secure.

## 2.1 Main assumptions

The urban space is represented by a line segment at the extremity of which lies a CBD where all jobs are located. Each location on this segment (denoted by its distance _x_ to the city center) has a unit mass of landowners, each endowed with one land plot.<sup>7</sup> Each landowner decides whether or not to sell his land plot to a potential migrant coming to the city, thereby extracting the migrant's willingness to pay for residence in that particular location. Because migrants will be working in the CBD, they value proximity to the city center. As migrant buyers are competing with one another, sellers sell their plots to the highest bidder. We consider an open-city model, in which buyers migrate to the city until the expected utility in the city (given by their expected disposable income) is equalized with the rural utility level _u_ .<sup>8</sup> In our model, land tenure is insecure for some plots in the sense that a buyer can lose his plot in the future with a non-zero probability. This probability may reect the numerous conicts over land ownership that are prevalent in many developing country cities, in particular those that result from unclear initial ownership of the land by the seller, which is ground for competing claims.<sup>9</sup> However, not all plots are insecure. There are two possible levels of tenure security _Q ∈{q,_ 1 _}_ , with _q <_ 1 . Insecure plots may be contested and have a probability _q_ of remaining in the hands of their buyer in the future, whereas secure plots are uncontested and have a probability 1 of remaining in the hands of their buyer. When a buyer is evicted from his plot, the plot is simply grabbed by an absentee landowner and can be resold to another migrant. We denote _π_ the exogenous proportion of secure plots before the city forms and assume it is uniform across all locations.

The key assumption in our model is the existence of an information asymmetry between sellers

> 7Our assumption of a unit of land per location makes our framework akin to Alonso (1964).

> 8We assume a linear utility function and a price of the composite good normalized to 1, so that the utility in the city dened as the consumption of the composite good is exactly equal to the expected disposable income.

> 9Other conicts may oppose heirs, customary owners and investors, private parties and public authorities. See Durand-Lasserve et al. (2015) for a full typology of urban land conicts.

6
(i.e; initial landowners) and buyers. Whereas sellers know the tenure security level of their plots (i.e., they know if there is a competing claim), migrant buyers cannot observe this characteristic before the transaction takes place. In what follows, we will refer to the initial owners of secure plots as 1 -owners and to the initial owners of insecure plots as _q_ -owners. If landowners decide not to sell their plot to a migrant, they simply keep it for agriculture and obtain a xed revenue equal to the agricultural land rent _Ra_ . We assume that _u ≤ Ra_ , reecting the fact that migrants are rural laborers who, by denition, cannot be paid above the agricultural land rent.

We present below the market behavior of buyers and sellers in each location and derive the resulting spatial extent of the urban land market.

# 2.2 Sellers and buyers' behavior

The sellers' decision to sell is modeled with a binary choice variable _P ∈{_ 0 _,_ 1 _},_ with _P_ = 1 if the seller transacts with a migrant, and _P_ = 0 if the seller does not participate in the land market. We denote _π_ ( _x_ ) the share of 1 -sellers among the sellers in location _x_ who decide to sell.<sup>10</sup> We consider that buyers have rational expectations and can fully anticipate the value _π_ ( _x_ ) .

In location _x_ , the buyer of a plot can expect to obtain a secure plot with probability _π_ ( _x_ ) and an insecure plot with probability 1 _− π_ ( _x_ ) . If the plot is insecure, it is lost with probability 1 _− q_ . For the buyer, the expected probability of keeping the plot is thus _π_ ( _x_ ) + (1 _− π_ ( _x_ )) _q_ and that of losing it is (1 _− π_ ( _x_ ))(1 _− q_ ) . Furthermore, the buyer knows that, if he is not evicted and is thus able to remain in the city, he will have utility _y − xt − R_ ( _x_ ) , corresponding to his urban income net of commuting costs and of the price paid for the plot (denoted _R_ ( _x_ ) ). If evicted and having to return to the rural area, the buyer is not reimbursed for the plot purchase and gets utility _u − R_ ( _x_ ) . It is then easy to see that the expected utility of a buyer purchasing a plot in _x_ is: We are now ready to derive the bidding behavior of buyers. Equating (1) with the rural utility level _u_ (given our open city assumption) and inverting the resulting equation in the land price, we obtain the bid-rent of a buyer of a plot located in _x_ :

> 10 _π_ ( _x_ ) may dier from _π_ if the proportions of secure and insecure plots sold in _x_ dier.

7
This bid-rent measures the buyer's willingness to pay for the plot to exactly attain equilibrium utility _u_ . Note that the bid-rent function depends on the buyer's expectation regarding tenure insecurity, so that the buyer's willingness to pay increases with _π_ ( _x_ ) , the fraction of 1 -sellers among sellers and with _q_ , the level of tenure security of risky plots .

As for sellers, their market participation decision will depend on the plot's location _x_ , its intrinsic tenure security level _Q_ and the market price _R_ ( _x_ ) , which we capture with the generic notation _P_ ( _x, Q, R_ ) . In turn, because the benet _B_ of a landowner will depend on his market participation decision and on the market prevailing price _R_ ( _x_ ) in location _x_ , we express it as _B_ ( _P, x, Q, R_ ) _≡ PR_ ( _x_ ) + (1 _− P_ ) _Ra_ . The formula expresses gross prot and simply states that landowners who do not participate in the land market ( _P_ = 0 ) obtain a benet of _B_ = _Ra_ , while landowners who participate in the market ( _P_ = 1 ) obtain a benet of _B_ = _R_ ( _x_ ) .<sup>11</sup>

# 2.3 The Competitive Equilibrium

Having characterized the behaviors of both sellers and buyers, we can now dene the spatial equilibrium. To do this, however, we need to introduce the additional notations _Lq_ ( _x_ ) and _L_ 1( _x_ ) for the respective quantities of insecure land and secure land that are transacted in _x_ . With these notations, the total quantity of land transacted in _x_ , can be decomposed as _L_ ( _x_ ) = _Lq_ ( _x_ ) + _L_ 1( _x_ ) .

The set of equilibrium conditions that dene the equilibrium are as follows: First, the equilibrium quantity of land that is sold in each location _x_ must be smaller than the initial unit land endowment in that location, which yields the following land use constraint: Then, in equilibrium, the market participation decision of the seller for each plot location, tenure security level and price, _P_<sup>_∗_</sup> ( _x, Q, R_ ) , maximizes his gross prot, which leads to the prot maximization condition: Additionally, due to the innite potential pool of migrants to the city, sellers are able to perfectly extract buyers' willingness to pay so that the land price is equal to the bid-rent in each location _x_ ,

> 11Considering that the seller gives up on agricultural production, the net prot from a sale is _PR_ ( _x_ )+(1 _− P_ ) _Ra − Ra_ .

8
taken at the rural utility level _u_ : Finally, we can directly express the city fringe or boundary denoted _xb_ as the location closest to the CBD such that the gross benets of 1 -sellers and _q_ -sellers accounting for their optimal market participation decisions are equal to the agricultural land rent _Ra_ .This can be written as: where _B_<sup>_∗_</sup> ( _x, Q, R_ ) _≡ B_ ( _P_<sup>_∗_</sup> ( _x, Q, R_ ) _, x, Q, R_ ) is the optimal payo (i.e. the gross prot evaluated at the optimal participation decision) of a seller of a plot in location _x_ , endowed with a plot of security level _Q_ , and selling at price _R_ . Since there is only one price for land irrespective of the tenure security level (given that buyer cannot distinguish between risky and secure plots), the condition boils down to _R_ ( _xb_ ) = _ψ_ ( _xb, u_ ) = _Ra_ , which is the standard city fringe condition. We have the following equilibrium denition: Denition 1: A competitive equilibrium is a set of market participation decisions ( _P_<sup>_∗_</sup> ) , prices in each location ( _R_ ( _x_ ) ), and a city fringe ( _xb_ ) that satises the system of equilibrium conditions (3), (4), (5) and (6).

Note that our equilibrium is dened ex-ante in the sense that agents make choices depending on their expectation that a conict may materialize, but before the existence of a conict can be observed. It also relies on rational expectations, in the sense that buyers and sellers know the model of the economy and are able to correctly assess the equilibrium proportion of secure plots among transacted plots. The main dierence with the standard adverse selection model (see Akerlof 1978) is that, in our model, uncertainty only directly aects buyers who may lose their plot if facing a risk of eviction.

To solve the system, we identify all possible combinations of sellers' _P_ -decisions that are Nash equilibria (i.e., the combinations in which no landowner would gain from revising his _P_ -decision given the _P_ -decisions of all other landowners). We show in Appendix A that a continuum of equilibria are possible. In other words, there is a continuum of _P_ -decisions that are compatible with our equilibrium 9
conditions (3)-(6). In these equilibria, all owners in the segment [0 _, xa_ ] participate in the market. These multiple equilibria only dier with respect to market participation decisions over the segment ] _xa, x_<sup>_∗_</sup> _a_<sup>[with</sup><sup>_x∗_</sup> _a_<sup>_≡_</sup><sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −Ra−u_)(onthissegmentanytransactedlandplotissoldatexactlyprice</sup> _Ra_ ). More specically, we show in Appendix A that each equilibrium in this continuum of equilibria is characterized by the number of 1 - and _q_ -sellers in each location _x ∈_ ] _xa, x_<sup>_∗_</sup> _a_<sup>[,whichmustverify</sup> ( _Lq_ ( _x_ ) _, L_ 1( _x_ )) _∈_ [0 _,_ 1 _− π_ [ _×_ [0 _, π_ [ and _B_ ( _P, x, Q, R_ ) = _Ra_ . Let us denote by _ELq,L_ 1 the equilibrium characterized by functions _Lq_ and _L_ 1 over the interval ] _xa, x_<sup>_∗_</sup> _a_<sup>[.WeshowinAppendixAthat</sup><sup>_EL_</sup> _q_<sup>_,L_</sup> 1<sup>is</sup> unstable if there exists _x ∈_ ] _xa, x_<sup>_∗_</sup> _a_<sup>[such that either</sup><sup>_Lq_(</sup><sup>_x_)</sup><sup>_>_0or</sup><sup>_L_1(</sup><sup>_x_)</sup><sup>_>_0and that any small deviation</sup> in _P_ -decisions from that equilibrium will always trigger a transition towards _E_ 0 _,_ 0 . This equilibrium is therefore the unique stable equilibrium. In that equilibrium, no landowner participates in the land market beyond _xa_ so that the city fringe is in _xb_ = _xa_ . Our results can be summarized in the following proposition: Proposition 1: There is a single stable equilibrium, in which all landowners between the city center and the city fringe _xa_ =<sup><u>1</u></sup> _t_ � _y − π_ + _qR_ (1 _<u>a−π</u>_ )<sup>_−u_</sup> � participate in the land market. The equilibrium price curve is _R_ ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − tx − u_ ) .<sup>12</sup>

# Proof: See Appendix A.

Observe that the land price curve is linear with slope _−_ [ _π_ + _q_ (1 _− π_ )] _t_ , where _π_ + _q_ (1 _− π_ ) is the probability for a buyer to keep a purchased plot. In equilibrium, when marginally moving outwards from the city center, the reduction in land price exactly compensates the increase in _expected_ transport costs (given the probability of keeping the plot and commuting to the city center rather than losing the plot and moving back to the rural area without the need to commute anymore). The equilibrium price slope becomes atter if plots become more insecure (lower _q_ ) or if sellers with secure plots are scarcer (lower _π_ ). As expected, when there is no tenure insecurity in the model (either because _q_ = 1 or _π_ = 1 ), the slope is equal to the certain marginal transport cost as in the standard version of the urban economics model with unit land consumption.

Inspection of the equilibrium city fringe formula for _xa_ shows that the city size decreases with the proportion of insecure plots ( 1 _− π_ ) and the level of tenure insecurity ( 1 _− q_ ). It is easy to understand

> 12Note that to avoid the degenerate case of an empty city, the model requires the exogenous parameters to verify _y − π_ + _qR_ (1 _<u>a−π</u>_ )<sup>_−u >_0.Fromnowon,weassumethatthisconditionissatised.</sup>

10
<!-- Start of picture text --> Land prices<br>R(x)<br>=<br>0La xr<br>All plots transacted<br><!-- End of picture text -->
# 2.4 Suboptimality of the equilibrium

The market equilibrium involves an externality, insofar as agents do not internalize the eect of their market participation decision on the composition of transacted plots, which in turn aects other agents' decisions. Following Fujita (1989), we dene the surplus as the city production (sum of wages) minus the costs to organize the city (transport costs, composite good consumption, and foregone agricultural production).<sup>15</sup> Although our framework to solve the equilibrium does not require to specify what happens to the plot after the buyer is evicted (since choices are made ex-ante, i.e. before the realization of a potential conict), we need to introduce assumptions regarding the use of the land after an eviction in order to be able to write the surplus formula. When the buyer of a land plot is evicted, we assume that the plot is grabbed by an absentee agent and resold at price _R_ ( _x_ ) to a new migrant worker. The process, however, may not be instantaneous. If there is some friction after the eviction, the plot will remain unoccupied for some time. We denote _θ ∈_ [0 _,_ 1] , the fraction of time during which an evicted plot is occupied. If _θ_ = 0 , the plot is immediately lled with a new occupant. If _θ_ = 1 , the plot occupant is never replaced. _θ_ is assumed to be the same for all land plots. In the competitive equilibrium, recognizing that the composite good consumption is _u_ , the surplus can be written as: When _θ_ = 0 , a plot which occupant was evicted is lost to city production and the surplus is: When _θ_ = 1 , evictions do not disrupt city production and the surplus is: As regards the optimal city conguration, it is easy to see that it corresponds to a situation where plots are allocated to a migrant until an optimal city boundary denoted _x_<sup>_∗_</sup> _a_<sup>(</sup><sup>_q, π, u_).Indeed,if a plot is</sup> not allocated, its contribution to the surplus is zero. If a secure plot is allocated, it contributes to the city surplus by an amount _y − xt − u − Ra_ . If an insecure plot is allocated, it contributes to the city

> 15This is mathematically equivalent to another denition of surplus that would consider the utility increment from migration to the city net of the opportunity cost of land use.

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surplus by an amount ( _q_ + _θ_ (1 _− q_ ))( _y − xt − u − Ra_ ) . _x_<sup>_∗_</sup> _a_<sup>isdenedastheboundarybeyondwhichno</sup> plots are allocated in the optimal city conguration. It is easy to see that _x_<sup>_∗_</sup> _a_<sup>=</sup><sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −Ra −u_)</sup><sup>_>xa_,</sup> and no informal plot will be sold beyond. The optimal surplus associated with this conguration is thus: The gap between the market equilibrium surplus Γ _θ_ ( _q, π, u_ ) and the optimal surplus Γ<sup>_∗_</sup> _θ_<sup>(</sup><sup>_q, π, u_)is</sup> caused by tenure insecurity ( _q_ and _π_ ), which decreases the size of the city to _xa < x_<sup>_∗_</sup> _a_<sup>.Whentenure</sup> insecurity is removed from the model, the surplus rises and the city size _xa_ grows to become optimal. 16 In contrast, suppressing production disruption does not make the city size optimal, as land owners do not take this parameter into account in their decisions.

Focusing on the optimal surplus, note that it is a function of the model's parameters _q, π_ and _u_ , as well as , _θ_ . Intuitively, a greater _u_ reduces both the optimal city size and the incremental gain from city production (as migration to the city is reduced), while a greater _π_ or _q_ increases both city size and city production. When _θ_ increases, however, only city production increases. The optimal surplus is maximum when tenure insecurity or production disruption are removed (i.e. _π_ = 1 , _q_ = 1 or _θ_ = 1 ). Indeed: We denote this maximum optimal surplus Γ<sup>_∗∗_</sup> and will use it as an optimality benchmark in the rest of the paper. It is reached by the market equilibrium surplus when tenure insecurity is suppressed for any level of production disruption:

# 3 Trust and registration as tenure security devices

The above section showed that land tenure insecurity and information asymmetry cause a market failure, whereby some landowners at the periphery of the city opt out of the market, leading to a sub-optimal urban equilibrium. Tenure insecurity and information asymmetry can be reduced through dierent institutional settings. In this section, we separately study the introduction in the benchmark

> 16It can easily be checked that Γ _∗θ_<sup>(1</sup><sup>_, π, u_) = Γ</sup><sup>_θ_(1</sup><sup>_, π, u_)</sup><sup>_≥_Γ</sup><sup>_θ_(</sup><sup>_q, π, u_)andΓ</sup><sup>_∗_</sup> _θ_<sup>(</sup><sup>_q,_1</sup><sup>_, u_) = Γ</sup><sup>_θ_(</sup><sup>_q,_1</sup><sup>_, u_)</sup><sup>_≥_Γ</sup><sup>_θ_(</sup><sup>_q, π, u_)forany</sup> _q ∈_ [0 _,_ 1] and _π ∈_ [0 _,_ 1] . When either _π_ = 1 or _q_ = 1 , we see that _xa_ = _x_<sup>_∗_</sup> _a_<sup>.</sup>

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model of two such institutional settings . The rst is a cadaster/registry of property rights, thanks to which landowners have their registered rights recognized and enforced. The second setting relies on bilateral trusted relationships between buyers and sellers, based on group kinship. We present these institutional settings sequentially before comparing them.

# 3.1 A model of registration

In the rst extension, we add a property rights registration system to the benchmark model. Registration extinguishes competing claims on a land plot and makes it fully secure. Because it is observable by all, it allows buyers to identify a fraction of secure plots (i.e., those which are registered) with certainty. In this extension, sellers have the possibility to register their land before entering a transaction, anticipating that a registered plot will sell at a premium (that capitalizes both the increase in tenure security and the removal of information asymmetry). We assume that there is a registration cost, _k_ , which is the same for all plots, irrespective of the initial tenure security level.

## 3.1.1 Sellers and buyers' behavior

An owner now has two choices. He chooses whether to participate in the market (decision _P ∈{_ 0 _,_ 1 _}_ ) and, conditional on participating in the market, whether to register his plot (decision _F ∈{_ 0 _,_ 1 _}_ ). If choosing _F_ = 1 , the tenure security level of the plot is reset at the value 1 . There are thus two sale situations depending on sellers' decisions, each commanding a dierent price. Applying the same bid-rent approach as in the previous section, we see that the buyer of a registered plot will be willing to pay _ψ_ ( _x, u|F_ = 1) = _y − tx − u_ to reach utility level _u_ . Similarly, the buyer of a non-registered plot will be willing to pay _ψ_ ( _x, u|F_ = 0) = _{π_ ( _x_ ) + _q_ (1 _− π_ ( _x_ )) _}_ ( _y − tx − u_ ) .

## 3.1.2 Competitive equilibrium

Given the additional features, of the model, the notations are slightly modied in comparison with the benchmark case. As before, we continue to use _L_ 1( _x_ ) and _Lq_ ( _x_ ) to denote the quantities of secure and risky land that are transacted, while _R_ ( _x_ ) continues to denote the price of informally transacted land. However, we now introduce the notation _Lf_ ( _x_ ) to denote the quantity of registered (and transacted) land in each location _x_ , and _Rf_ ( _x_ ) to denote the price of formally registered land. We now allow the market participation and the registration decisions to depend on location, the intrinsic tenure security level, and both formal and informal market prices, with the notations _P_ ( _x, Q, Rf , R_ ) and 14
_F_ ( _x, Q, Rf , R_ ) . As for the seller's gross prot, it is now also generically a function of the registration status of the land and of both formal and informal land prices, hence the notation _B_ ( _P, F, x, Q, Rf , R_ ) . We can now adapt our denition of the competitive equilibrium as follows: Denition 2: A competitive equilibrium is a set of market participation and registration decisions, prices in each location _x_ and a city fringe that satises the following equilibrium conditions: where _B_<sup>_∗_</sup> ( _x, Q, Rf , R_ ) _≡ B_ ( _P_<sup>_∗_</sup> ( _x, Q, Rf , R_ ) _, F_<sup>_∗_</sup> ( _x, Q, Rf , R_ ) _, x, Q, Rf , R_ ) is the optimized payo (i.e., taken at the optimal participation and formalization decisions) of a seller of a plot of security level _Q_ in location _x_ selling at price _R_ if the plot is informal and _Rf_ if it is formal.

The equilibrium conditions are very similar to (3)-(6), but they also account for the registration decision, registered land, and registered land prices. As in the benchmark version of the model, the city fringe condition allows for one of the two owner types ( 1 - or _q_ -owners) to not participate in the market and comes down to equating the bid rent of the last seller to the agricultural land rent. There is also an additional equation that determines the price of formal land as the upper envelope of formal bid-rents in places where land is registered by sellers. As previously, we solve for the competitive equilibrium by identifying stable Nash equilibria (see Appendix B).<sup>17</sup> The equilibrium structure of the city depends on the level of the registration cost as summarized in the following proposition:

> 17In what follows, we only mention stable equilibria. All the stable and unstable equilibria are presented and discussed in the Appendix.

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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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<!-- Start of picture text --> Case 1 (k < k) Case 2 (k << k <k)<br>Land prices Land prices<br>Ry(2)<br>Ry(2)<br>:<br>pee<br>0 i(k) 2 0 Jone one | (k) Zoneone x<br>[ wm | All plots transacted<br><!-- End of picture text -->
# 3.1.3 Surplus gain from property rights

We can derive the surplus in the city under the registration system. Taking into account, as before, the possibility of productive disruption following an eviction, the surplus is given by: Comparing the competitive surpluses with and without the land registration system ( Λ and Γ ) yields the following property: Proposition 3: The equilibrium surplus under the registration system is always strictly greater than in the benchmark model for _k < k_<sup> ̄</sup> and it is equal to the surplus in the benchmark model for _k ≥ k._<sup> ̄</sup> ˆ Proof: When _k_ = 0 , Λ _−_ Γ is strictly positive since 1 _> π_ +( _q_ + _θ_ (1 _− q_ ))(1 _− π_ ) and _x_ ( _k_ ) _> xa_ . On the interval [0 _, k_ <u>]</u> , we have Λ _−_ Γ = �0 _x_ ˆ( _k_ )( _y−xt−k−u−Ra_ ) _dx−_ �0 _xa_<sup>(</sup><sup>_π_+(</sup><sup>_q_+</sup><sup>_θ_(1</sup><sup>_−q_))(1</sup><sup>_−π_))(</sup><sup>_y−xt−u−Ra_)</sup><sup>_dx_.</sup> The gap decreases with _k_ , as<sup>_∂_</sup><sup><u>(Λ</u></sup> _∂k_<sup>_−_Γ)</sup> = _−_<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y−k−u −Ra_)</sup><sup>_<_0.Next,letusconsiderΛ</sup><sup>_−_Γfor</sup> _k ∈_ <u>[</u> _<u>k,</u> k_<sup> ̄</sup> ] . We have Λ _−_ Γ = �0 _x_ ˇ( _k_ )[ _y − xt − k − u − Ra −_ ( _π_ + ( _q_ + _θ_ (1 _− q_ ))(1 _− π_ ))( _y − xt − u − Ra_ )] _dx_ . Its derivative is _<u>∂</u>_ <u>Λ</u> _∂k−_ <u>Γ</u> = _−_<sup><u>1</u></sup> _t_ � _y − u −_ (1 _− θ_ ) � (1 _−q_ )(1 _<u>k</u> −π_ )<sup>_−Ra_</sup> �� , which is negative until _k_ reaches _k_ 0 _≡_ (1 _− π_ )(1 _− q_ ) � _<u>y</u>_ 1 _−−uθ_<sup>+</sup><sup>_Ra_</sup> � and positive after. As _k_ 0 _> k_<sup> ̄</sup> , Λ _−_ Γ decreases with _k_ on the interval <u>[</u> _<u>k,</u> k_<sup> ̄</sup> ] and reaches 0 in _k_ = _k_<sup> ̄</sup> . Thus, Λ _−_ Γ _>_ 0 for _k < k_<sup> ̄</sup> and Λ _−_ Γ = 0 for _k ≥ k_<sup> ̄</sup> . This completes the proof of Proposition 3.

To understand Proposition 3, observe that _k_ can be viewed as the extra cost to avoid eviction and the associated production loss. When _k_ is suciently small, some degree of formalization occurs and increases the surplus. Interestingly, the dominance of the registration system does not necessarily require the city to be fully formal (as in Case 2, which is only partially formal). Note, however, that although the introduction of property rights is an improvement over the benchmark model if the registration cost is low enough, it does not completely remove the market failure. In fact, we show in Appendix B that the market equilibrium city structure and the optimal city structure only coincide 18
when _k_ = 0 .<sup>21</sup> When the registration cost is low enough ( 0 _< k ≤_ _<u>k</u>_ <u>),</u> the equilibrium conguration is completely formal (which solves both the tenure insecurity and information asymmetry problems), and yet, market participation is still reduced at the periphery of the city because of the strictly positive registration cost.

It is also remarkable that the optimum city conguration for the registration model (see Appendix B) actually requires the presence of an informal zone, as long as _k >_ 0 . This is because registration of peripheral plots involves a registration cost that outweighs the incremental production gain net of commuting costs. In the competitive equilibrium, there is an informal zone (between _x_ ˇ( _k_ ) and _xa_ ), but its size is sub-optimal. This is due to an externality associated with registration, as landowners choose to register based on the anticipation of buyers' willingness to pay for a secure plot rather than on buyers' contribution to production in the city.<sup>22</sup>

# 3.2 A model of trust

We now present a second extension of the benchmark model where landowners do not have the possibility to register their plot (i.e., they cannot remove the risk on their plot) but may instead choose whom they transact with in response to the information asymmetry. More precisely, each individual owner may decide whether to transact with a potential migrant with whom he has an ethnic relationship that involves some amount of trust. Borrowing the language of the anthropological literature on joking relationships, we refer to this behavior as transacting with a cousin (versus transacting with a non-cousin).<sup>23</sup> In our setting, there is no need to dene groups and specify their numbers, as we just focus on whether landowners transact with a cousin or not (with an innite pool of cousins potentially supplied by migration to the city). Cousinage relationships (the existence of a trusted relationship) are known and observable by all agents. Our only assumption is that tenure insecurity is location-specic but not group-specic, implying that, in each location, the proportions of risky and secure plots are the same among all groups.

> We denote by _C ∈{c, nc}_ the landowner's decision to sell to a cousin ( _C_ = _c_ ) or to a non-cousin ( _C_ = _nc_ ). As in the benchmark model, buyers do not know the risk associated with the plot they are

> 21In that case, not only is the market equilibrium optimal, but it reaches the maximum optimal surplus Γ _∗∗_ of the benchmark model with no tenure insecurity (as property rights can be obtained for free).

> 22The result that the optimal city structure may require some level of informality when formal property rights are costly to establish was rst derived in Cai et al. (2018). Also note that in the presence of agglomeration eects, the externality might be attenuated if sellers anticipate an increase in their own labor income from migration to the city.

> 23Although we refer to cousinage relationships, this version of our model encompasses the case of any society where trusted relationships exist within and across groups of individuals.

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purchasing ( _Q ∈{q,_ 1 _}_ ). Mirroring the literature on ethnic groups and social sanctions (see Fearon and Laitin 1996, La Ferrara 2003, Habyarimana, et al. 2007), the key assumption in this setting is that selling a low-security plot to a cousin will always be punished with penalty _J >_ 0 . Therefore, sellers have an incentive to be more honest in their transactions with cousins than in their transactions with non-cousins.<sup>24</sup> The social penalty reects the ostracism imposed on individuals who betray trust among cousins as codied in the cousinage institution. Given our assumption that tenure security is only location-specic, we denote _π_<sup>_c_</sup> ( _x_ ) the proportion of 1 -sellers in _x_ , as a fraction of all sellers who transact with a cousin. Similarly, _π_<sup>_nc_</sup> ( _x_ ) is the proportion of 1 -sellers in _x_ , as a fraction of all sellers who transact with a non-cousin.

Observe that land markets are now inter-personal (as opposed to the impersonal land markets presented in the benchmark and in the registration versions of the model). Also note that, because dierent levels of trust exist between cousins and between non-cousins, there are now two prices for plots, depending on whether the transaction involves cousins or non-cousins. We denote these interpersonal prices _R_<sup>_c_</sup> ( _x_ ) and _R_<sup>_nc_</sup> ( _x_ ) when the transaction involves cousins and non-cousins respectively.

# 3.2.1 Sellers and buyers' behavior

The expected utilities of a buyer buying from a cousin seller or from a non-cousin seller are respectively: Because we have an open city, migration will occur until buyers obtain the same utility level as in the rural area (which we still denote _u_ ). Inverting the above utility functions in the land price gives us the bid-rent functions in each location of the city, both for transactions among cousins and among non-cousins: Anticipating buyers' willingness to pay (given by the above bid-rent functions), landowners choose whether to participate in the market (decision _P ∈{_ 0 _,_ 1 _}_ ) and whom to sell to (decision _C ∈{c, nc}_ ) . Let us now detail the sellers' prot associated with each decision. If a seller decides not to participate

> 24For simplicity, _J_ is the same among all cousin groups.

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in the land market, he receives the agricultural rent _Ra_ . An owner of a secure plot selling to a cousin buyer ( _C_ = _c, Q_ = 1 ) receives a payment _R_<sup>_c_</sup> ( _x_ ) . An owner of an insecure plot selling to a cousin buyer ( _C_ = _c, Q_ = _q_ ) receives a payment _R_<sup>_c_</sup> ( _x_ ) , but faces the social penalty _J_ that reduces his/her benet to _R_<sup>_c_</sup> ( _x_ ) _− J_ .<sup>25</sup> Finally, an owner selling to a non-cousin buyer receives a payment _R_<sup>_nc_</sup> ( _x_ ) and there is no social penalty if the transacted plot is insecure.

# 3.2.2 Competitive equilibrium

We can now adapt the equilibrium denition to the inter-personal market setting. The decision to participate in the market is now a function of the interpersonal prices. It can be denoted _P_ ( _x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) . The decision to sell to a cousin or a non-cousin is also a function of the same arguments and is denoted _C_ ( _x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) . The prot of a seller can now be generically expressed as _B_ ( _P, C, x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) . In this version of the model, we decompose transacted land, not only according to its intrinsic tenure insecurity, but also according to the potential cousinage relationship between buyers and sellers. This requires the introduction of the notations _L_<sup>_c_</sup> ( _x_ ) and _L_<sup>_nc_</sup> ( _x_ ) for land transacted between cousins and non-cousins respectively. With these additional notations, we have the following equilibrium denition: Denition 3: An equilibrium is a set of market participation decisions, cousinage decisions, prices in each location _x_ and a city fringe that satises the following equilibrium conditions: where _B_<sup>_∗_</sup> ( _x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) _≡ B_ ( _P_<sup>_∗_</sup> ( _x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) _, C_<sup>_∗_</sup> ( _x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) _, x, Q, R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) is the optimized payo (i.e., for the optimal participation and cousinage decisions) of a seller of a plot of security level _Q_ in location _x_ selling at price _R_<sup>_c_</sup> if the buyer is a cousin and at price _R_<sup>_nc_</sup> if he is not.

> 25 _J_ captures the disutility experienced by landowners when they are punished. Although it can be non-monetary in nature, landowners behave as if their monetary benet were reduced by _J_ .

21
Condition (14) states that the quantity of transacted land must be smaller than the initial endowment in location _x_ . It also provides the decomposition of the total amount of transacted land by tenure security, as well as by cousinage. Condition (15) characterizes the optimal market participation and cousinage decisions of sellers. Conditions (16) and (17) reect sellers' extraction of buyers' willingness to pay under dierent cousinage situations. Condition (18) characterizes the city fringe.

Before solving the equilibrium, we derive the following two lemmas regarding cousinage decisions.

Lemma 1: In a stable equilibrium, 1 -owners only sell to cousin buyers.

Proof : See in Appendix C.

Lemma 1 implies that it is only possible to acquire a secure plot if transacting with a cousin. Yet, transactions between cousins involve information asymmetry, as insecure plots may also be sold to cousin buyers.

Lemma 2: In a stable equilibrium, if there are transactions between non-cousins, they must always involve insecure plots ( _Q_ = _q_ ).

Proof : This is a direct consequence of Lemma 1. Because 1 -owners never sell to non-cousins, any transaction between non-cousins must therefore involve insecure plots ( _Q_ = _q_ ).

Lemma 2 implies that there is no information asymmetry in transactions between non-cousins. With Lemmas 1 and 2, we see that transactions between cousins pool risky and non-risky plots, whereas transactions between non-cousins clearly separate a subset of risky plots. With these lemmas in mind, let us now solve for the equilibrium. We show that the city structure depends on the value of the social penalty _J_ (see full details in Appendix C). Intuitively, the main qualitative dierence when increasing the social penalty is that _q_ -owners tend to sell more to non-cousins (in order to avoid the social penalty) and are more likely to drop out of the market. For ease of presentation and without major loss in generality, we will focus, in the rest of this subsection, on the case where _J ≤_ _<u>J</u>_ = _πRa_ 1 _−q_ _<u>q</u>_<sup>,which</sup> encompasses situations in which cousinage is relatively mild, as practiced in urban areas in Senegal for instance (as observed by Smith, 2004).<sup>26</sup> In that case, however, transactions only occur among

> 26As we will see later in the subsection, the institution of cousinage is socially justied when _J_ is suciently small.

22
cousins. We present the other cases where _J > J_ and where non-cousins may transact with each other in Appendix C.

We have the following proposition: Proposition 4: Let us denote the social penalty threshold _<u>J</u>_ = _πRa_ 1 _−q_ _<u>q</u>_<sup>,andthezoneboundary</sup> thresholds _<u>x</u>_ <u>(</u> _J_ ) =<sup><u>1</u></sup> _t_ � _y − π_ (1 _<u>R−a</u>_ <u>+</u> _q_ )+ _<u>J</u> q_<sup>_−u_</sup> � and _x_ ̄( _J_ ) =<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −_(</sup><sup>_Ra_+</sup><sup>_J_)</sup><sup>_−u_).Thecityextendsuntilthe</sup> city boundary _x_<sup>_∗_</sup> _a_<sup>andisorganizedinthethreefollowingzones,movingoutwardfromthecitycenter:</sup>

- Zone 1 (fully residential): On [0 _, x_ <u>(</u> _J_ )[ , all landowners (irrespective of the tenure security level of their plot) participate in the land market and exclusively sell to cousins. The unique price in each location _x_ is _R_<sup>_c_</sup> ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − xt − u_ ) .

- Zone 2 (mixed agricultural and residential, partial market participation of _q_ -sellers): On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_  ̄ _x_ ( _J_ )[ , all 1 -sellers and a share of _q_ -sellers participate in the land market and exclusively sell to their respective cousins. The mass of _q_ -sellers selling to cousins in _x_ is _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x, J_)=</sup> _π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u>

- _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_.Theuniquepriceineachlocation</sup><sup>_x_is</sup><sup>_Rc_(</sup><sup>_x_) =</sup><sup>_Ra_+</sup><sup>_J_.</sup>

- Zone 3 (mixed agricultural/residential, _q_ -sellers dropping out of the market): On ] ̄ _x_ ( _J_ ) _, x_<sup>_∗_</sup> _a_<sup>[,all1-sellersparticipateinthelandmarketandselltotheircousins,whereasall</sup><sup>_q_-</sup> sellers keep their plots under agricultural use. The unique price in location _x_ is _R_<sup>_c_</sup> ( _x_ ) = _y−xt−u_ .

Proof: See the proof of the Extended Proposition 4 in Appendix C.

Given Lemmas 1 and 2, _q_ -owners participating in the market face a trade-o between selling to a non-cousin, and selling to a cousin at a higher price while facing the social penalty _J_ . Since the penalty is suciently low, _q_ -owners always prefer to sell to their cousins.<sup>27</sup> In terms of city size, we see that the city extends all the way to _x_<sup>_∗_</sup> _a_<sup>_> xa_.Indeed,thecousinageinstitutionreducestheasymmetry</sup> of information as compared to the benchmark model, so that some 1 -owners now have an incentive to sell further away from the city center (to their cousins). This leads to a larger city, with plots under urban and agricultural use coexisting at the periphery (i.e., not all landowners decide to participate in the market).

> 27We derive in Appendix C all the other city congurations for _J >_ _<u>J</u>_ . In four of these cases, there are zones where some or all _q_ -owners sell to both cousins and non-cousins. We also show the intuitive result that the number of _q_ -owners selling to cousins decreases when the penalty increases.

23
<!-- Start of picture text --> Land prices<br>R°(x)<br>R,+J R(x)<br>pS ;<br>0 ax(J) z(J) x xr<br>; Zone 1 Zone 2 , Zones,<br>All plots transacted<br>Pipe<br>q-transactions - aNd otsSOME! Only 1-plots<br><!-- End of picture text -->
# 3.2.3 Surplus gains from cousinage

We now derive the competitive surplus of the city, which can be expressed as a function of _J_ . For instance, in the case _J < J,_ it can be written as:<sup>30</sup> For _J > J,_ the formula for Φ changes with the city congurations presented in Appendix Figure 9. Φ is a continuous and decreasing function of _J_ . The surplus dierence between the cousinage and benchmark competitive equilibria, Φ _−_ Γ , is thus also continuous and a decreasing function of _J_ . We show in Appendix C that Φ _−_ Γ _>_ 0 when _J_ is close to 0 and Φ _−_ Γ _<_ 0 in _J_ = _J_<sup> ̄</sup> . We therefore have the following proposition.

Proposition 5: There exists a penalty threshold _J_ 0( _θ_ ) _∈_ ]0 _, J_<sup> ̄</sup> [ such that, for 0 _< J < J_ 0( _θ_ ) (respectively _J > J_ 0( _θ_ ) ), cousinage increases (respectively decreases) the city surplus in comparison with the benchmark model.

# Proof: See Appendix C.

The intuition underpinning the proposition is as follows: Low values of the social penalty increase the surplus (in comparison with the benchmark model), as they incentivize participation to the market and cousinage relationships that address information asymmetry through trusted transactions among cousins. When the penalty increases, however, the market participation of owners of insecure plots is reduced (since _<u>x</u>_ <u>(</u> _J_ ) , _x_ ̄( _J_ ) and _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x, J_)arealldecreasingfunctionof</sup><sup>_J_),whichtendstomitigatethe</sup> benets from cousinage, as too few plots are sold on the market. For larger values of the penalty, the latter eect dominates the former, and the surplus might be lower under the trust model than under the benchmark model. Another way to look at this result is to note that the merit of cousinage is to introduce trust among cousins (through the threat of punishing deception), but that the threat itself needs to be small, as it tends to deter market participation.

> 30Note that, because _J_ is non-monetary, it is not directly accounted for in the city surplus denition and only aects this surplus indirectly, through the modication of landowners' decisions.

25
# 3.3 Theoretical dominance: trust or registration?

We can now compare the surplus under the registration and the trust models. It is intuitive to see that the registration system has a greater potential to increase the surplus because it removes both tenure insecurity and asymmetry of information. On the contrary, reliance on cousinage can only, at best, remove information asymmetry by allowing buyers to better assess the quality of the plot they purchase, so that sellers of secure plots can get a higher benet and also sell at the periphery of the city. When the formalization cost _k_ tends towards zero, the competitive equilibrium under the registration model yields an optimal city size _x_<sup>_∗_</sup> _a_<sup>(i.e.,thecitysizeunderthebenchmarkmodelwith</sup> no tenure insecurity) with full market participation of all landowners throughout the city. The trust model also yields a city of similar size, but some landowners nd it more protable to keep their land under agricultural use and do not participate in the market. When the formalization cost _k_ tends towards zero, tenure insecurity is completely suppressed through registration, whereas this is never the case in the trust model for any intensity of the cousinage penalty. Therefore, when registration costs tend towards zero, the registration system dominates the trust system. However, considering more realistic situations where the registration cost is non zero, the comparison of the two property right systems becomes more complex. In fact, their ranking depends on the relative size of the registration cost _k_ and the social penalty _J_ . This result is detailed in the following proposition: Proposition 6: The ranking of the registration and trust models (in terms of competitive surplus attained) depends on the relative intensities of the registration cost _k_ and the cousinage penalty _J_ . For low values of _k_ , the registration system is always preferable (irrespective of the value of the social penalty _J_ ). For high values of _k_ , there exists a threshold for the cousinage penalty below which (respectively above which) the trust model performs better (respectively worse) than the registration model. Mathematically, it means that there exists a positive threshold _k_<sup>ˆ</sup> ( _θ_ ) _∈_ ]0 _, k_<sup> ̄</sup> ] such that:

- If _k < k_<sup>ˆ</sup> ( _θ_ ) , the competitive surplus is always greater under the registration model, irrespective of the value of the cousinage penalty _J_ ;

-  ̃

- _•_ If _k_<sup>ˆ</sup> ( _θ_ ) _< k <_ + _∞_ , there exists a threshold _J_<sup> ̃</sup> ( _k, θ_ ) such that, if _J < J_ ( _k, θ_ ) (respectively _J > J_<sup> ̃</sup> ( _k, θ_ ) ),<sup>31</sup> the competitive surplus under the trust model is greater (respectively smaller) than the competitive surplus under the registration model.

> 31In Appendix C, we show that _J_  ̃( _k, θ_ ) is increasing in _k_ . This means that the larger the registration cost, the larger the value of the social penalty threshold below which the trust model is preferable to the registration model.

26
<!-- Start of picture text --> J<br>J CI Registration<br>Jo(9) dominates<br>| Trust<br>|| S dominates<br>I<br>IK<br>FkyA<br>|<br>k<br>0 k(0)A k<br><!-- End of picture text -->
# 4 A hybrid model with trust and registration

In the preceding section, we sequentially presented two types of property rights regimes: an impersonal system based on the registration of land property rights, and an interpersonal system based on trusted relationships between land buyers and sellers. In fact, both regimes usually co-exist in today's subSaharan African cities, due to the fact that the introduction of formal/modern property rights that allow for impersonal exchange has not replaced the traditional institutions that make interpersonal exchange possible.

In this section, we investigate what this coexistence of systems implies in terms of city structure and surplus. To do this, we combine the features of both our registration and trust models in a single urban land use model. As shown in Appendix D, we nd that the overall structure of the city depends on both the value of the registration cost _k_ and the cousinage penalty _J_ . For the sake of brevity, we only present results for the most likely but most complex case where the formalization cost is high ( _<u>k</u> < k < k_<sup> ̄</sup> ) and where _J_ is low ( _J < J_ <u>,</u> as in Section 3).<sup>32</sup>

## 4.1 Competitive equilibrium

In the hybrid version of the model, landowners may decide to participate in the land market or not. If participating, they may decide to register their land or to sell it informally. If selling it informally, they may choose to sell it to a cousin or to a non-cousin.<sup>33</sup> The three decisions of landowners can be written as functions of location, the plot's intrinsic level of tenure security, the formal price of land, and the informal prices of land sold to cousins and to non-cousins. The benet of a seller is a function of the same variables and of his market participation, registration and cousinage decisions. This leads to the following updated version of the equilibrium denition:

> 32Throughout this section, we assume, in addition to the previous sections' assumptions, that _y − u > Ra πq_ (11 _−π_ )<sup>.</sup> This somewhat stronger hypothesis reduces the number of cases to be studied without changing the intuition of the model (See Appendix D for details).

> 33Observe that cousinage and registration decisions are exclusive of one another. Once a landowner registers his plot, transactions occur at arm's length, so that ethnic bilateral relations do not play any role.

28
Denition 4: An equilibrium is a set of market participation, registration, and cousinage decisions, prices in each location _x_ and a city fringe that satises the following equilibrium conditions: where _B_<sup>_∗_</sup> ( _x, Q, Rf , R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) _≡ B_ ( _P_<sup>_∗_</sup> _, F_<sup>_∗_</sup> _, C_<sup>_∗_</sup> _, x, Q, Rf , R_<sup>_c_</sup> _, R_<sup>_nc_</sup> ) is the optimized payo (i.e. after taking the optimal participation, formalization and cousinage decisions) of a seller of a plot of security level _Q_ in location _x_ selling at price _Rf_ if the plot is formal, _R_<sup>_c_</sup> if the plot is informal and the buyer is a cousin, and at price _R_<sup>_nc_</sup> if the plot is informal and the buyer is not a cousin.

We now have 6 equilibrium conditions. As before, condition (19) says that the quantity of land sold must be smaller than the initial endowment in location _x_ . Condition (20) characterizes the optimal market participation, registration and cousinage decisions of sellers. Conditions (21)-(23) reect sellers' extraction of buyers' willingness to pay under the dierent land tenure situations (as regards security and registration) and social link congurations, where _L_<sup>_C_</sup> ( _x_ ) is the quantity of land transacted informally with a cousin ( _C_ = _c_ ) or a non-cousin buyer ( _C_ = _nc_ ).<sup>34</sup> Condition (24) characterizes the city fringe.

Before presenting the equilibrium, observe that Lemmas 1 and 2 remain valid in the hybrid model (i.e., 1-owners always sell to their cousins and transactions between non-cousins always involve insecure plots). The structure of the competitive equilibrium city now depends on both the intensity of the cousinage penalty and on the registration cost, as illustrated by the following proposition (see Appendix D for the full and detailed resolution and for the resolution of the equilibrium for other values of _k_ and _J_ ):

> 34Observe that _Lq_ ( _x_ ) + _L_ 1( _x_ ) = _Lc_ ( _x_ ) + _Lnc_ ( _x_ )

29
Proposition 7: In equilibrium, the city is organized in four zones. Denoting the boundary zone threshold ̊ _x_ ( _k, J_ ) =<sup><u>1</u></sup> _t_ � _y −_ (1 _−kq_ )(1 _−J−π_ )<sup>_−u_</sup> � , we have:

- Zone 1 <u>(mixed formal</u> and informal residential zone): On [0 _,_  ̊ _x_ ( _k, J_ )[ , all owners (irrespective of the initial tenure security level of their plot) participate in the market. Some _q_ -owners register their plot before the sale (in quantity _Lfq_ ( _x, k, J_ ) = 1 _−_ (1 _−πq_ <u>(1)(</u> _−yq−_ <u>)(</u> _xty−−xtu_ )+ _−uJ_ <u>)</u> _−k_<sup>),thoughsome</sup> do not and sell exclusively to their cousins. 1 -owners do not register their secure plots and sell them exclusively to their cousins. The informal price in each location _x_ is _R_<sup>_c_</sup> ( _x_ ) = _y−xt−u−k_ + _J_ and the formal price for registered plots is _Rf_ ( _x_ ) = _y − xt − u_ .

- Zone <u>2 (informal</u> residential zone, <u>full</u> market <u>participation):</u> On ] ̊ _x_ ( _k, J_ ) _, x_ <u>(</u> _J_ )[ , all owners ( _q_ - and 1 -owners) sell their plot informally and exclusively to cousins. In each location _x_ , the price for these informal sales is _R_<sup>_c_</sup> ( _x_ ) = ( _π_ (1 _− q_ ) + _q_ )( _y − xt − u_ ) .

- Zone <u>3 (mixed</u> informal residential and agricultural zone, <u>partial</u> market <u>participation</u> of _<u>q</u>_ -owners): On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_  ̄ _x_ ( _J_ )[ , all 1 -owners and some _q_ -owners sell their plots exclusively to cousins. The rest of _q_ -owners drop out of the market. The mass of _q_ -owners selling their plot in _π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u>

- _x_ to cousins is _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x, J_) =</sup> _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_.Thepriceineachlocation</sup><sup>_x_is</sup><sup>_Rc_(</sup><sup>_x_) =</sup><sup>_Ra_+</sup><sup>_J_.</sup>

- Zone <u>4 (mixed</u> informal residential and agricultural zone, all _<u>q</u>_ -owners dropping out of the market): On ] ̄ _x_ ( _J_ ) _, x_<sup>_∗_</sup> _a_<sup>[,all1-ownersselltheirplotexclusivelytocousinsandall</sup> _q_ -owners drop out of the market. The price in each location _x_ is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ .

- The city boundary is at _xb_ = _x_<sup>_∗_</sup> _a_<sup>=</sup><sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −u_].</sup>

# Proof: See Appendix D.

The structure of the city and the corresponding equilibrium land prices are represented on Figure 5.<sup>35</sup> In the central residential zone (Zone 1, which has mixed registered and non-registered plots), the registered price curve has slope _−t_ , reecting the standard trade-o between proximity to the center and land prices under full tenure security. There is a constant markup between the registered price curve and non-registered price curve equal to _k − J_ . This ensures that _q_ -sellers are indierent

> 35In Appendix D, we present a gure that plots the payos of sellers underlying their participation, registration and cousinage decisions (see Figure 18).

30
<!-- Start of picture text --> Land prices<br>Rj(x)<br>—t ce<br>kJ<br>Ra + J R°(x)<br>_ Zone1 . Zone2 _ Zone3 _ Zone4 .<br>Registration by | No registration No registration No<br>some q-sellers - All plots transacted) A\| 1- and some} registration<br>Otherwith transactionscousins |onlyqjransactionswith cousins transactedq-plots Onlytransacted1-plots<br><!-- End of picture text -->
decreasing function of _k_ and an increasing function of _J_ , whereas _Lfq_ ( _x_ ) is a decreasing function of _k_ and an increasing function of _J_ . It follows that an increase in _k_ reduces both the zone over which plots are registered (Zone 1) and the proportion of landowners registering their plot in each location, resulting in an unambiguous reduction in the overall number of registered plots. Landowners who do not register their plots anymore while remaining in Zone 1 all resort to bilateral trusted relationships. Landowners who are now within the boundaries of Zone 2 also all resort to trusted relationships. The resulting eect is an unambiguous increase in the overall number of transactions under trusted relationships. Let us now focus on an increase in _J_ . Considering that ̊ _x_ ( _k, J_ ) is shifted to the right, _<u>x</u>_ <u>(</u> _J_ ) and _x_ ̄( _J_ ) are shifted to the left, _L_<sup>_c_</sup> _q,S_ =1<sup>(</sup><sup>_x, J_)isreducedand</sup><sup>_Lqf_(</sup><sup>_x, k, J_)isincreased,itiseasy</sup> to see that fewer landowners resort to trusted relationships and more to registering their plot.

The above proposition illustrates how, in a context of costly registration, social relationships can partially address information asymmetry issues at the periphery of the city in the hybrid model, as it did in the trust version of the model. This is in line with the persistence of trusted relationships in the periphery of sub-Saharan African cities (as documented by Smith 2004) where registration is relatively scarce (see Selod and Tobin 2018, and Picard and Selod 2020). It also predicts that, if registration becomes more aordable over time, the role of ethnic relationships governing land transaction could be phased out. In the polar case with _k_ = 0 , ethnic relationships are not mobilized ˆ anymore. Mathematically, we see that, when _k_ tends towards zero, _x_ ( _k_ ) =<sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −k −u_]tends</sup> towards _x_<sup>_∗_</sup> _a_<sup>andtheinformalzoneofthecitydisappears.</sup> Finally, we try to compare the surplus under the hybrid model with that of the trust and registration models. Recall that the model involves several externalities associated with the agents' private decisions to participate in the market, register and transact with cousins. It is thus not straightforward to predict how the hybrid model will rank in comparison with the other two models. Although the comparison is mathematically intricate, we are able to show the following proposition for small values of the cousinage penalty:

# Proposition 9:

- When the registration cost is suciently low, the registration model leads to a greater surplus than the hybrid model, which itself leads to a greater surplus than the trust model.

- When registration cost is suciently high, the hybrid and trust models provide greater surpluses than the registration model.

32
## Proof: See Appendix D

The intuition for the rst part of the rst item in Proposition 9 is straightforward. When registration costs are low, adding the possibility of registration to the trust model allows for an ecient substitution of registration to trusted relationships. This substitution, which occurs close to the city center reduces both the risk of conict and information asymmetry, whereas trusted relationships only addressed information asymmetry without reducing the risk of conict. The possibility of cheap registration allows all owners of risky plots to enter the market, with a fraction of them deciding to register. The second part of the rst item in Proposition 9 sheds light on the desirability of a full registration system when the registration cost becomes low. In that case, the possibility of circumventing registration through trusted relationships is harmful as it leaves tenure risks intact. In this context, it makes sense for governments to make registration compulsory.

On the contrary, the second item in Proposition 9 reects the fact that, when registration costs are high, adding the possibility of establishing trusted relationships in the registration model allows some owners to forego costly registration for trusted relationships that relatively eciently addresses information asymmetry and encourages more low-risk plot owners to sell at the outskirt of the city. In the case of developing countries where registration costs are likely to be high, our results thus indicate that the existence of trusted relationships can be an ecient substitute to an inecient land administration.<sup>36</sup>

# 5 Conclusion

In this paper, we proposed a series of models to shed light on alternative ways whereby households respond to the fundamental issue of tenure insecurity in sub-Saharan African cities. We rst developed a pure registration model with information asymmetry between buyers and sellers that produced a series of results. First, the model made explicit how formal property rights protect land owners by making land tenure secure, but also how they facilitate transactions by reducing asymmetries of information between potential buyers and sellers (as all market participants know that formally recognized property rights cannot be contested). The model also showed that when registration costs

> 36Finally, note that it is technically dicult to compare the surpluses of the hybrid and trust models. But it is possible to exhibit sucient conditions for the hybrid model to dominate the trust model in the vicinity of _k_<sup> ̄</sup> .

33
are high, the proportion of formal land plots decreases with distance to the city center, a pattern that is observed in many sub-Saharan African cities (see e.g., Durand-Lasserve et al., 2015, Bertrand, 2016, World Bank, 2019 for evidence of this in the case of Bamako, Mali and Yaounde, Cameroon). Finally, the model predicted that at the periphery of the city, plots are pooled on the informal periurban land market irrespective of their intrinsic risk, and that the existence of information asymmetry leads to a reduction in the city size, compared to the case with perfect information or with no formalization cost. As in Picard and Selod (2020), the latter result reects a market failure that stems from information asymmetry in a context of tenure insecurity.

We then developed a model without registration but with ethnic matching along trusted relationships and interpersonal obligations between sellers and buyers. The pure trust model showed that, in some cases, trust can remove information asymmetry by generating a separating equilibrium where, on the informal market, riskless plots are sold to allied buyers and risky plots to non-allied buyers. Trusted relationships, however, cannot address the intrinsic problem of tenure insecurity (which requires a formal property right system). The size of the city is also greater because, although some owners of risky plots drop out of the market, owners of riskless plots can get a higher prot from their sale and therefore do not drop out of the market. Further comparing our trust and our registration settings, we found that the registration system is potentially more ecient, because it can reduce tenure insecurity in addition to addressing information asymmetry. The registration cost, however, tends to limit market transactions at the periphery of cities so that, when registration costs are very high, the pure registration model is not necessarily the best option.

Eventually, we developed a hybrid model where both registration and trusted relationships are possible, leading to the following conclusions: First, the model predicts that registration will occur in central parts of the city, whereas trust will be relied upon mainly in peri-urban areas. This is consistent with the observation that land transactions in peri-urban areas often involve interpersonal relationships. Second, we found that trusted relationships are a substitute to registration: A higher registration cost leads more people to engage in ethnic matching. Conversely, if policies are able to bring down the cost of registration, the model predicts that ethnic matching will be phased out. Although we focused on the role of registration cost in deriving our result, the transition towards a full registration system could be accelerated by cultural change and the weakening of the norms governing kinship relationships (Canut and Smith 2006). Finally, our hybrid model showed that when registration costs are high, allowing households to transact informally along trusted relationships leads to a greater 34
economic surplus. This second best result implies that, as long as registration costs remain high, informal transactions need not be discouraged and can even improve the eciency of the economy. In a sense, this mirrors an ongoing debate in the policy world regarding the role played by the dierent tenure situations along a continuum of land rights (UN Habitat, 2012) and the contributions of these tenure situations to the overall welfare. Proponents of this approach have claimed that the promotion of freehold titles as the unique acceptable solution to hold land has been misguided and our results lend some credit to their position. Even though the continuum approach focuses on the benets of weaker but more aordable documentation to legitimize property, our focus on one of the extremities of the continuum where land is held without any documentation shows that informal transactions can be benecial when made according to trusted relationships. In fact, it should not be forgotten that land tenure is above all a social relationship and that registration is just an impersonal means of recognition of this relationship. However, as property rights become more aordable, the pure registration model becomes preferable and informal land transactions reduce rather than increase the surplus. In that context, the establishment of a fully formal land system will result in secure, denser, and more productive cities. We leave the study of underlying factors governing these changes and the speed at which they could happen for future research.

35
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37
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38
# 6 Appendix A - Benchmark model

## 6.1 Proof of Proposition 1 - Competitive equilibrium in the benchmark model

To derive the competitive equilibrium, we study, in each _x_ , all possible combinations of participation decisions that 1 - and _q_ -owners may take to satisfy (4) subject to (5). We then study the stability of these congurations and retain only the stable one. We nally verify that this stable conguration veries the equilibrium conditions (3)-(6).

Sellers' participation decisions and spatial city conguration We consider the three possible cases in terms of 1 -sellers' participation decisions, i.e. _L_ 1( _x_ ) = _π_ , _L_ 1( _x_ ) _∈_ ]0 _, π_ [ or _L_ 1( _x_ ) = 0 . For each case, we then derive the implications for the participation decisions of _q_ -sellers and nd the set of compatible city locations for these participation decisions.

- Let's start with the case _L_ 1( _x_ ) = _π_ , which means that all 1 -owners participate in the market. Because _q_ -owners face the same payo function as 1 -owners (since they cannot be distinguished from one another), all of them also participate in the market so that _Lq_ ( _x_ ) = 1 _− π_ . Therefore, _L_ 1( _x_ <u>)</u>

- the proportion of 1 -sellers among all sellers in _x_ , _π_ ( _x_ ) _≡ L_ 1( _x_ )+ _Lq_ ( _x_ )<sup>,is equal to</sup><sup>_π_.Plugging this</sup> expression into the participation constraint of 1 -sellers _B_ ( _P_ = 1 _, x, Q_ = 1 _, R_ ) _≥ Ra_ simplies to _x ≤_<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −_</sup> _π_ + _qR_ (1 _<u>a−π</u>_ )<sup>_−u_) =</sup><sup>_xa_asdenedinProposition1.Wehaveshownthat:</sup> _L_ 1( _x_ ) = _π ⇒_ ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = ( _π,_ 1 _− π_ ) _⇒ x ≤ xa_

_•_ Let's consider the second case _L_ 1( _x_ ) _∈_ ]0 _, π_ [ , which means that only a fraction of 1 -owners located in _x_ sell their land, requiring indierence between participation and non-participation with _B_ ( _P_ = 1 _, x, Q_ = 1 _, R_ ) = _{π_ ( _x_ ) + _q_ (1 _− π_ ( _x_ )) _}_ ( _y − xt − u_ ) = _Ra_ . Because _q_ -owners face the same payo as 1 -owners, they are also indierent between participation and non-participation in the market. The above indierence condition provides an explicit formula for _π_ ( _x_ ) and thus for the ratio _LLq_ 1(( _xx_ ))<sup>=</sup> _R_ <u>(</u> _<u>ya−−xtq</u>_ ( _−y−u−xtR−au_ <u>))</u><sup>.Observethatthenumeratorinthisratiorepresentsthe</sup> maximum net gain that a seller can obtain from a sale in location _x_ (since the buyer of a plot would be willing to pay _y − xt − u_ if he knew for sure that the plot is secure). Given that plots are transacted in _x_ under information asymmetry, the numerator of<sup>_L_</sup> _L_<sup>_<u>q</u>_</sup> 1(<sup><u>(</u></sup> _x_<sup>_x_</sup> )<sup><u>)</u>isnecessarilypositive,</sup> which implies _x <_<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −Ra −u_)</sup><sup>_≡x∗_</sup> _a_<sup>.Furthermore,inordertohave</sup><sup>_L_</sup> _L_<sup>_<u>q</u>_</sup> 1(<sup><u>(</u></sup> _x_<sup>_x_</sup> )<sup><u>)</u></sup><sup>_>_0,wemustalso</sup> 39
have _Ra − q_ ( _y − xt − u_ ) _>_ 0 , which requires that _x >_<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −_</sup><sup>_<u>R</u>_</sup> _q_<sup>_<u>a</u>−u ≡x_</sup> _a_<sup>_q_).Wehaveshownthat:</sup> _L_ 1( _x_ ) _∈_ ]0 _, π_ [ _⇒_ ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) _∈_ ]0 _, π_ [ _×_ ]0 _,_ 1 _− π_ [ _⇒ x ∈_ ] _x_<sup>_q_</sup> _a_<sup>_, x∗_</sup> _a_<sup>[</sup>

- In the third case, _L_ 1( _x_ ) = 0 , which means that 1 -owners prefer not to sell. As _q_ -owners have the same payos as 1 -owners, they also prefer not to sell, so that _L_ 1( _x_ ) = _Lq_ ( _x_ ) = 0 . Observe that we are in a polar case where the function _π_ ( _x_ ) is actually not dened. From a buyer's perspective, given the shares of secure and insecure plots in location _x_ , if a plot were to be oered on the market, it would be a secure plot with probability _π_ . Non-participation thus requires _Ra ≥_ ( _π_ + (1 _− π_ ) _q_ )( _y − xt − u_ ) , where the RHS is the willingness to pay of a buyer in _x_ . The latter inequality boils down to _x ≥ xa_ . We have shown that:

We have derived necessary conditions for the three above cases. Since it can easily be checked that _x_<sup>_q_</sup> _a_<sup>_< xa< x∗_</sup> _a_<sup>,thisimpliesthefollowingspatialconguration:</sup>

- For any _x ≤ x_<sup>_q_</sup> _a_<sup>,wehave(</sup><sup>_L_1(</sup><sup>_x_)</sup><sup>_, Lq_(</sup><sup>_x_)) = (</sup><sup>_π,_1</sup><sup>_−π_).</sup>

- For any _x ≥ x_<sup>_∗_</sup> _a_<sup>,wehave(</sup><sup>_L_1(</sup><sup>_x_)</sup><sup>_, Lq_(</sup><sup>_x_)) = (0</sup><sup>_,_0).</sup>

- On the interval _x ∈_ ] _x_<sup>_q_</sup> _a_<sup>_, xa_[, one may encounter any(</sup><sup>_L_1(</sup><sup>_x_)</sup><sup>_, Lq_(</sup><sup>_x_))</sup><sup>_∈_]0</sup><sup>_, π_[</sup><sup>_×_]0</sup><sup>_,_1</sup><sup>_−π_[(second case)</sup> or ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = ( _π,_ 1 _− π_ ) (rst case).

- For _x_ = _xa_ , one may encounter any ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) _∈_ ]0 _, π_ [ _×_ ]0 _,_ 1 _−π_ [ (second case) or ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = ( _π,_ 1 _− π_ ) (rst case) or ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = (0 _,_ 0) (third case).

- On the interval _x ∈_ ] _xa, x_<sup>_∗_</sup> _a_<sup>[, one may encounter any(</sup><sup>_L_1(</sup><sup>_x_)</sup><sup>_, Lq_(</sup><sup>_x_))</sup><sup>_∈_]0</sup><sup>_, π_[</sup><sup>_×_]0</sup><sup>_,_1</sup><sup>_−π_[(second case)</sup> or ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = (0 _,_ 0) (third case).

Stability of the dierent congurations The multiplicity of solutions for _L_ 1( _x_ ) and _Lq_ ( _x_ ) on _x ∈_ ] _x_<sup>_q_</sup> _a_<sup>_, x∗_</sup> _a_<sup>[implies that we potentially have a continuum of equilibria.To study the stability of each of</sup> the possible combinations identied, we look at whether each combination is robust to a small deviation in the participation decisions made by sellers in _x_ . We have three cases, depending on whether we have full-participation, no-participation or partial participation in the market. We have the following results for each one of these cases: 40
- The full-participation case ( ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = ( _π,_ 1 _− π_ ) ) is stable on [0 _, xa_ ] . To show this, observe that the participation constraint of owners is _{π_ + _q_ (1 _− π_ ) _}_ ( _y − xt − u_ ) _> Ra_ . If a mass _ε_ of owners stops selling, owners will now compare _{π_ ( _x_ ) + _q_ (1 _− π_ ( _x_ )) _}_ ( _y − xt − u_ ) and _Ra_ to decide whether to participate in the market. As buyers will expect deviant owners to include _επ_ 1 -owners and _ε_ (1 _− π_ ) _q_ -owners, _π_ ( _x_ ) remains unchanged and equal to _π_ . It follows that the market participation constraint is unchanged so that deviant sellers will come back to their initial decision.

- The noparticipation case ( ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = (0 _,_ 0) ) is stable on ] _xa, x_<sup>_∗_</sup> _a_<sup>].Toshowthis,observe</sup> that this case is characterized by the participation constraint _Ra >_ ( _π_ + (1 _− π_ ) _q_ )( _y − xt − u_ ) , where the RHS is the expected plot price in case of a land sale (given the proportions of 1 and _q_ -owners). If a mass _ε_ of owners starts selling, buyers will expect these deviant owners to include _επ_ 1 -owners and _ε_ (1 _−π_ ) _q_ -owners, leading to _π_ ( _x_ ) = _π_ . As in the previous case, the participation constraint is unchanged, so that deviant owners come back to their initial decision.

- The partial participation cases ( ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) _∈_ ]0 _, π_ [ _×_ ]0 _,_ 1 _− π_ [ ) are not stable on ] _x_<sup>_q_</sup> _a_<sup>_, x∗_</sup> _a_<sup>[.To</sup> _L_ 1( _x_ <u>)(1</u> _−q_ <u>)</u>

- show this, observe that these cases are characterized by equality � _L_ 1( _x_ )+ _Lq_ ( _x_ )<sup>+</sup><sup>_q_</sup> � ( _y − xt − u_ ) = _Ra_ which reects owners' indierence between participating and not participating in the market. We need to consider two sub-cases here, depending on whether _x ∈_ ] _x_<sup>_q_</sup> _a_<sup>_, xa_[or</sup><sup>_x ∈_]</sup><sup>_xa, x∗_</sup> _a_<sup>[.</sup>

- First, consider _x < xa_ . This is equivalent to [ _π_ (1 _− q_ ) + _q_ ] ( _y − xt − u_ ) _> Ra_ . And, as we know that [ _π_ ( _x_ )(1 _− q_ ) + _q_ ] ( _y−xt−u_ ) = _Ra_ , we clearly have: _π_ ( _x_ ) _< π_ . Consider now that _π−L_ 1( _x_ <u>)</u>

- a mass _ε_ of owners start selling. Buyers expect that 1 _−L_ 1( _x_ ) _−Lq_ ( _x_ )<sup>_ε_of them are1-sellers and</sup> that 1 _−_ 1 _L−_ 1 _π_ ( _x−_ ) _L−qL_ <u>(</u> _xq_ () _x_ )<sup>_ε_are</sup><sup>_q_-sellers.37Denoting</sup><sup>_η_=</sup> 1 _−L_ 1( _xε_ ) _−Lq_ ( _x_ )<sup>, the price of a land plot be-</sup> comes [ _Π_ ( _x, η_ )(1 _− q_ ) + _q_ ] ( _y−xt−u_ ) where _Π_ ( _x, η_ ) = _L_ 1( _x_ )+ _η_ ( _π−LL_ 11(( _xx_ <u>)+))+</u> _<u>η</u>_ <u>(</u> _Lπq−_ ( _xL_ )+1( _xη_ <u>))(1</u> _−π−Lq_ ( _x_ ))<sup>.</sup> It can easily be shown that<sup>_<u>∂Π</u>_</sup> _∂η_<sup>=</sup> [( _πL_ <u>(1</u> _L_ (1 _x_ <u>()+</u> _x_ <u>)+</u> _LqL_ ( _<u>qx</u>_ <u>())(1</u> _x_ <u>))</u> _−−Lη_ )+1( _xη_ <u>)]</u><sup>2</sup><sup>_>_0because</sup><sup>_π_(</sup><sup>_x_)</sup><sup>_<π_.Thenew</sup> price of land is therefore increased by the deviation and becomes strictly greater than _Ra_ . Consequently, the deviation triggers a cumulative process whereby all owners enter the market until we reach the stable conguration with full participation ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = ( _π,_ 1 _−π_ ) .

- Second, consider _x > xa_ . This is equivalent to [ _π_ (1 _− q_ ) + _q_ ] ( _y − xt − u_ ) _< Ra_ . And, as we know that [ _π_ ( _x_ )(1 _− q_ ) + _q_ ] ( _y − xt − u_ ) = _Ra_ , we clearly have _π_ ( _x_ ) _> π_ . Consider that a

> 37This is because all owners that were initially not selling are equally likely to start selling. The formula is obtained by recognizing that a mass _π − L_ 1( _x_ ) of 1 -owners and 1 _− π − Lq_ ( _x_ ) of _q_ -owners was initially not participating in the market.

41
mass _ε_ of owners start selling. Here again, buyers expect that 1 _−Lπ_ 1 _−_ ( _xL_ )1 _−_ <u>(</u> _xL_ <u>)</u> _q_ ( _x_ )<sup>_ε_ofthemare</sup> 1 -sellers and that 1 _−_ 1 _L−_ 1 _π_ ( _x−_ ) _L−qL_ <u>(</u> _xq_ () _x_ )<sup>_ε_are</sup><sup>_q_-sellers.Thepriceofalandplotbecomes</sup><sup>_Π_(</sup><sup>_x, η_),</sup> as dened in the previous case. However, we now have<sup>_<u>∂Π</u>_</sup> _∂η_<sup>_<_0because</sup><sup>_π_(</sup><sup>_x_)</sup><sup>_>π_.The</sup> new price of land is therefore decreased by the deviation and becomes strictly smaller than _Ra_ . This triggers a cumulative process whereby all owners stop participating in the market until we reach the stable equilibrium with no participation ( _L_ 1( _x_ ) _, Lq_ ( _x_ )) = (0 _,_ 0) .

On each interval for _x_ , we select the only stable cases. This leads to a unique possible conguration with full sale of all 1 -plots and _q_ -plots on [0 _, xa_ ] and no sale on ] _xa,_ + _∞_ [ .

Compatibility with equilibrium conditions It is easy to verify that the selected stable Paretooptimal conguration satises the 4 equilibrium conditions for each _x ∈_ [0 _, xa_ ] :

- _L_ 1( _x_ ) + _Lq_ ( _x_ ) = 1 so that (3) is veried.

- _B_ ( _P_ = 1 _|x, Q ∈{q,_ 1 _}, R_ ) = ( _π_ (1 _− q_ ) + _q_ )( _y − tx − u_ ) _> Ra_ so that (4) is veried.

- _R_ ( _x_ ) = ( _π_ (1 _− q_ ) + _q_ )( _y − tx − u_ ) so that (5) is veried.

- _π_ (1 _− q_ ) + _q_ )( _yu − txa − u_ ) = _Ra_ so that (6) is veried.

42
Payoff <!-- Start of picture text --> P=l R(x)<br>—(w#+q(1l—7))t<br>Ra<br>P=0<br>0) La r<br>All plots transacted |<br><!-- End of picture text -->
and _q_ -owners may take to satisfy (9) subject to (10 and 11). In each location, we determine which combinations of participation and formalization decisions are possible and study their stability. We then select the unique stable conguration and verify that it satises the equilibrium conditions (8)(12).

Sellers' participation and registration decisions and spatial city conguration In a fashion similar to the proof of Proposition 1 for the benchmark model, we show below that the decisions of all sellers in _x_ can be uniquely characterized by the pair of variables ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) given that uniquely determine _Lfq_ ( _x_ ) and _Lq_ ( _x_ ) . We now analyze the seven possible combinations of decisions, i.e.: ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _π,_ 0) , ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, π_ ) , ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α, π − α_ ) , ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _,_ 0) , ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α,_ 0) , ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, α_ ) and ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α, β_ ) where _α >_ 0 , _β >_ 0 and _α_ + _β ∈_ ]0 _, π_ [ :

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _π,_ 0) (i.e., all 1 -owners register their plot and sell), then the payo maximization constraint, the symmetry between 1 - and _q_ -owners' decisions (because of their identical payos), and the equivalence _x_ ˇ( _k_ ) _< x_ ˆ( _k_ ) _⇔ k > k_ imply that:

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, π_ ) (i.e., all 1 -owners sell informally), then the payo maximization constraint and the equivalence _x_ ˇ( _k_ ) _< xa ⇔ k > k_ imply:

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α, π−α_ ) with _α ∈_ ]0 _, π_ [ (i.e., all 1 -owners sell, some informally and some after _L_ 1( _x_ <u>)</u> _k_

- registering their plot), then the payo maximization constraint implies: _Lq_ ( _x_ ) = (1 _−q_ )( _y−xt−u_ ) _−k_<sup>.</sup> The conditions _Lq_ ( _x_ ) _>_ 0 and _L_ 1( _x_ ) _< π_ are incompatible so that this combination is in fact not possible.

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _,_ 0) (i.e., no 1 -owner participates in the market), then the payo maximization constraint and the equivalence _xa > x_ ˆ( _k_ ) _⇔ k > k_ imply:

44
- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α,_ 0) with _α ∈_ ]0 _, π_ [ (i.e., some 1 -owners sell after registering their plot), indierence between not participating in the market and selling after registering implies that the ˆ

- interval for _x_ is a singleton that corresponds to the limit of the formal zone, _x_ ( _k_ ) . Because we have a continuous framework, we can overlook this border case of measure zero.

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, α_ ) where _α ∈_ ]0 _, π_ [ (i.e., some 1 -owners sell informally), then the payo maximization constraint and the conditions _π > L_ 1( _x_ ) _>_ 0 and 1 _− π > Lq_ ( _x_ ) _>_ 0 imply:

- ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_ 1( _x_ ) _, Lq_ ( _x_ )) = (0 _,_ 0 _, α, γ_ ( _α_ ))

where _<u>x</u>_ _~~α~~_<sup>=</sup><sup><u>1</u></sup> _t_ � _y − Ra q_ (11 _−−ππ_ <u>+)+</u> _<u>αα</u>_<sup>_−u_</sup> � , _x_ ̄ _α_ =<sup><u>1</u></sup> _t_ � _y − Ra ππ_ ++ _qγγ_ <u>((</u> _αα_ <u>))</u><sup>_−u_</sup> � . Note that this interval for _x_ is always included in ] _x_<sup>_q_</sup> _a_<sup>_, x∗_</sup> _a_<sup>[.</sup>

- If ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _α, β_ ) with _α >_ 0 , _β >_ 0 and _α_ + _β ∈_ ]0 _, π_ [ (i.e., some 1 -owners participate in the market, some of them sell informally, and some after registering their plot). If this case ˆ

- occurs, it will only be for the singleton _x_ = _x_ ( _k_ ) . We discount this case of measure zero.

Stability of the dierent congurations We now determine which of the four remaining cases are robust to a small deviation in the decisions of sellers (which cases are stable). The proof follows a reasoning similar to that of the benchmark case (see Appendix A).

- The full registration and full participation case ( ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = ( _π,_ 0) ) is stable on [0 _,_ ˇ _x_ ( _k_ )[ if _k > k_ (idem on [0 _,_ ˆ _x_ ( _k_ )[ if _k < k_ <u>).</u> To show this, we consider deviations on the registration and participation decisions:

- If a mass _ε_ of owners stop registering to sell informally, then buyers expect that there are _επ_ 1 -owners and _ε_ (1 _− π_ ) _q_ -owners. Therefore, _π_ ( _x_ ) = _π_ and the same payo maximization constraint, _y − xt − u − k >_ ( _π_ (1 _− q_ ) + _q_ )( _y − xt − u_ ) and _y − xt − u − k > Ra_ , are still veried. Therefore, deviant owners come back to their initial registration decisions.

- If a mass _ε_ of owners stops participating in the market, land prices are unchanged at _y − xt − u_ . Therefore, deviant owners come back to their initial decisions.

45
- The no-registration and full participation case (i.e., ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, π_ ) ), in zone ]ˇ _x_ ( _k_ ) _, xa_ [ if _k > k_ <u>,</u> is stable. This is because:

- If a mass _ε_ of owners start registering their plot before selling, then buyers expect them to be in proportions _επ_ 1 -owners and _ε_ (1 _− π_ ) _q_ -owners. Thus, _π_ ( _x_ ) remains equal to _π_ and the payo maximization constraint is unchanged. Therefore, deviant owners come back to their initial decisions.

- If a mass _ε_ of owners stop participating in the market, then buyers expect them to be in proportions _επ_ 1 -owners and _ε_ (1 _− π_ ) _q_ -owners. Thus, _π_ ( _x_ ) remains equal to _π_ and the payo maximization constraint is unchanged. Therefore, deviant owners come back to their initial decisions.

- The no-participation case (i.e., ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _,_ 0) ), in zone ] _xa,_ + _∞_ [ if _k >_ _<u>k</u>_ and in zone ]ˆ _x_ ( _k_ ) _,_ + _∞_ [ if _k < k_ is stable. Indeed we see that:

- If a mass _ε_ of owners start selling on the informal market, then buyers will expect them to be in proportions _επ_ 1 -owners and _ε_ (1 _− π_ ) _q_ -owners. Therefore, _π_ ( _x_ ) = _π_ . The payo maximization constraint is unchanged and deviant owners come back to their initial decisions.

- If a mass _ε_ of owners start selling on the formal market, then the payo maximization constraint is unchanged and deviant owners come back to their initial decisions.

- The no-registration but partial participation case (i.e., ( _Lf_ 1( _x_ ) _, L_ 1( _x_ )) = (0 _, α_ ) with _α ∈_ ]0 _, π_ [ ), possible when max ( _<u>x</u>_ _~~α~~_<sup>_,_ˆ</sup><sup>_x_(</sup><sup>_k_))</sup><sup>_<x<x_ ̄</sup><sup>_α_withthepayomaximizationconstraint(</sup><sup>_y−xt−_</sup> _u_ ) � _LL_ 11(( _xx_ )+)(1 _L−q_ ( _<u>qx</u>_ <u>))</u><sup>+</sup><sup>_q_</sup> � = _Ra_ , is not stable, for the same reason as in the benchmark model.

We select the only stable combination on each interval:

- When _k > k_ <u>,</u> we have:

- Full registration and full participation on [0 _,_ ˇ _x_ ( _k_ )[

- No formalization but full participation on ]ˇ _x_ ( _k_ ) _, xa_ [

- No participation on ] _xa,_ + _∞_ [

46
<!-- Start of picture text --> Case 1 (k < k) Case 2 (k < k < k)<br>Payoff Payoff<br>(P=1,F=1)<br>Ry(a) —k<br>(P=1,F =1) - R,(«)—k<br>(P=1,F=0) 7 (P=1,F=0) Rt)<br>Ke *<br>= ee P=0 —(7+q(1—7))t<br>LSS<br>0 i(k) x 0 &(k) La x<br>SEEN ont S$Zone 1 Zone2<br>[= All plots transacted<br><!-- End of picture text -->
<!-- Start of picture text --> No registration<br>All plots transacted<br>0 7(k,@) ry<br><!-- End of picture text -->
Given that we have, by denition: the equality of payos implies _π_<sup>_c_</sup> ( _x_ ) = _π_<sup>_nc_</sup> ( _x_ ) . Then, the payo of _q_ -sellers selling to cousins is _ψ_ ( _x, u|C_ = _c_ ) _− J_ and the payo of _q_ -sellers selling to non-cousins is _ψ_ ( _x, u|C_ = _nc_ ) = _ψ_ ( _x, u|C_ = _c_ ) _._ Thus, _q_ -sellers all prefer to sell to non-cousins and do so. Therefore, _π_<sup>_c_</sup> ( _x_ ) = 1 (all sellers selling to cousins are 1 -sellers), while _π_<sup>_nc_</sup> ( _x_ ) _<_ 1 (because 1 -sellers only make up a limited proportion of the sellers selling to non-cousins). This contradicts _π_<sup>_c_</sup> ( _x_ ) = _π_<sup>_nc_</sup> ( _x_ ) and thus our initial assumption.

- If there is a stable equilibrium in which 1 -sellers sell only to non-cousins, then we have _π_<sup>_nc_</sup> ( _x_ ) _> π_<sup>_c_</sup> ( _x_ ) = 0 and the payo of 1 -sellers is strictly larger when selling to a non-cousin than when selling to a cousin, which means that:

Then, the payo of _q_ -sellers selling to cousins is _ψ_ ( _x, u|C_ = _c_ ) _−J_ and the payo of _q_ -sellers selling to non-cousins is _ψ_ ( _x, u|C_ = _nc_ ) , which is strictly larger than _ψ_ ( _x, u|C_ = _c_ ) _− J_ . Thus, _q_ -sellers all prefer to sell to non-cousins and do so. Therefore, all sellers sell to non-cousins and _π_<sup>_nc_</sup> ( _x_ ) = _π_ . A deviating 1 -seller selling to a cousin would get a payo of _{π_ + _q_ (1 _− π_ ) _}_ ( _y − tx − u_ ) , as the buyer would assume that a probability _π_ for the seller to be a 1 -seller. This payo is exactly equal to _ψ_ ( _x, u|C_ = _nc_ ) , so that the deviating seller would not come back to selling to a non-cousin and would increase _π_<sup>_c_</sup> ( _x_ ) to 1 , thereby triggering a transition away from the equilibrium where 1 -sellers sell only to cousins. Thus, this equilibrium would be unstable, which contradicts our initial assumption.

Both cases have led to a contradiction. Consequently, at a stable equilibrium, 1 -sellers only sell to cousins.

49
# 8.2 Proof of Proposition 4 - Competitive equilibrium in the trust model

Here, we provide a more complete version of Proposition 4. Proposition 4 corresponds to the rst case (low value of the social penalty) highlighted in the Extended Proposition 4.

## Extended Proposition 4

Extended Proposition 4:Denoting the social penalty thresholds _<u>J</u>_ = _πRa_ 1 _−q_ _<u>q</u>_ and _J_<sup> ̄</sup> = _Ra_ 1 _−q_ _<u>q</u>_<sup>,and</sup> _<u>Ra</u>_ <u>+</u> _<u>J</u>_ ̄ the zone boundary thresholds _<u>x</u>_ <u>(</u> _J_ ) =<sup><u>1</u></sup> _t_ � _y − π_ (1 _−q_ )+ _q_<sup>_−u_</sup> � and _x_ ( _J_ ) =<sup><u>1</u></sup> _t_<sup>(</sup><sup>_y −_(</sup><sup>_Ra_+</sup><sup>_J_)</sup><sup>_−u_),thereare</sup> ve possible cases, depending on the magnitude of the cousinage penalty:<sup>38</sup>

- <u>Case</u> 1: _<u>J < J</u>_ <u>.</u> The city boundary is at _x_<sup>_∗_</sup> _a_<sup>andthecityisorganizedinthethreefollowing</sup> zones, moving outward from the city center:

- Zone 1 (fully residential): On [0 _, x_ <u>(</u> _J_ )[ , all landowners (irrespective of the tenure security level of their plot) participate in the land market and exclusively sell to cousins. The unique price in each location _x_ is _R_<sup>_c_</sup> ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − xt − u_ ) .

- Zone 2 (mixed agricultural and residential, partial market participation of _q_ - sellers): On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_  ̄ _x_ ( _J_ )[ , all 1 -sellers and a share of _q_ -sellers participate in the land market and exclusively sell to their respective cousins. The mass of _q_ -sellers selling to cousins in _x π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u>

- is _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x, J_) =</sup> _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_.Theuniquepriceineachlocation</sup><sup>_x_is</sup><sup>_Rc_(</sup><sup>_x_) =</sup><sup>_Ra_+</sup><sup>_J_.</sup>

- Zone 3 (mixed agricultural/residential, _q_ -sellers dropping out of the market): On ] ̄ _x_ ( _J_ ) _, x_<sup>_∗_</sup> _a_<sup>[,all1-sellersparticipateinthelandmarketandselltotheircousins,whereas</sup> all _q_ -sellers keep their plots under agricultural use. The unique price in location _x_ is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ .

- <u>Case 2:</u> _<u>J < J ≤ J</u>_<sup> ̄</sup> . The city boundary is at _x_<sup>_∗_</sup> _a_<sup>andthecityisorganizedinfourzones,moving</sup> outwards from the city center: (i) a zone where all 1 -owners and _q_ -owners sell to a cousin; (ii) a zone where all 1 -owners sell to a cousin and all _q_ -owners sell (a fraction of them selling to cousins and the rest to non-cousins); (iii) a zone where all 1 -owners and a share of _q_ -owners sell to their respective cousins; and (iv) a zone where all 1 -owners sell to their cousins whereas _q_ -owners do not participate in the land market.

> 38Here we assume that _y −_<sup>_<u>R</u>_</sup> _qa − u >_ 0 , which is necessary to exclude unrealistic cases where risky plots never get sold under high values of _J_ . It is a more restrictive assumption than our previous condition _y − π_ + _qR_ (1 _<u>a−π</u>_ )<sup>_−u >_0fora</sup> non-degenerate city to exist under the benchmark model.

50
- <u>Case 3:</u> _π_ <u>(1</u> _−_ _<u>q</u>_ <u>)(</u> _<u>y −</u> u_ <u>)</u> _<u>> J > J</u>_<sup> ̄</sup> . The city boundary is at _x_<sup>_∗_</sup> _a_<sup>andthecityisorganizedinfour</sup> zones, moving outwards from the city center: (i) a zone where all 1 -owners and _q_ -owners sell to a cousin; (ii) a zone where all 1 -owners sell to a cousin and all _q_ -owners sell (a fraction selling to cousins and another fraction to non-cousins); (iii) a zone where all 1 -owners sell to their cousins and _q_ -owners all sell to non-cousins; and (iv) a zone where all 1 -owners sell to their cousins, whereas _q_ -owners do not participate in the land market.

- <u>Case 4: (1</u> _−_ _<u>q</u>_ <u>)(</u> _<u>y −</u> u_ <u>)</u> _<u>> J > π</u>_ <u>(1</u> _−_ _<u>q</u>_ <u>)(</u> _<u>y −</u> u_ <u>)</u> : The city structure is identical to Case 3, but without zone (i).

- <u>Case 5:</u> _<u>J ></u>_ <u>(1</u> _−_ _<u>q</u>_ <u>)(</u> _<u>y −</u> u_ <u>)</u> : The city structure is identical to Case 3, but without zones (i) and (ii).

Proof of Extended Proposition 4 For this proof, we denote _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)thequantityofsecurelandsoldtocousins,</sup><sup>_Lc_</sup> _q_<sup>(</sup><sup>_x_)thequantityof</sup> insecure land sold to cousins and _L_<sup>_nc_</sup> ( _x_ ) the quantity of (insecure) land sold to non-cousins. Note that we have _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_) +</sup><sup>_Lc_</sup> _q_<sup>(</sup><sup>_x_) +</sup><sup>_Lnc_(</sup><sup>_x_) =</sup><sup>_L_1(</sup><sup>_x_) +</sup><sup>_Lq_(</sup><sup>_x_).</sup> To derive the competitive equilibrium, we study, in each _x_ , all possible combinations of participation and cousinage decisions that 1 - and _q_ -owners may take to satisfy (15) subject to (16 and 17). We then study the stability of these combinations. In some locations, we will see that more than one combination is possible and stable. In that case, we select the Pareto-dominant combination, i.e. the combination that unambiguously benets owners the most (as 1 -owners and _q_ -owners prefer the same combinations). We check that this conguration satises the equilibrium conditions (14)-(18).

Sellers' participation and cousinage decisions and spatial city conguration Given Lemmas 1 and 2, it is clear that the decisions taken by all sellers in location _x_ can be uniquely characterized by the triple ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)).Wedistinguishthefollowingsituations:</sup>

- _<u>J</u>_

- _•_ If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π,_1</sup><sup>_−π,_0),then,denoting</sup><sup>_x_</sup> �( _J_ ) _≡_<sup><u>1</u></sup> _t_ � _y − π_ (1 _−q_ )<sup>_−u_</sup> � , the payo maximization constraint implies:

51
- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π, α,_1</sup><sup>_−π −α_)where</sup><sup>_α∈_]0</sup><sup>_,_1</sup><sup>_−π_[,thenthepayomaximization</sup> constraint implies _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x_) =</sup><sup>_π_</sup><sup><u>(1</u></sup><sup>_−q_</sup><sup><u>)(</u></sup> _J_<sup>_<u>y−xt−u</u>_</sup><sup><u>)</u></sup> _− π_ and:

- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (</sup><sup>_π,_0</sup><sup>_,_0),thenthepayomaximizationconstraintimplies:</sup>

- ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (</sup><sup>_π,_0</sup><sup>_,_0)</sup><sup>_⇒_(</sup><sup>_J>_</sup> _J_ and _x_<sup>_∗_</sup> _a_<sup>_> x > xq_</sup> _a_<sup>)or(</sup><sup>_J<_</sup> _J_ and _x_<sup>_∗_</sup> _a_<sup>_> x >x_ ̄(</sup><sup>_J_))</sup>

- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_α,_0</sup><sup>_,_0)where</sup><sup>_α∈_]0</sup><sup>_, π_[,thenitiseasytoseethatthepayomaxi-</sup> mization constraint implies that the case is in fact not possible.

- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (0</sup><sup>_,_0</sup><sup>_,_0),thenthepayomaximizationconstraintimplies:</sup>

- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (</sup><sup>_π, α,_0)where</sup><sup>_α ∈_]0</sup><sup>_,_1</sup><sup>_−π_[,thenthepayomaximizationconstraint</sup> _π_ <u>(1</u> _−q_ <u>)(</u> _<u>yu−xt−u</u>_ <u>)</u>

- implies _L_<sup>_c_</sup> _q,S_ =1<sup>(</sup><sup>_x_) =</sup> _Ra_ + _J−q_ ( _yu−xt−u_ )<sup>_−π_and:</sup>

- If ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π, α, β_)where</sup><sup>_α>_0,</sup><sup>_β>_0and</sup><sup>_α_+</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[,thenthepayo</sup> maximization constraint implies that it is possible only on the singleton _x_ = _x_<sup>_q_</sup> _a_<sup>.Wediscard</sup> this case of measure zero.

52
Stability of the dierent congurations Using the same approach as before, it can easily be shown that:

- All corner combinations (where all 1 -owners in a given location adopt the same decisions, and all _q_ -owners adopt the same decisions) are stable. As before, this is done by showing that a small enough deviation in participation and cousinage decisions does not change the strict ranking of owners' decisions so that they return to their initially optimal decisions.

- We now show that the case ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (</sup><sup>_π, α,_1</sup><sup>_−π −α_)where</sup><sup>_α ∈_]0</sup><sup>_,_1</sup><sup>_−π_[is stable</sup> on <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_  ̄ _x_ ( _J_ )[ if _J > J_<sup> ̄</sup> (respectively on <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _, x_<sup>_q_</sup> _a_<sup>[if</sup><sup>_J_ ̄</sup><sup>_> J> J_</sup> ):

- If a mass _ε_ of owners who used to sell to cousins stop selling (or start selling to non-cousins), buyers expect these deviant owners to include _ε α_ + _<u>π</u> π_<sup>1-ownersand</sup><sup>_ε_</sup> _α_ + _<u>απ</u>_<sup>_q_-owners.There-</sup> fore, the land price for transactions between cousins and the land price for transactions between non-cousins remain unaltered. All payos and the payo maximization constraint are preserved, so that deviant owners come back to their initial decisions.

- If a mass _ε_ of owners who used to sell to non-cousins stop selling, land prices and thus payos remain unaltered. Therefore, deviant owners come back to their initial decisions.

- If a mass _ε_ of owners who used to sell to non-cousins start selling to cousins (those can only be _q_ -owners), the land price for transactions between cousins is reduced from _π_ <u>(1</u> _−q_ <u>)</u>

- � _α_ + _π_ + _q_ � ( _y − xt − u_ ) _− J_ (which is equal to _q_ ( _y − xt − u_ ) since _q_ -owners are indierent _π_ <u>(1</u> _−q_ <u>)</u>

- between selling to cousins and non-cousins) to � _α_ + _ε_ + _π_<sup>+</sup><sup>_q_</sup> � ( _y − xt − u_ ) _− J_ . It is easy to see that the new price is lower than the former price, so that _q_ -owners now strictly prefer selling to non-cousins, while 1 -owners still prefer selling to cousins. Therefore, 1 -owners do not change their decisions and some _q_ -owners shift from selling to cousins to selling to non-cousins. This cumulative shift lasts until the benet obtained by _q_ -owners when selling to cousins becomes equal to their benet when selling to non-cousins. At this point, we are back to the initial combination of owners' decisions.

- The case ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π, α,_0)for</sup><sup>_α∈_]0</sup><sup>_,_1</sup><sup>_−π_[isstableon]</sup><sup>_xq_</sup> _a_<sup>_,_</sup> _<u>x</u>_ <u>(</u> _J_ )[ if _J > J >_ _<u>J</u>_ and on <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_ _<u>x</u>_ <u>(</u> _J_ )[ if _<u>J</u> > J_ :

- If a mass _ε_ of owners stop selling to cousins, then buyers assume that deviant owners include

53
- _ε α_ + _<u>π</u> π_<sup>1-owners and</sup><sup>_ε_</sup> _α_ + _<u>απ</u>_<sup>_q_-owners.Therefore, the price of land transacted between cousins</sup> is unaltered, payos are preserved and deviant owners come back to their initial decisions.

- If a mass _ε_ of owners starts selling to non-cousins, payos are unaltered and they come back to their initial decisions.

- If a mass _ε_ of owners starts selling to cousins, then they must be _q_ -owners as all 1 - owners are already selling. The price of land transacted between cousins is reduced from � _πα_ <u>(1+</u> _−πq_ <u>)</u> + _q_ � ( _y − xt − u_ ) _− J_ (which is equal to _Ra_ ) to � _απ_ <u>(+1</u> _ε−_ + _<u>qπ</u>_ <u>)</u><sup>+</sup><sup>_q_</sup> � ( _y − xt − u_ ) _− J_ , which is strictly below the agricultural rent _Ra_ . Therefore, although 1 -owners still prefer selling to cousins, _q_ -owners now strictly prefer keeping their land under agricultural use. Consequently, 1 -owners do not change their decisions and some _q_ -owners shift from selling to cousins to keeping their land under agricultural use. This cumulative shift lasts until the benet obtained by _q_ -owners when selling to cousins becomes equal to the agricultural rent. At this point, we are back to the initial combination of owners' decisions.

Selection of the Pareto-dominant conguration In the periphery of the city, several stable combinations are possible, depending on the relative size of _J_ and _Ra_ . These combinations include the no-participation case and either the partial or the total participation case. As before, we selected the situation unambiguously preferred by owners. We therefore obtain the following city structure, depending on _J_ :

- If the cousinage penalty _J_ is low, i.e. _J < J_ <u>,</u> then the city is organized in three zones:

- Zone 1: On [0 _, x_ <u>(</u> _J_ )[ , all owners participate in the market and sell their plot to a cousin. The price on that segment is _R_<sup>_c_</sup> ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − tx − u_ ) .

- Zone 2: On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _,_ _<u>x</u>_ <u>(</u> _J_ )[ , all 1 -owners and a share of _q_ -owners sell their plot to cousins. _π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u>

- The number of _q_ -owners selling their plot in _x_ is _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x_)=</sup> _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_.Theprice</sup> is _R_<sup>_c_</sup> ( _x_ ) = _Ra_ + _J_ .

- Zone 3: On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _, x_<sup>_∗_</sup> _a_<sup>[,all1-ownersselltheirplottocousins,whileall</sup><sup>_q_-ownerskeeptheir</sup> plot for agricultural use. The price is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ .

- The city boundary is at _x_<sup>_∗_</sup> _a_<sup>=</sup><sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −u_].</sup>

54
- If the cousinage penalty _J_ is intermediate, i.e. _<u>J</u> < J < J_ , then the city is organized in four zones:

- Zone 1: On [0 _, x_ �( _J_ )[ , all owners participate in the market and sell their plot to a cousin. The price on that segment is _R_<sup>_c_</sup> ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − tx − u_ ) .

- Zone 2: On [ _x_ �( _J_ ) _, x_<sup>_q_</sup> _a_<sup>[,allownersparticipateinthemarket.1-ownersonlyselltocousins,</sup> while _q_ -owners sell both to cousins and non-cousins. The prices on that segment are _R_<sup>_c_</sup> ( _x_ ) = _q_ ( _y − tx − u_ ) + _J_ and _R_<sup>_nc_</sup> ( _x_ ) = _q_ ( _yu − xt − u_ ) .

- Zone 3: On ] _x_<sup>_q_</sup> _a_<sup>_,_</sup> _<u>x</u>_ <u>(</u> _J_ )[ , all 1 -owners and a share of _q_ -owners sell their plot. They all sell to _π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u>

- cousins. The number of _q_ -owners selling their plot in _x_ is _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x_)=</sup> _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_.</sup> The price is _R_<sup>_c_</sup> ( _x_ ) = _Ra_ + _J_ .

- Zone 4: On <u>]</u> _<u>x</u>_ <u>(</u> _J_ ) _, x_<sup>_∗_</sup> _a_<sup>[,all1-ownersselltheirplottocousins,whileall</sup><sup>_q_-ownerskeeptheir</sup> plot for agricultural use. The price is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ .

- The city boundary is at _x_<sup>_∗_</sup> _a_<sup>=</sup><sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −u_].</sup>

- If the cousinage penalty _J_ is high, i.e. _J_<sup> ̄</sup> _< J_ , then the city is organized in four zones:

- Zone 1: On [0 _, x_ �( _J_ )[ , all owners participate in the market and sell their plot to a cousin. The price on that segment is _R_<sup>_c_</sup> ( _x_ ) = [ _π_ + _q_ (1 _− π_ )] ( _y − xt − u_ ) .

- Zone 2: On [ _x_ �( _J_ ) _,_ � _x_ ( _J_ )[ , all owners participate in the market. 1 -owners only sell to cousins, while _q_ -owners sell both to cousins and non-cousins. The prices on that segment are _R_<sup>_c_</sup> ( _x_ ) = _q_ ( _y − xt − u_ ) + _J_ and _R_<sup>_nc_</sup> ( _x_ ) = _q_ ( _yu − xt − u_ ) .

- Zone 3: On [ _x_ �( _J_ ) _, x_<sup>_q_</sup> _a_<sup>[,allownersparticipateinthemarket.1-ownersonlyselltocousins,</sup> while _q_ -owners only sell to non-cousins. The price of secure plots on that segment is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ and the price of insecure plots is _R_<sup>_nc_</sup> ( _x_ ) = _q_ ( _y − xt − u_ ) .

- Zone 4: On ] _x_<sup>_q_</sup> _a_<sup>_, x∗_</sup> _a_<sup>[,all1-ownersselltheirplottocousins,whileall</sup><sup>_q_-ownerskeepitfor</sup> agricultural use. The price is _R_<sup>_c_</sup> ( _x_ ) = _y − xt − u_ .

- The city boundary is at _x_<sup>_∗_</sup> _a_<sup>=</sup><sup><u>1</u></sup> _t_<sup>[</sup><sup>_y −Ra −u_].</sup>

55
<!-- Start of picture text --> ; mom<br>0 Ta<br>0 Za x,<br>J 0 (J) T(J) ry<br>Case 2 q-transactions<br>with both<br>J 0 z(J) xf z(J) ry<br>Case 3 q-trans. only<br>rans with non-<br>cousins<br>m(1—q)(y—u) 0 z(J) z(J) x4 ry<br>Case 4 q-trans. q-trans.with. non-only<br>(1—q)(y—u) 0 7(J) rf x<br>Case5 q-trans. only<br>with non-<br>cousins<br>J 0 «4 xy<br><!-- End of picture text -->
<!-- Start of picture text --> Payoff<br>—(7+q(1—q7))t<br>(P=1,C=c,Q=1) R°(x)<br>(P=1,C=c,Q=4q) Ra td R(x)<br>- |—-----__<br> pw“at | re<br>0 x(J) z(J) at £<br>; Zone 1 HK Zone2 \ Zone 3 ;<br>All plots transacted<br>All 1- and some Only 1-plots<br>y W Sins transacted transacted<br><!-- End of picture text -->
_π_ <u>(1</u> _−q_ <u>)(</u> _<u>y−xt−u</u>_ <u>)</u> where _Ra_ + _J−q_ ( _y−xt−u_ )<sup>_−π_clearlydecreaseswith</sup><sup>_J_.</sup> Furthermore, the surplus under the trust model is strictly greater than the surplus in the benchmark model when _J →_ 0<sup>+</sup> . Indeed, in that case, the equilibrium city in the trust model replicates the equilibrium city in the benchmark model, while adding an additional built-up zone [ _xa, x_<sup>_∗_</sup> _a_<sup>].Eventually,</sup> when _J ≥ J_ , the surplus is independent on _J_ and equal to: which is strictly below the surplus of the benchmark model.<sup>39</sup>

# 8.5 Proof of Proposition 6

Let us dene Ω( _k, J_ ) _≡_ Λ( _k_ ) _−_ Φ( _J_ ) . It is clear that _ψ_ is a continuous function of _k_ and _J_ . It decreases with _k_ and increases with _J_ . Additionally, it is clear that Ω(0 _,_ + _∞_ ) _>_ 0 , Ω(0 _,_ 0<sup>+</sup> ) _>_ 0 , Ω(+ _∞,_ 0<sup>+</sup> ) _<_ 0 and Ω(+ _∞,_ + _∞_ ) _≥_ 0 .

Ω(0 _,_ 0<sup>+</sup> ) _>_ 0 , Ω(+ _∞,_ 0<sup>+</sup> ) = Ω( _k,_<sup> ̄</sup> 0<sup>+</sup> ) _<_ 0 and Ω( _.,_ 0<sup>+</sup> ) increasing implies that there exists _k_<sup>ˆ</sup> ( _θ_ ) _∈_ ]0 _, k_<sup> ̄</sup> ] such that Ω( _k_<sup>ˆ</sup> ( _θ_ ) _,_ 0<sup>+</sup> ) = 0 .

For all _k ∈_ [0 _, k_<sup>ˆ</sup> ( _θ_ )[ , we have Ω( _k,_ 0<sup>+</sup> ) _>_ 0 . Thus, Ω( _k, J_ ) _>_ 0 for all _J_ and the registering system is always better than the social norm (i.e., whatever the intensity of the social norm).

For all _k ∈_ ] _k_<sup>ˆ</sup> ( _θ_ ) _,_ + _∞_ [ , we have Ω( _k,_ 0<sup>+</sup> ) _<_ 0 , Ω( _k,_ + _∞_ ) _>_ 0 (because Ω(+ _∞,_ + _∞_ ) _>_ 0 ) and Ω( _k, ._ ) is increasing. Thus, there exists _J_<sup> ̃</sup> ( _k, θ_ ) such that Ω( _k, J_ ) _<_ 0 for all _J < J_<sup> ̃</sup> ( _k, θ_ ) and Ω( _k, J_ ) _>_ 0 for all _J > J_<sup> ̃</sup> ( _k, θ_ ) . It is clear that _J_<sup> ̃</sup> is an increasing function of _k_ , as Ω( _., ._ ) increases with _k_ , and that _J_<sup> ̃</sup> is constant for _k > k_<sup> ̄</sup> .

Consequently, the registering system is preferable to the social norm only if the social penalty is high enough and the social norm is preferable if the social penalty is low enough.

> 39Because Φ _−_ Γ = _−_<sup><u>(1</u></sup><sup>_−π_</sup><sup><u>)</u></sup><sup>_π_</sup><sup><u>(1</u></sup><sup>_−q_</sup><sup><u>)2</u></sup><sup>_R_</sup> _<u>a</u>_<sup>2</sup><sup><u>(</u></sup> 2<sup>_<u>q</u>_</sup> _q_<sup><u>(</u></sup><sup>_π_2+</sup> ( _π_<sup>_<u>q</u>_</sup> +<sup><u>(1</u></sup> _q_<sup>_−_</sup> (1<sup>_π_</sup> _−_<sup><u>))</u></sup> _π_<sup>+</sup> ))<sup><u>(21</u></sup> _t_<sup>_−q_</sup><sup><u>)(</u></sup><sup>_π_+2(1</sup><sup>_−π_</sup><sup><u>)</u></sup><sup>_<u>q</u>_</sup><sup><u>)</u></sup><sup>_θ_</sup><sup><u>)</u></sup> _<_ 0 .

58
# 9 Appendix D - Hybrid model

## 9.1 Proof of Proposition 7 - Competitive equilibrium in the hybrid model

Similarly to the previous versions of the model, to derive the competitive equilibrium, we study, in each _x_ , all possible combinations of participation, registration and cousinage decisions that 1 - and _q_ -owners may take to satisfy (20) subject to (21, 22 and 23). We then study their stability. In some locations, we will see that more than one combination is possible and stable. In that case, we select the Pareto-dominant combination that unambiguously benets owners the most (we will see that 1 -owners and _q_ -owners prefer the same combinations).<sup>40</sup> We then verify that the selected conguration satises the equilibrium conditions (19)-(24).

Determination of sellers' possible participation and cousinage decisions Given Lemmas 1 and 2, which still hold in this hybrid version of the model, it is clear that the decisions taken by all sellers in location _x_ can be uniquely characterized by the 5-uple of variables ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup> _, L_<sup>_nc_</sup> ( _x_ )) . We therefore can have the following combinations:

- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_,_0</sup><sup>_, π,_1</sup><sup>_−π,_0),thepayomaximizationconstraint</sup> has dierent implications depending on the values of _k_ and _J_ :

> ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (0</sup><sup>_,_0</sup><sup>_, π,_1</sup><sup>_−π,_0)isimpossible</sup><sup>_._</sup>

> 40We assume, without changing the model's main results, that _k_  ̄ _> Ra_<sup>1</sup> _πq_<sup>_−q,_whichallowstoreducethenumberof</sup> possible cases to be studied.

59
- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_,_0</sup><sup>_, π,_0</sup><sup>_,_1</sup><sup>_−π_),thentheimplicationsofthepayo</sup>  ̈ <u>1</u> _<u>k</u>_

- maximization constraint also depends on _k_ and _J_ . Denoting _x_ ( _k_ ) _≡ t_ � _y −_ 1 _−q_<sup>_−u_</sup> � , these implications are:

- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π,_0</sup><sup>_,_0</sup><sup>_,_0</sup><sup>_,_1</sup><sup>_−π_),thenthepayomaximizationcon-</sup> straint implies that this case is impossible.

- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_π,_1</sup><sup>_−π,_0</sup><sup>_,_0</sup><sup>_,_0),thenthepayomaximizationcon-</sup> straint implies:

60
- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_, β, π,_0</sup><sup>_,_1</sup><sup>_−π−β_)where</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[,thenthe</sup>  ̈

- payo maximization constraint implies that it is only possible on the singleton _x_ = _x_ ( _k_ ) . As this case has measure zero, we disregard it.

- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_,_0</sup><sup>_, π,_1</sup><sup>_−π−β, β_)where</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[,thenthe</sup> payo maximization constraint implies:

61
- If ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_α,_0</sup><sup>_, π−α,_1</sup><sup>_−π−β, β_)where</sup><sup>_α∈_]0</sup><sup>_, π_[and</sup><sup>_β∈_</sup> ]0 _,_ 1 _− π_ [ , then the payo maximization constraint implies that this case is possible on at most one singleton. We disregard this case because it is of measure zero.

62
- In all other combinations of decisions where some _q_ -owners participate in the market (proportion _L_<sup>_c_</sup> _q_<sup>(</sup><sup>_x_) +</sup><sup>_Lnc_(</sup><sup>_x_) +</sup><sup>_Lfq_(</sup><sup>_x_)</sup><sup>_∈_]0</sup><sup>_,_1</sup><sup>_−π_[)andall1-ownersparticipateintheinformal</sup> market ( _L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_) +</sup><sup>_Lf_1(</sup><sup>_x_)=</sup><sup>_π_),thepayomaximizationconstraintimpliesthatthecorre-</sup> sponding interval for _x_ is reduced to a singleton or the empty set.

- The payo maximization constraint implies that the six following cases are reduced to a singleton or the empty set:

63
- ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_α, β, π−α,_1</sup><sup>_−π−β,_0)where</sup><sup>_α∈_]0</sup><sup>_, π_[and</sup> _β ∈_ ]0 _,_ 1 _− π_ [

- ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(</sup><sup>_α, β, π−α,_0</sup><sup>_,_1</sup><sup>_−π−β_)where</sup><sup>_α∈_]0</sup><sup>_, π_[and</sup> _β ∈_ ]0 _,_ 1 _− π_ [

- ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_)) = (</sup><sup>_π, β,_0</sup><sup>_,_0</sup><sup>_,_1</sup><sup>_−π −β_)where</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[</sup>

- 1 -sellers do not participate in the market but _q_ -sellers do.

- _q_ -owners do not participate in the market and 1 -owners participate, at least partially, with cousinage decisions dierent from the cases previously analyzed.

- Stability of the equilibria Using the same approach as before, it can easily be shown that:

- All corner combinations (where all 1 -owners in a given location adopt the same decisions, and all _q_ -owners adopt the same decisions) are stable. As before, this is done by showing that a small enough deviation in participation and cousinage decisions does not change the strict ranking of owners' decisions so that they return to their initially optimal decisions.

- The case ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_, β, π,_1</sup><sup>_−π−β,_0)where</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[is</sup> stable:

- If a mass _ε_ of owners stop selling to cousins (i.e. they start formalizing, or they start selling to non-cousins, or they stop selling altogether), then buyers assume that deviant owners include _ε_ 1 _−πβ_<sup>1-owners and</sup><sup>_ε_1</sup><sup>_−_</sup> 1 _−_<sup>_π−_</sup> _β_<sup>_<u>β</u>_</sup> _q_ -owners. Therefore, the price of land transacted between cousins is unaltered, payos are preserved and deviant owners come back to their initial decisions.

- If a mass _ε_ of owners start selling to cousins (i.e. if a mass _ε_ of _q_ -owners stops formalizing), then buyers assume that deviant owners include only _q_ -owners, as 1 -owners were already all selling to cousins. Therefore, the price of land transacted between cousins is reduced and it becomes strictly more attractive for _q_ -owners to sell after registration than to sell informally to cousins (while it remains more attractive for 1 -owners to sell informally to cousins, because they do not face the social penalty). Therefore, the mass of _q_ -owners selling informally to cousins decreases in favor of the mass of _q_ -owners selling after registration

64
until _q_ -owners become indierent between the two options. We are back to the initial conguration.

- Other deviations to owners' behaviors (e.g. when a mass _ε_ of owners stop formalizing and sell to non-cousins) do not aect land prices (in our example, it is _q_ -owners who start to sell to non-cousins, which does not aect the price of informal land traded between noncousins). Because land prices are not aected, these deviations do not aect the ranking of payos and the deviations are reversed back to the initial conguration.

- The case ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_,_0</sup><sup>_, π, β,_1</sup><sup>_−π−β_)where</sup><sup>_β∈_]0</sup><sup>_,_1</sup><sup>_−π_[is</sup> unstable (the proof is essentially the same as in previous sections and is available upon request).

- The case ( _Lf_ 1( _x_ ) _, Lfq_ ( _x_ ) _, L_<sup>_c_</sup> 1<sup>(</sup><sup>_x_)</sup><sup>_, Lc_</sup> _q_<sup>(</sup><sup>_x_)</sup><sup>_, Lnc_(</sup><sup>_x_))=(0</sup><sup>_,_0</sup><sup>_, π, β,_0)where</sup><sup>_β∈_]0,1-</sup><sup>_π_[isstable(the</sup> proof is the same as in the previous sections and is available upon request).

The stable congurations are presented in Figures 11, 12 and 13 for all possible combinations of _k_ and _J_ values. In those gures, we indicate with 1 : and _q_ : the decision of 1 -owners and _q_ -owners respectively, where _C_ = _c_ and _C_ = _nc_ refer to their decision to sell informally to cousins and non-cousins respectively, _f_ refers to registering and participating in the market, _no sale_ refers to staying out of the market, and _partial_ qualies any of the above decisions to indicate that only a fraction of 1 -owners or _q_ -owners take that decision. For instance, on the rst graph of Figure 11, 1 : _C_ = _c, q_ : _C_ = _nc_ indicates that all 1 -owners informally participate in the market and sell to cousins and all _q_ -owners informally participate in the market and sell to non-cousins.

Selection of the Pareto-dominant congurations When several stable congurations are possible for given values of _J_ and _k_ , we select the one that benets owners the most (as 1 -owners and _q_ -owners prefer the same combinations). The Pareto-dominant conguration is highlighted in red for each _k_ and _J_ value. For example, when _k > Ra_ 1 _πq−q_<sup>and</sup><sup>_<u>J</u>_</sup> _> J > πRa_ 1 _−π_ 1 _−_ (1 _<u>q−q</u>_ )<sup>(i.e.bottomleftgraph</sup> in Figure 11), there are two possible congurations on _x ∈_ [0 _,_ ˇ _x_ ( _k_ )] , one in which all 1 -owners register their plot and one in which they sell informally. They prefer the second conguration where they sell informally, because it allows them to avoid incurring the registration cost.

Compatibility with equilibrium conditions It is easy to verify that the stable and Paretodominant conguration satises the 6 equilibrium conditions (19)-(24).

We present the spatial structure of the city for all values of _k_ and _J_ on Figures 14, 15, 16 and 17.

65
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Consequently, denoting Θ( _k, J_ ) the surplus in the hybrid model, it is clear that, when _k_ = 0 and _J →_ 0<sup>+</sup> , we have Θ(0 _, J_ ) _−_ Φ( _J_ ) _>_ 0 and the surplus in the hybrid model is greater than in the trust model. As both city surpluses are clearly continuous for _J ∈_ ]0 _,_ + _∞_ [ and _k ∈_ ]0 _,_ + _∞_ [ , this result holds for _J_ suciently small and _k_ suciently small.

When _k_ = 0 and _J →_ 0<sup>+</sup> also, the registration model city corresponds to a fully formal zone between 0 and _x_<sup>_∗_</sup> _a_<sup>.Asallplotsareregisteredbeforesaleatnocost,tenureinsecuritydisappearsandthe</sup> surplus reaches its optimum. Therefore, it is clear that the surplus in the registration model is greater than in the hybrid model ( Θ( _k, J_ ) _−_ Λ( _k_ ) _<_ 0 ) when _k_ = 0 and _J →_ 0<sup>+</sup> . As all city surpluses are continuous for _J ∈_ ]0 _,_ + _∞_ [ and _k ∈_ ]0 _,_ + _∞_ [ , this result holds for _J_ suciently small and _k_ suciently small.

When _k_ = _k_<sup> ̄</sup> and _J →_ 0<sup>+</sup> , the cities in the hybrid model and in the trust model are the same and yield the same surplus.

When _k_ = _k_<sup> ̄</sup> and _J →_ 0<sup>+</sup> , we have _x_ ˇ( _k_ ) = 0 . The city in the registration model is fully informal (i.e., yellow zone between location 0 and _xa_ ) and the city in the hybrid model is fully informal between location 0 and _xa_ (i.e., yellow zone) and informal with partial participation in the market between _xa_ and _x_<sup>_∗_</sup> _a_<sup>(i.e.,lightgreenzone).Consequently,thecityinthehybridmodelcorrespondstothecity</sup> in the registration model, plus an additional zone in the periphery. Thus, the city surplus is greater in the hybrid than in the registration model ( Θ( _k, J_ ) _−_ Λ( _k_ ) _>_ 0 ) when _k_ = _k_<sup> ̄</sup> and _J →_ 0<sup>+</sup> . As all city surpluses are clearly continuous for _J ∈_ ]0 _,_ + _∞_ [ and _k ∈_ ]0 _,_ + _∞_ [ , this result holds for _J_ suciently small and _k_ suciently large.

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