WPS8482 Policy Research Working Paper 8482 The Informal City Harris Selod Lara Tobin Development Economics Development Research Group June 2018 Policy Research Working Paper 8482 Abstract This paper proposes a theory of urban land use with endog- informality consistent with developing country cities. It enous property rights. Socially heterogeneous households also highlights non-trivial effects of land administration compete for where to live in the city and choose the type of reforms in the presence of pecuniary externalities, possibly property rights they purchase from a land administration explaining why elites may have an interest in maintain- which collects fees in inequitable ways. The model gener- ing inequitable land administrations that insulate them ates predictions regarding sorting and spatial patterns of from competition for land from the rest of the population. 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/research. The authors may be contacted at hselod@worldbank.org and lara.tobin@m4x.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 The Informal City∗ Harris Selod† and Lara Tobin‡ JEL classification: R14, R52 and P48 Keywords: Land markets; Property rights; Land administration; Tenure security; Multiple sales ∗ We are grateful to Asao Ando, Jan Brueckner, Denis Cogneau, Ma¨ ylis Durand-Lasserve, Gilles Duranton, Karen Macours, Stephen Sheppard, Jacques-Fran¸ cois Thisse, Thierry Verdier, and the participants to seminars and conferences where this paper was presented for useful discussions. We acknowledge funding from the World Bank’s Knowledge for Change Program and the Multi Donor Trust Fund for Sustainable Urban Development. The views expressed in this paper are those of the authors and do not necessarily reflect those of the World Bank, its Board of Directors or the countries they represent. † Harris Selod: Energy and Environment Team, Development Research Group, The World Bank. Email: hselod@worldbank.org. ‡ ere de la coh´ Lara Tobin: Minist` esion des territoires, France. Email: lara.tobin@m4x.org. 1 1 Introduction The main forces that shape the spatial structure of cities have long been identified by urban economic theory. In the standard land-use model of urban economics (Alonso, 1964; Muth, 1969; Mills, 1972), heterogeneous households compete for where to live in the city while trading-off accessibility for land consumption under endogenously determined land prices. This spatial equilibrium representation of cities accounts for universal regularities such as decreasing population densities, a negative land price gradient moving away from city centers, and residential stratification by income (Fujita, 1989; Anas et al., 1998). This canonical model, however, implicitly assumes costless and clearly defined property rights. Its validity is thus challenged in many developing country cities where land tenure informality and insecurity is more often the norm than the exception (Marx et al., 2013; UN-Habitat, 2010). The objective of our paper is to fill this gap and build an urban land use theory which can account for informal property rights and associated tenure insecurity. Our framework has three important ingredients observed in many developing countries: a multiplicity of tenure options ranging from very insecure to very secure (UN-Habitat, 2012), a risk of conflict in the presence of multiple sales of a same plot (Durand-Lasserve et al., 2015), and costly access to secure tenure (Wehrmann, 2008), which is often blamed for driving households into informality (Udry, 2011; Collier and Venables, 2013). We focus on the trade-off households face between establishing costly property rights and facing the risk of dispossession in a context where unclear property rights may result in multiple sales. While most models so far have focused on a binary vision of residential informality (i.e., formal versus informal), it is important to acknowledge that there is in fact a wide range of tenure situations in the cities of developing countries, depending on legal documentation held by the “owner” of a plot (Bruce, 1998; Becker, 2013; Durand-Lasserve et al., 2013). Taken together, these situations constitute a “continuum” of options for holding land under various levels of tenure security (UN-Habitat, 2012). The existence of this continuum also implies that different 2 tenure situations often coexist within a same city, an issue which has largely remained unexplored and requires systemic analyses land markets, such as the one we are proposing here. Secondly, the model needs to specify the exact mechanisms whereby the lack of property rights or lack of enforcement of property rights results in tenure insecurity. So far, the literature has exclusively focused on the risk of eviction of squatters who occupy someone else’s land (Jimenez, 1984; Turnbull, 2008; Brueckner and Selod, 2009). Although squatting is an important topic, it only concerns a particular form of tenure insecurity. In many cities, the main source of tenure insecurity that affects the poor as well as a large fraction of the middle class is of a different nature as it often stems from the multiplicity of claims (either because of multiple sales to different buyers, or because multiple claimants each managed to obtain a property right for the same plot) (Wehrmann, 2008; Durand-Lasserve et al., 2015). We retain multiple sales as the source of insecurity in our model, resulting in households paying for urban land before knowing whether or not they will be able to exercise their rights.1 Lastly, the theory should account for the costly processes faced by households to obtain property rights from the land administration.2 We thus allow the land administration to be clientelistic, possibly collecting fees in inequitable ways that favor some households and harm others. The legal and illegal fees 1 A recent study on land markets in Ghana ranked “double sales of land by traditional owners” as the number one problem in Accra (Omirin and Antwi, 2004). In Benin, it was estimated that 20 percent of urban plots were subject to a conflict (OCS, 2006). The problem of multiple sales is so pervasive in that country that a specific provision has been inserted in the recent land code to punish authors of multiple sales with a heavy fine and a jail sentence of up 5 years. In Mali, in the sole district of Bamako, at least 12,000 cases of double allocations of use rights or of superimposition of property titles have been identified (Coulibaly, 2009). Multiple sales are actually clogging tribunals as an estimated 80% of cases in Mali are related to land (Camara, 2012). In Togo, land-related conflicts represent 70% of tribunal cases (Kakpo, 2018). 2 e (2007) reports an example in a locality at the outskirts In the case of Mali for instance, Djir´ of the Bamako district where the average cost of obtaining a property title is 725, 000 CFA Francs (about $1, 500). This includes the payment for the topographical survey, the permit fees (valid for five years), the fees for demarcation, registration, notice of public inquiry, the signature of the sub-prefect and the village chief, and the Land Office registration stamp... In comparison, the price of land in that locality is on average 225, 000 CFA Francs per hectare (about USD 450) (Djir´ e, 2007; Bouju, 2009). 3 necessary to establish property rights are documented in a few descriptive studies, which also insist on the role of social connections and corruption in the land administration, which can be determining factors in obtaining a plot, regularizing tenure and speeding up the process of obtaining a property right (Bertrand, 1995, 2006, 1998; Djir´ e, 2007). In our model we allow the “cost of tenure” (the fee paid to the land administration to obtain a property right) to increase with the level of tenure security provided by the purchased right and to decrease with household’s social connexion to the land administration. Our setting is the first to account for the continuum of property rights, multiple sales and a dysfunctional land administration in a unified land-market equilibrium which applies to cities in developing countries. It makes it possible to under- stand the non-trivial interactions between land administration practices, house- hold tenure choices, and exposure to multiple sales in a context where land prices and city structure are endogenously determined, and thus to understand the gen- eral equilibrium effects of urban land tenure policies. Section 2 presents relevant insights from the literature on the urban land market models. Section 3 presents the model while section 4 studies how changes in the parameters of the model affect city structure. Section 5 concludes. 2 Insights from urban land market models The theoretical literature on the coexistence of formal and informal types of land use was initiated by Jimenez (1985) and Hoy and Jimenez (1991) who analyzed squatting in a partial equilibrium perspective. These papers present a framework of tenure choice under uncertainty, where in the case of an eviction from a squat, households have to relocate in the formal market. In these “eviction models”, the probability of eviction – a direct measure of tenure insecurity – is endogenous but the price of housing in the formal market is exogenous and not affected by the informal sector. This is a strong assumption, especially considering that a large fraction of the population usually occupies land informally. Relaxing the assumption of a fixed formal land price, Turnbull (2008) considers 4 a stochastic (but still exogenous) formal land price on which the probability of eviction depends. Brueckner and Selod (2009) are the first to adopt a general equilibrium perspective by modeling the effect of squatter settlements on formal land prices in a context of fixed land supply. In their setting, inflation of the squatter settlement “squeezes” the formal market, leading to an increase in the price of land in the formal sector up to the point where landowners would find it profitable to evict squatters. Subsequent work considers the case of squatting on public land instead of private land (Shah, 2014) and introduce competition among squatter organizers (Brueckner, 2013). Other general equilibrium models (Alves, 2016; Cavalcanti et al., 2018) study the formation of slums and study the role of barriers to formalization and regularization policies on informality. None of these models, however, offer an explicit representation of the urban space within a city. An exception is Henderson et al. (2016) who propose a spatial model where developers may switch from informal to formal development over time as a response to rising land values but they do not account for insecurity of property rights. These drawbacks are somehow addressed in Cai et al. (2018) who study the dynamics of property rights in a city. Their representation of the urban space, however, is simplified as they do not account for the direct capitalization of tenure insecurity in land rents.3 In a nutshell, the above contributions focus on specific and sometimes simplified definitions of informality, and, with the exception of (Cai et al., 2018), none adopt the continuous spatial framework of urban economics.4 Our model fills these two important gaps as it determines an equilibrium land use with an endogenous choice of tenure among a continuum of options. 3 In their dynamic stochastic setting, they model each sub-period as an instantaneous land market equilibrium where land is held with certainty. 4 There are empirical papers on the measurement of the tenure security and tradability premia which allow for more diverse tenure situations (Jimenez, 1982, 1984; Friedman et al., 1998; Lanjouw and Levy, 2002; Kim, 2004) as well as papers on the impacts of land tenure formalization in an urban context (Field, 2007; Di Tella et al., 2007; Galiani and Schargrodsky, 2010). Very few papers have analyzed the causes of residential informality (Hidalgo et al., 2010). 5 3 The model 3.1 The setup The economy has a city and a rural area and an overall population of mass N . The city is linear and represented by a segment with a Central Business District (CBD) located at one origin. The rural area encompasses all non-urban locations and does not have an explicit spatial representation. The city is open, meaning that the equilibrium number of households in the city is endogenously determined by migration to the urban area. We assume that there is one unit of land in each point and that each urban household consumes a fixed quantity of land, normalized to one. Within a distance x from the CBD, there are thus exactly x units of land and x households. The model also has absentee landlords who extract rents from households who are competing on the land market.5 A household located at a distance x from the CBD (i.e. in location x) pays an endogenous rent R(x) to an absentee landlord. It also has one member who commutes to the CBD to work at a cost xt, where t is the unit transport cost. All workers in the economy earn the same exogenous urban wage yu when residing in the city. We assume that the urban wage is significantly higher than the income yr of households residing in the rural area who can only engage in low-productivity agricultural activities. When residing in the rural area, households incur a cost Ra to access land but do not pay any commuting cost since they derive all their in- come from on-farm activities and do not need to travel to the city in order to work. Our model departs from standard urban land use models as it allows a same plot of land to be sold to more than one household, generating conflicts and tenure insecurity. Buyers reduce tenure insecurity by purchasing property rights from a land administration offering a menu of options that are more or less efficient in deterring multiple sales. This continuum reflects the different steps along the formalization process towards a fully-fledged ownership right. It can 5 Landlords are defined in a broad sense and refer to all individuals, groups or institutions that make land available for housing. They can be the primary owners of the land, private developers, or public authorities, and, in practice, are often a combination of these different stakeholders. 6 include situations where a plot is held with a simple sales certificate, different levels of administrative documents which do not provide a statutory right but instead confer a “presumption of right”, various types of revocable temporary permits to occupy, and, at the end of the spectrum, a registered ownership title. If a buyer obtains a registered property right, the nature and number of checks that are made before the right can be delivered makes it unlikely–although not impossible–that multiple sales will happen. Formally, we assume that there exists a continuum of tenure situations, each characterized by a level of tenure security measured by the probability π ∈ [πmin , πmax ] that a right holder can successfully discourage multiple sales. When π = πmin , the probability of keeping the purchased plot is lowest, corresponding to a situation in which the household has no recognized right and the probability of occurrence of multiple sales is highest. π = πmax corresponds to the most secure form of tenure for which multiple sales are scarcest. Intermediate values of π correspond to property rights with moderate levels of tenure security. Different households face different costs to obtain property rights. This is because social status and personal acquaintances are crucial in determining the level of bribes households may have to disburse to obtain land documentation.6 We model this by having each household characterized by its ability to interact with the land administration, as measured by its “social distance” e from the administration, with e uniformly distributed over [0, 1] across the population. To purchase a property right providing a level π of tenure security, a type-e household incurs a cost C (π, e). This “tenure cost function” is increasing in π as more secure property rights come at higher fees, and in e as more socially distant households face higher fees to acquire a given level of tenure security.7 We further assume that the marginal cost of tenure security is increasing with distance to the admin- ∂2C istration, implying ∂π∂e > 0, and that the cost of tenure security is convex in π . 6 See Van der Molen and Tuladhar (2007) for a description of corruption in land administra- tions in a selection of Asian, African, and Central and Eastern European countries. 7 C (π, e) could be regarded as a tenure security premium captured by the land administration. This is consistent with bureaucrats designing complex systems to induce agents to transfer some of the rents to them (Antwi and Adams, 2003). 7 The tenure cost function is represented on Figure 1 below for two different agents.8 C Π,e Π Πmin Πmax Figure 1: Tenure cost functions of well-connected (e=.1) and poorly-connected (e=.9) households We can now present the timing of the model. In a first stage, households decide whether to purchase land in the city and the type of property rights (if any at all) they purchase from the land administration. Because of multiple sales, conflicts emerge over land. In a second stage, each plot of land is adjudicated to one buyer only. Households who bought land in the first stage remain in the city if they are able to enforce their property right and stay in the rural area otherwise. A type-e household who bought a plot in location x with tenure security π in the first stage has a probability π of retaining its plot during the second stage. If so, the household then faces the second-stage budget constraint given by Equation (1), in which zu is the consumption of a composite good taken as the numeraire. zu + xt + R(x) + C (π, e) ≤ yu (1) With probability 1 − π , the household loses the plot it purchased and remain in the rural area. It faces the second-stage budget constraint given by Equation (2) in which zr is the consumption of the composite good. The amount initially paid 8 Figure 1 and all the other figures in this section are drawn for the specifications and parameter values used in our base case simulation (see Section 4 below). 8 to buy the plot and the property right are both lost. zr + Ra + R(x) + C (π, e) ≤ yr (2) Under the assumption that the consumption of land is fixed, the only endogenous argument in the household’s utility function is its consumption of the composite good z . We assume without loss of generality that u(z ) = z . The expected utility of a household purchasing land in the city (at location x) is therefore: E(u) = πzu + (1 − π )zr (3) In this setting, a type-e household purchasing land in a given location x, chooses its level of tenure security π and its anticipated consumption levels zu and zr in each state of the nature so as to maximize its expected utility function (3) subject to budget constraints (1) and (2). Recognizing that budget constraints must be saturated at the optimum, the household’s optimization program conditional on x and R(x) simplifies to the choice of π which maximizes the objective function: E(u)(π, x, e) = π [yu − xt] + (1 − π ) [yr − Ra ] − R(x) − C (π, e) (4) Solving the model requires identifying, in equilibrium, which households purchase land in the city, where, and with which level of tenure security. It also requires determining the expected utilities of all households, and the profile of land rents prevailing in the city. Note that our equilibrium definition may have several households choose a same location ex-ante due to the possibility of multiple sales but, ex-post, after the conflicts arising from multiple claims are resolved, only a fraction of the initial buyers will effectively reside in the city. The equilibrium is determined ex-ante as all decisions are made in the first stage, anticipating outcomes in the second stage. We solve the model in two steps. In a first step, we parametrize by x and determine, for a given household of type e, the optimal choice of π as function of x and R(x). In a second step, we account for competition for land in a context of multiple sales and determine the equilibrium mapping between household types and land 9 purchase locations (including multiple purchases of a same plot). We then derive all the endogenous variables of the model. These two steps are presented sequentially in the two subsections below. 3.2 Tenure choice Let us denote π ∗ (x, e) the optimal level of tenure security chosen by a type-e household in location x. π ∗ (x, e) is a solution to the maximization problem: max E(u)(π, x, e) (5) π ∈[πmin ,πmax ] Since the tenure cost function is convex in π , the expected utility function (4) is concave in π and therefore has a unique maximum reached in π = π ∗ (x, e) ∈ [πmin , πmax ]. If π ∗ (x, e), is an interior solution to the maximization problem, it must verify the first order condition obtained from differentiating Equation (4): ∂C yu − xt − (yr − Ra ) = (π, e) (6) ∂π The above equation simply states that the optimal level of tenure security must equate the marginal cost of tenure security with the marginal gain from tenure security improvement. Because the tenure cost and the land rent are paid in both states of nature, the gain associated with a marginal increase in tenure security is simply the difference between the urban wage net of commuting costs and the rural wage net of the agricultural land rent as expressed by Equation (6). If the marginal gain and the marginal cost of tenure security do not intersect over [πmin , πmax ] then π ∗ (x, e) is a corner solution. If the marginal gain is greater (respectively smaller) than the marginal cost of tenure security over [πmin , πmax ], then the optimal level of tenure security is π ∗ (x, e) = πmax (respectively π ∗ (x, e) = πmin ). Differentiating Equation (6) with respect to x and e, we derive proposition 1 (proof in Appendix A). Proposition 1. The demand for tenure security is non-increasing with phys- ical distance to the CBD and social distance to the land administration. Where π ∗ 10 is differentiable, we have: ∂π ∗ (x, e) ∂π ∗ (x, e) ≤ 0 and ≤0 ∂x ∂e The intuition for these results is straightforward: Consider first a household of given type e. Inspection of Equation (6) shows that the relative gain from residing in the city as opposed to the rural area, yu − xt − yr + Ra , increases with proximity to the CBD. The closer the location to the CBD, the greater the saving on commuting costs and thus the stronger the incentive to seek secure tenure so as to be able to stay in the city. Now consider a location x, because the ∂2C marginal cost of tenure security is increasing in e ( ∂e∂π > 0), a household which is socially more distant from the land administration will choose to purchase property rights that are less secure than a household which has better connections. We now derive the household’s demand for tenure security throughout the city which, due to possible corner solutions in the optimization program, may be characterized by a piecewise function (proof in Appendix A). Proposition 2. The household tenure choice function x → π ∗ (x, e) can be defined by parts over at most three zones in the city: - a central zone defined by x ≤ x(e) where a type-e household chooses π ∗ (x, e) = πmax , - an intermediate zone defined by x(e) < x < x(e) where a type-e household chooses π ∗ (x, e) ∈ ]πmin , πmax [, given by Equation (6), - a peripheral zone defined by x(e) ≤ x where a type-e household chooses π ∗ (x, e) = πmin . Proposition 2 states that there are threshold locations, x(e) and x(e), below and beyond which a household’s optimal tenure choice is a corner solution. In locations up to x(e), transport costs are small and the relative gains to residing in the city area high, so that the household chooses the highest level of tenure security. In locations beyond x(e) transport costs are high and relative gains to being in the 11 urban area low, the household therefore chooses the lowest level of tenure security. Between these two thresholds, the household chooses intermediate values of tenure security. The optimal tenure choice π ∗ (x, e) is represented on Figure 2a below as a function of distance to the CBD for a given value of e. Π x,e x Πmax xe xe Πmin x e xe xe Π Πmax Πmin Π Πmax Π Πmin (a) Demand for tenure security as a (b) Tenure threshold locations x(e) and function of location (e = 0.2) x(e) as functions of e Figure 2: Household demand for tenure security Appendix A explicits how these functions are derived from the maximization of Equation (5) and shows that x(e) and x(e) are decreasing in e. Observe that the three zones may not necessarily exist for all values of e. For some sufficiently high e for instance, the central and intermediate zones may not exist if for all values x ≥ 0, π ∗ (x, e) = πmin . Figure 2b graphs tenure threshold locations as functions of household type. For each location x plotted on the vertical axis, it shows which household type e plotted on the horizontal axis would choose the minimum, intermediate, and maximum tenure security options. 12 3.3 The Land Market Equilibrium Now that we have determined how tenure choice is affected by location and type, we solve the land market equilibrium. We proceed in four steps. We first determine how much each household is willing to pay for a plot in all possible locations. Taking into account competition on the land market, we then derive the overall city structure (i.e. household locations and tenure zones), the city fringe beyond which no plots are purchased, and finally land prices and household utilities. 3.3.1 Bid Rents The bid rent of a type-e household is defined as the maximum payment that the household would be willing to make to purchase a plot in location x in order to be indifferent between all locations and achieve a given level of expected utility ν (e). Inverting the indirect utility function (4), we obtain the bid rent as a function of π ∗ (x, e), presented in the previous subsection. The bid rent function expresses the maximum land rent that an individual of type e would be willing to pay to reside in location x so as to achieve utility ν (e) and taking into account its optimization of tenure security. It is expressed as Equation (7). Ψe (x, ν (e)) = π ∗ (x, e)[yu − xt] + [1 − π ∗ (x, e)][yr − Ra ] − C (π ∗ (x, e), e) − ν (e) (7) Differentiating the bid-rent function with respect to x results in Proposition 3. Proposition 3. Bid-rent functions are downward sloping and convex ∂ Ψe (x, ν (e)) = −π ∗ (x, e)t < 0 (8) ∂x Proposition 3 states that a household located in x is willing to pay more for a location marginally closer to the CBD because of the expected marginal saving in commuting costs under the household’s optimal level of tenure security. This is a modified version of the Alonso-Muth-Mills condition which states that, under complete tenure security, savings in land consumption should exactly compensate 13 for transport costs (Fujita, 1989).9 In our setting, introducing tenure insecurity, makes bid-rents flatter compared to the standard monocentric model without tenure insecurity,10 which will have a dampening effect on equilibrium land prices. As a result of Proposition 2, the bid rent Ψe can be defined piecewise over at most three zones in the city: - In the central zone defined by x ≤ x(e), the type-e household chooses π ∗ (x, e) = πmax , the slope of the bid rent is −πmax t and the bid rent is a linear function of x. - In the peripheral zone defined by x(e) ≤ x, the household chooses π ∗ = πmin , the slope of the bid rent is −πmin t and the bid rent is a linear function of x. - For x(e) < x < x(e), the bid rent is strictly convex.11 Over the portion of the city where the household would choose intermediate levels of tenure security, the convexity of the bid-rent function reflects the willingness to pay a higher land price for a plot marginally closer to the CBD, with the increment compensating for the marginal increase in tenure security in addition to the marginal saving in transport costs. This is represented on Figure 3 below. To continue our demonstration, we now establish the following Lemma by differentiating Equation (8) with respect to e, and using the fact that π ∗ is a non-increasing function of e (see Proposition 1). The intuition is that lower-type households demand higher levels of tenure security due to their advantage in terms of land administration fees. They thus have an incentive to bid more for a location marginally closer to the CBD. 9 In the standard urban economics model, the Alonso-Muth-Mills condition strictly speaking actually equates the marginal transport cost increment with the marginal saving in land con- sumption. Our result in Equation (8) follows from the assumption that land consumption is constant and normalized to unity and accounts for the likelihood of keeping the plot (and having to commute to the city center). The standard urban economics model can thus be viewed as a specific case of our broader model under which tenure security is certain. 10 Ψe Constraining π (x, e) to be equal to 1 would lead the standard condition that ∂∂x = −t 11 ∂ 2 Ψe ∂π ∗ ∂π ∗ Since ∂x2 (x, ν (e)) = −t ∂x > 0 with ∂x < 0 following Proposition 1. 14 xe xe x xe xe Figure 3: The bid rent function for household e = 0.2 Lemma 1. Wherever they may intersect, the bid rent of a lower type-e house- hold has a steeper or same slope than that of a higher type-e household. ∂ 2 Ψe ∂π ∗ (x, ν (e)) = −t (x, e) ≥ 0 (9) ∂x∂e ∂e We can now characterize the land market equilibrium. The requirement in competitive land use models that land gets allocated to the highest bid also holds in the presence of multiple sales.12 When comparing purchase location choices, we use the standard result that, in equilibrium, agents are ranked by order of bid-rent steepness: agents with steeper bid-rents bid away other agents to more remote locations (Fujita, 1989). Under Lemma 1, households purchase plots in order of increasing type. Although this is indeed true in our model, the determination of the spatial equilibrium is more complex than in the standard case and requires addressing three specific issues. First, we are dealing with a continuum of households instead of a discrete number of agents, and must compare a continuum of bid rents. Building on an assignment problem first analyzed by Beckmann (1969), several papers in the urban economics literature have developed methods to ensure that the hypothesized mapping be- tween types and space results from competition in the land market (Brueckner et al., 2002; Selod and Zenou, 2003; Brueckner and Selod, 2006; Behrens et al., 2014). We resort to a similar approach. Second, the model has multiple sales, 12 In the model, multiples sales occur at identical prices to ensure profit maximization of sellers. 15 which means that there are more transactions than plots available in the city (there are more buyers than sellers). In order to define the equilibrium mapping between households purchases and locations, we will need to account for the way the risk of multiple sales is attenuated by the purchase of property rights and how these choices translate into land use. Third, the unambiguous ranking of house- holds according to type is valid only for bid rents that intersect over isolated points. In our model bid rents may overlap over a whole interval. This happens in zones of the city where the optimal tenure choice of households yields a common corner solution. Over such zones, the bid rents all have the same slope (see Equation (8) and Figure 3) irrespective of household type. It is possible to determine which households purchase a plot in that zone but not their exact location within the zone. This indeterminacy means that there can be an infinity of spatial configura- tions in equilibrium. Fortunately, this will not affect the model in any significant way since these equilibria share the same general spatial structure of the city and all the other endogenous variables of the model will take the exact same value. 3.3.2 City structure We now identify the spatial distribution of tenure situations throughout the city as well as the mapping between households and locations of purchased land. Confronting the piecewise bid rents of households described in Proposition 2 and the condition that households must be ranked in decreasing order of their bid rents’ steepness, we obtain Proposition 4 (proof in Appendix B). Proposition 4. In equilibrium, the city is divided into at most three different “tenure zones” : - A “secure tenure zone” in the central section of the city (x ∈ [0, x]) where households with the strongest social ties to the land administration (e ≤ e) reside and purchase the most secure type of property rights (π ∗ = πmax ). - A “precarious tenure zone” in the intermediate section of the city13 (x ∈ x, x[) where households with intermediate values of e (e ∈]e, e[) 13 The term “precarious” is used in a country like Mali for instance to designate a range of 16 reside and purchase property rights with intermediate levels of tenure security π ∗ (x, e). - An “insecure tenure zone” in the peripheral section of the city (for x = x to the city fringe) where the households with the weakest ties to the administration (e ≥ e) reside and purchase the lowest level of tenure security from the administration (π ∗ = πmin ). A city may exhibit either one, two or three zones depending on the model’s parameters and the exact location of households within the secure zone and the insecure zone (if they exist) are undetermined. Without loss of generality, we consider only the equilibrium in which households are ordered by increasing e. With this ordering, and accounting for multiple sales, we derive Proposition 5. Proposition 5. The equilibrium mapping x(e) between types and locations verifies the differential equation and initial conditions described by Equation (10). dx = π ∗ (x(e), e)N and x(0) = 0 (10) de Proposition 5 maps types and purchase locations considering that N de buyers, with type between e and e + de, purchase land dx located between between x(e) and x(e) + dx. For an infinitesimal de, each plot is bought with tenure security π ∗ (x(e), e). Equating the quantity of available land with the number of “suc- cessful buyers” results in dx = π ∗ (x(e), e)N de. Solving this differential equation requires an initial condition given by x(0) = 0 since we consider the case in which households are ordered by type. We now determine the thresholds e, x, e and x which characterize city struc- ture. The resolution is presented graphically on Figure 4 which superimposes the mapping x(e) and the tenure threshold functions represented on Figure 2b. The intersection between x(e) and x(e) gives e and x. Figure 4 shows that for any household with e < e, we have x(e) < x(e) and π ∗ (x, e) = πmax . Therefore, x(e) = πmax N e for e < e.The intersection of x(e) with x(e) gives e and x. For a revocable permits to occupy (known as “precarious titles”), which stand in between the absence of any statutory right (very insecure tenure) and the registered ownership title (very secure tenure). 17 x xe xe x x xe e e e Figure 4: Determination of tenure category zones in the city household with e > e, we have x(e) < x(e), such that they purchase land in the insecure portion of the city and have tenure security πmin . For e > e, x(e) is a linear function of slope N πmin . Finally, for households with e < e < e, we have x(e) < x(e) < x(e), implying that they reside in the precarious portion of the city and demand intermediate levels of tenure security. 3.3.3 City composition and urban fringe Figure 4 represents the type/location mapping as if all households in the economy purchased a plot in the city. There is no reason for this to be the case in equilibrium and we need to determine which households purchase land and the spatial extent of the corresponding market (which will define the true domain of definition for the mapping function). There are two conditions that characterize the city fringe. First, by definition of the land market equilibrium, there is no discontinuity in rural and urban land prices at the city fringe xf . The second condition is the open city assumption, requiring that household of type ep located at the urban fringe at the equilibrium (or the “last” household purchasing land in the city) is indifferent between purchasing a plot at the urban fringe and not attempting migration. These conditions are given by Equation (11). R(xf ) = Ra and ν (ep ) = yr − Ra (11) To show the existence of ep , we write the indirect utility of a household e purchasing 18 a plot at the city fringe, using Equation (11) and the mapping function x(e). ν (e) |R(x(e))=Ra = π ∗ (x(e), e)[yu −x(e)t]+[1−π ∗ (x(e), e)][yr −Ra ]−Ra −C (π ∗ (x(e), e), e) We then differentiate this expression with respect to e. Using the envelope theorem, we obtain Equation (12) which is is always negative given that dx de (e) > 0. dν dx ∂C |R(x(e))=Ra = −π ∗ (x(e), e) (e)t − <0 (12) de de ∂e The unique value ep is such that all households with a smaller e purchase land in the urban area, whereas all others do not attempt migration.14 Depending on the value of ep relative to e and e as previously defined, the sections of the city described in Proposition 4 may or may not exist. ep determines the city fringe xf through Equation (13).15 xf = x(ep ) (13) The urban fringe is represented on Figure 5. It is not possible to come up with an analytical expression for the city fringe except in the particular case when the cost of buying minimal tenure security is the same for all households (C (πmin , e) = C (πmin )) and the three tenure zones exist (e < ep ). In this case we can derive an explicit value for xf by solving Equation (11) considering that R(xf ) can be written as Ψe (x, ν (e)) (as given by Equation (7) with x = xf , π ∗ = πmin and ν (e) = yr − Ra ).16 3.3.4 Land prices and utilities We determine the utilities ν (e) and the land-rent curve R(x) (which is defined as the upper envelope of bid rents in equilibrium). The utilities of a continuum of households are determined using an equilibrium condition which states that the 14 ep is equal to 1 if for all values of e, ν (e) |R(x(e))=Ra > yu − Ra . 15 Because of unit land consumption, xf is also the number of “successful buyers”. 16 We obtain xf = yu − t yr − Ra (1−πmin )+C (πmin ) tπmin , where the second term is negative. In this particular case, urban extent is thus smaller than in the standard model with complete tenure security where it can be shown that the city extends up to xf = yr − t yu . This property bears no generality as simulations outside this particular case exhibit situations in which the urban extent is larger in our model. 19 x xf x x e e e ep Figure 5: Determination of the city fringe mapping x(e) is consistent with land being allocated in equilibrium to highest bids: R(x) = max Ψ[x, ν (e)] |x=x(e) (14) e This reflects competition for land and trivially holds on the secure and insecure portions of the city since households which purchase land in those zones have the same bid-rents. In what follows, we assume that all three zones exist17 (e < ep ) and consider each zone sequentially, starting with the most remote, and apply land-rent and expected utility continuity at the thresholds identified in Proposition 4. The insecure tenure zone In the insecure zone [x, xf ], households of type e ∈ [e, ep ] purchase land with the lowest level of tenure security πmin . Equation (15) gives their bid-rent. Ψ(x, ν (e)) = πmin [yu − xt] + [1 − πmin ][yr − Ra ] − C (πmin , e) − ν (e) (15) Since all households have the same bid rent over [x, xf ], Equation (15) is also the analytical expression for R(x) on this portion of the city. The border condition equates the urban and rural rents at the fringe, such that rent and utility are given 17 A similar approach can be applied in the other configurations with less than three zones. 20 by Equation (16) with K1 = πmin (yu − xf t) + (1 − πmin )(yr − Ra ) − Ra . R(x) = πmin (xf − x)t + Ra and ν (e) = −C (πmin , e) + K1 (16) The precarious tenure zone Households of type e ∈ [e, e] purchase land in the precarious section of the city [x, x] and choose intermediate values for π ∗ . Each household’s location is given by the mapping x(e) and must satisfy Equation (14). We derive the first order condition of Equation (14) by considering Equation (7) for x = x(e) and differentiating the resulting expression with respect to e.18 Using the envelope theorem, we find that the first order condition of Equation (14) is equivalent to Equation (17). dν ∂C ∗ (e) = − [π (x(e), e), e] (17) de ∂e Solving this differential equation determines ν (e) for all e ∈ [e, e] and enables us to determine R(x), which is continuous in x due to the continuity of ν in e. e ∂C ∗ ∀e ∈ [e, e], ν (e) = − [π (x(e), e), e] + K2 (18) e ∂e The function ν (.), defined in Equation (18), is continuous such that K2 = −C (πmin , e) + K1 . The mapping is an invertible function over [e, e] and its in- verse is e(x), such that the bid-rent over [x, x] is given by Equation (19). R(x) = π ∗ (x, e(x))[yu − xt]+[1 − π ∗ (x, e(x))][yr − Ra ] − C (π ∗ (x, e(x)), e(x)) − ν (e(x)) (19) The secure tenure zone Households of type e ∈ [0, e] purchase land over [0, x] and choose π ∗ = πmax . Their bid rents are given by Equation (20). Ψ(x, ν (e)) = πmax [yu − xt] + [1 − πmax ][yr − Ra ] − C (πmax , e) − ν (e) (20) Since their bid rents are confounded in the secure zone, we equate Ψ(x, ν (e)) 18 Differentiating a second time with respect to e, rearranging the terms, and using (10), we can show that the second order condition always holds. 21 and Ψ(x, ν (e)). This yields ν (e) + C (πmax , e) = ν (e) + C (πmax , e) where ν (e) is provided by the limit when e → e+ of the function ν over [e, e]. Plugging ν (e) + C (πmax , e) back into Equation (20) and using Equation (18) to express e ν (e), we obtain Equation (21) with K3 = C (πmax , e) + e ∂C∂e [π ∗ (x(e), e), e] + K2 . R(x) = πmax (yu −xt)+(1−πmax )(yr −Ra )−K3 and ν (e) = −C (πmax , e)+K3 (21) The land market equilibrium is represented in Figure 6 below.19 Other aggregates from the model such as the utilities, the land administration’s revenue and the total income of landowners are presented in Appendix C. Rx x Secure x Precarious x Insecure xf Figure 6: The equilibrium land rent R(x) and the threshold values x, x, and xf 19 The predictions of our model that more secure forms of tenure are located closer to the city center and that the price gradient is steeper for secure forms than for insecure forms of tenure are confirmed by an empirical study of land transfers in Bamako, Mali (Durand-Lasserve et al., 2015). To our knowledge, it is the only study so far which has documented the spatial patterns of the continuum of tenure rights at the scale of a single city. 22 4 Comparative statics 4.1 Population, transport costs and wage differentials We discuss here the effects on city patterns of marginal changes in the parameters of the model (see the demonstrations in Appendix D). We consider a growing population (dN > 0), decreasing unit transport cost (dt < 0) and a rising urban income relative to the rural income (d(yu − yr ) > 0), which effects are presented in Table 1 below. We are able to determine how changes in the values of these parameters affect the size and the composition of the secure zone of the city but the effects on the precarious and insecure zones are in several instances ambiguous. We find that an increase in the overall population leads to an increase in the size x of the secure section of the city but to a decrease in the share e of households purchasing secure property rights. This is because an increase in N increases the number of well-connected households purchasing land close to the city center, pushing away other households towards the periphery of the city where they choose lower levels of tenure security. If transport costs are reduced, or the urban rural wage differential is increased, the size of the formal section of the city will also increase but the share of households in the secure section of the city will in this case increase rather than decrease. This is because these variations in transport costs or wages increase the gains a household can expect from purchasing land in the city, providing households with an incentive to purchase higher levels of tenure security for any given location, which inflates both the zone over which households purchase secure rights and the share of households doing so. In the specific case where the city has three zones, we show that city size xf increases with the urban rural wage differential and as the unit transport cost decreases (as the city becomes more attractive). On the other hand, the share ep of households purchasing land in the city decreases as N or yu − yr increase, or when t decreases, reflecting the fact that households with a smaller e are taking up more space (either because they are more numerous or because they demand more secure tenure). 23 Table 1: Comparative statics Population Transport costs Wage differential dN > 0 dt < 0 d(yu − yr ) > 0 dx + + + de − + + dx + + + de − ? ? dxf 0 + + dep − − − General case for x, e, x and e; city with three zones for xf and ep . 4.2 Land administration policies The key parameter in the model which affects city patterns is C (π, e), which describes the cost paid by a household e to establish a property right ensuring security π over a plot of land. This parameter is a function such that it is difficult to say anything about how variations of C affect the city structure and the land market in the general case. To get some grip on this question, we parameterize C and resort to simulations to illustrate the predictions of our model and the likely general equilibrium effects of policies. We specify the cost function C (e, π ) as a function of two parameters K and θ which affect the “level” and “equity” of the cost function. The details of the simulations are given in Appendix E. The base case scenario is constructed assuming that the administration serves the interests of a small elite. Examples of such clientelism abound in many different countries (Van der Molen and Tuladhar, 2007; Deininger and Feder, 2009) and it makes sense to make this situation the initial stage or benchmark from which we will explore the desirability and feasibility of policies. The base case is obtained by maximizing the expected welfare of the best-connected household e = 0. We find 24 that the well-connected households benefit from a land administration that treats households as unequally as possible. This is due to a pecuniary externality: the more expensive tenure security is made for higher types, the smaller their demand for tenure security and the more depressed their bid rents (because of the negative tenure insecurity premium which materializes in the form of a flatter bid-rent). In this context, it is thus easier for lower types to bid away higher types to the periphery of the city without having to bid too much for central locations (prices are depressed throughout the city and in fine the land rent in x = 0 is low). In a sense, the land administration makes land available under advantageous conditions for the elite while protecting them from the competition of others on the formal segment of the market. 4.2.1 More affordable property rights It is often argued that excessive land administration fees push people into infor- mality. To explore this argument within the framework or our model, we simulate a policy that makes fees smaller without making them less inequitable. There are three interesting impacts we can highlight from those simulations (see Appendix E for the details). Making land administration fees more affordable results in: 1. An increase in (and displacement of ) the spatial extent of the secure and precarious sections at the expense of the insecure zone. 2. An increase in land rents associated with tenure security improvement. This comes from the shrinking of the insecure portion of the city, which pushes all prices up, and the capitalization in land prices of more secure tenures over the precarious and secure portions of the city. 3. A disparate effect on the expected utility. The expected utility of households in the “social vicinity” of the land administration decreases because of the higher land rents they now face. The expected utility of households residing in the secure and precarious sections of the city increases, as the decrease in the cost of acquiring secure property rights outweighs increase in land rent they face. 25 4.2.2 More equitable property rights It is often advocated to increase transparency in the land administration in order to avoid bribes and discretionary treatment. We simulate such a policy by making the land administration more equitable, without making them less expensive for the median household, but increasing them for the low e households and decreasing them for the high e households (see Appendix E for the details). Making land administration fees more equitable results in: 1. A larger precarious section of the city at the expense of both the spatial extent of the secure and insecure sections of the city. 2. A nuanced effect on rent curves. At the periphery of the city, locations held under insecure tenure under the base case are now held under precarious tenure, which raises the price of land for these plots and exerts an upward pressure on land rents in more central locations. But because of increased fees, households who formerly had secure tenure now demand lower tenure security. There is thus a countervailing force towards the decrease in land prices for central locations. 3. A nuanced effect on expected utility. Households who have switched from insecure to precarious tenure (in the middle of the distribution) experience an increase in expected utility, whereas household with lower types face a decrease in their expected utility due to an increase in tenure costs (which may result in the choice of a lower level of tenure security) which is not completely compensated by the decrease in land rents in more central locations. 26 5 Conclusion In this paper, we introduced an endogenous choice of property rights in the standard urban land use model. To our knowledge, this is the first attempt to build a theory of residential informality that allows for a continuous spatial representation of the urban space under endogenously determined land prices. The result is an augmented monocentric land use model which generates the standard urban economics model as a particular case under the assumption of costless and fully secure property rights. Considering the more general case where property rights are costly and provide only limited tenure security, we proposed a suitable framework to analyze urban land markets and land tenure patterns in developing countries. Our approach makes the following four contributions: First, analyzing land tenure from a systemic perspective shifts the focus from an analysis of informal land markets in isolation from formal land markets to one where the formal and the informal sectors interact: In our model, although households may hold land under a continuum of tenure situations (from very insecure to very secure), they compete for land over the same city. A previous theoretical paper to have explored such interactions was Brueckner and Selod (2009) who argued, in a context of inelastic land supply, that the greater the informal squatting sector in a city, the greater the prices in the formal sector because of a “squeezing effect” (due to squatters settlements using up scarce land in the city). The assumptions in the present model, however, are different as land supply is not restricted and because informal dwellers are not squatters but actually pay a competitive price to use land under a chosen level of tenure security. Interestingly, holding city size constant, the predictions of our spatial model differ from Brueckner and Selod (2009): Because of competition for land and the lower value put by the market on insecure tenure, informal tenure in peripheral locations contributes to depressing rather than increasing land prices in central locations where formal housing locates.20 20 Cai et al. (2018) also model the interaction between formal and informal land use in a spatial equilibrium but this is done in a simplified framework that does not account for the direct capitalization of tenure insecurity in land prices. 27 A second lesson to be derived from the model is the central role of land administration fees in shaping cities. In line with recent research measuring the elasticity of demand for property rights with respect to regularization or registration fees (Monkkonen, 2012; Ali et al., 2013), our model suggests that land administration fees, which can reach the same order of magnitude as land prices, can significantly alter the tenure choices and location decisions of households within a city. The systemic approach also highlighted that land policies may entail responses from land markets that affect households beyond the targeted populations, with possible trade-offs regarding who benefits from the policy. An important implication for policy makers should thus be whether pricing by the land administration is compatible with urban planning objectives. Further studies on the willingness to pay of households for property rights, and on the impact of land administration fees on land markets will certainly be needed. A third lesson is that clientelism in the land administration may determine who has access to secure land in the city. In our model, a political elite benefits from imposing costly fees to others in order to weaken competition for land in the formal sector. This result is in line with the theoretical finding of Sonin (2003) who, in a different context, argues that the rich and the poor have different preferences for property rights institutions. Although political scientists and urban economists have long studied how cities serve political clienteles (Ades and Glaeser, 1995), discrimination in the land administration and how it affects city patterns and composition has been little studied. More generally, empirical studies that identify the losers and winners from the complex processes and barriers to obtain secure land in the cities of developing countries would be very useful. In this respect, Durand-Lasserve et al. 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The optimization program is given by Equation (22). maximize E(u) = πzu + (1 − π )zr zu ,zr ,π subject to zu + xt + R(x) + C (π, e) ≤ yu zr + Ra + R(x) + C (π, e) ≤ yr πmin ≤ π ≤ πmax zu ≥ 0, zr ≥ 0 Recognizing that the two budget constraints must be binding at the optimum and considering only parameters of the model which ensure that the consumption of the composite good at equilibrium is positive, the program simplifies to a maximization of expected utility E(u)(π, x, e) with respect to π as set out in Equation (4). Under our assumption that the tenure cost function C (π, e) is convex with respect to π , the expected utility function E(u)(π, x, e) is concave and there- fore has a unique maximum over the segment [πmin , πmax ]. Observing that ∂ E(u) ∂π (π, x, e) = yu − xt − yr + Ra − ∂C∂π (π, e), there are three cases depending on whether the solution π ∗ (x, e) is an interior solution or one of two corner solutions: - If there exists a value π ∗ ∈]πmin , πmax [ which satisfies the first order condition yu − xt − yr + Ra = ∂C ∂π (π ∗ , e), it maximizes the expected utility since the latter is concave. This corresponds to the case in which the expected utility increases over [πmin , π ∗ ] and decreases over [π ∗ , πmax ]. - If yu − xt − yr + Ra ≤ ∂C ∂π (πmin , e), then yu − xt − yr + Ra ≤ ∂C ∂π (π, e) for ∂C all values of π ∈ [πmin , πmax ] since ∂π is an increasing function of π . This means that ∂ E (u) ∂π (π, x, e) < 0 and implies that E(u)(π, x, e) decreases over ∗ [πmin , πmax ]. π = πmin therefore maximizes the expected utility. . 34 - If yu −xt−yr +Ra ≥ ∂C ∂π (πmax , e), then yu −xt−yr +Ra ≥ ∂C ∂π (π, e) for all values ∂C of π ∈ [πmin , πmax ] because ∂π is an increasing function of π . E(u)(π, x, e) increases over [πmin , πmax ] and π ∗ = πmax maximizes the expected utility. yu x1 t yr Ra yu x2 t yr Ra C Π e,Π yu x3 t yr Ra Π Πmin Π e, x2 Πmax Figure 7: Graphical determination of π ∗ (x, e) for e = 0.2 and three different values of x : x1 = 0, x2 = N 2 and x3 = N . We can now present the proofs of Propositions 1 and 2, starting with Proposition 2. Proof of Proposition 2 For a given x and e, whether we have an interior or a corner solution to the program (5) depends on whether or not yu − xt − yr + Ra and ∂C ∂π (π, e) intersect over [πmin , πmax ]. Figure 7 illustrates this for e = 0.2, three different values of x, and the specific quadratic tenure cost function used throughout the paper for illustration purposes. The curve π → ∂C ∂π (π, e) is the upward sloping one. The curve π → yu − xt − yr + Ra is a horizontal line which shifts downwards with higher values of x. - For a small value of x (case 1 on the graph with x = x1 = 0), π → ∂C ∂π (π, e) is always below the horizontal line π → yu − xt − yr + Ra and does not intersect with it. In this case, the solution is π ∗ = πmax . - For an intermediate value of x (case 2 with x = x2 = N 2 ) the two curves intersect and their intersection provides the optimal tenure choice for this household and location. 35 - For a high value of x (case 3 with with x = x3 = N ), π → ∂C∂π (π, e) is always above the horizontal line π → yu − xt − yr + Ra leading to the corner solution π ∗ = πmin . We can thus define two threshold values x(e) and x ¯(e), such that for x ≤ x(e), the optimal solution is always πmax . For x ≥ x ¯(e), the optimal solution is always πmin . Between these two threshold values, the optimal solution is an interior solution. Proof of Proposition 1 The intuition for Proposition 1 can be understood from inspection of Figure 7. Since the line π → yu − xt − yr + Ra shifts downwards with greater values of x, ∂2C π ∗ (x, e) is a decreasing function of x. We have assumed that ∂π∂e > 0, which ∂C means that the curve π → ∂π (π, e) shifts upwards with greater values of e, which implies that π ∗ (x, e) is also a decreasing function of e. Analytically, we can apply the implicit function theorem to the first order condition (6) for x ∈]x(e), x ¯(e)[. We obtain: ∂π ∗ −t (x, e) = ∂2C <0 (22) ∂x ∂π 2 (π ∗ (x, e), e) ∂2C ∂π ∗ − ∂π∂e (π ∗ (x, e), e) (x, e) = ∂2C <0 (23) ∂e ∂π 2 (π ∗ (x, e), e) 2 Both these equations are perfectly defined as we have assumed ∂ C ∂π 2 (π ∗ (x, e), e) to ¯(e)[, x → π ∗ (x, e) is constant be strictly positive. Outside of the interval ]x(e), x and the derivative is equal to 0. The derivative of π ∗ (x, e) with respect to x is not ¯(e). defined in either x(e) or x 36 B Land market equilibrium Proof of Proposition 4 In land use models, spatial patterns are determined by the comparison of bid rent functions, with households with the steepest bid rents locating closer to the CBD (Fujita, 1989). In our model however, the bid rent functions of different households cannot systematically be ranked by order of decreasing steepness. This is because for some locations x, households of different types may have the same demand for tenure security and therefore have bid rents of the same slope (see Equation (8)). Although the equilibrium location of these households cannot be determined precisely, it is nevertheless possible to determine the relative positions of three groups of households defined as follows: - Group 1 includes all households who choose π ∗ = πmax at equilibrium (x(e) is the equilibrium location of a type-e household) G1 = {e|π ∗ (x(e), e) = πmax } - Group 2 includes all households who choose π ∗ ∈]πmin , πmax [ G2 = {e|π ∗ (x(e), e) ∈]πmin , πmax [} - Group 3 includes all households who choose π ∗ = πmin G3 = {e|π ∗ (x(e), e) = πmin } In equilibrium G1 , G2 or G3 could be empty sets. We will show that all households in group 1 have smaller values of e than all households in group 2, who in turn all have smaller values of e than all households in group 3. ∀e1 ∈ G1 , e2 ∈ G2 , e3 ∈ G3 : e1 < e2 < e3 37 We will also show that all individuals in group 1 are located at the vicinity of the city center and households in group 3 at the periphery of the city while those in group 2 are located in the middle. ∀e1 ∈ G1 , e2 ∈ G2 , e3 ∈ G3 : x(e1 ) < x(e2 ) < x(e3 ) Finally, although we will be unable to identify the exact location of households in groups 1 and 3, we will show that the individuals in group 2 locate in order of increasing e moving outwards from the CBD. ∀{ea , eb } ∈ G2 , ea < eb ⇔ x(ea ) < x(eb ) Let us start by considering the different ways bid-rent functions can intersect. We have shown that the bid rent of a type-e household is linear over [0, x(e)], strictly convex over [x(e), x x(e), +∞[. From Lemma 1, as ¯(e)], and linear over [¯ also illustrated by Figure 8,21 we know that the bid rent of the household with the smaller e dominates (eventually weakly) to the left of the intersection (in the direction of the CBD) whereas the bid rent of the household with the larger e dominates (eventually weakly) to the right of the intersection (in the direction of the city periphery). Figures 8a and 8b illustrate the intersection of two bid rents confounded over either of their linear segments: - On Figure 8a, the bid rent Ψe1 dominates Ψe2 weakly: over [0, x(e1 )[, we have Ψe2 < Ψe1 , and over [x(e1 ), +∞[, we have Ψe2 = Ψe1 . - On Figure 8b, the bid rent Ψe2 dominates Ψe1 weakly: over [0, x(e2 )[, we have Ψe2 = Ψe1 , and over [x(e2 ), +∞[, we have Ψe2 > Ψe1 . Figure 8c illustrates two bid rents intersecting in a single point xi : - The bid rent Ψe1 dominates Ψe2 over [0, xi [ ∀x < xi , Ψe2 (x) < Ψe1 (x) 21 Note that the bid rents in Figure 8 are not represented for the households’ respective equilib- rium utilities but for arbitrary levels of utility that generate the different cases for intersection. 38 - The bid rent Ψe2 dominates Ψe1 over ]xi , +∞[ ∀x > xi , Ψe2 (x) > Ψe1 (x) x x x e1 x e1 x e2 x e2 x e1 x, e2 x x x x x e1 x e2 xi (a) Weak dominance (b) Weak dominance (c) Strong dominance Figure 8: Intersections between the bid rents of households e1 = 0 (blue) and e2 = 0.1 (red). With this in mind, let us now consider the case where, in equilibrium, a type-e1 household is in group 1 and a type-e2 household is in group 2 as defined above, and let us show that e2 is necessarily greater than e1 . We demonstrate our result reductio ad absurdum by assuming that e2 is smaller than e1 and showing that it is impossible. First of all, by definition of group 1, household of type e1 chooses π = πmax at an equilibrium location x(e1 ). Given that e1 chooses πmax in position x(e1 ), it must be that x(e1 ) < x(e1 ) < x(e2 ) because we have assumed that e2 < e1 and x(.) is a decreasing function. So in x(e1 ), both bid rents Ψ(e1 ) and Ψ(e2 ) are linear and of slope −tπmax . Furthermore, since x(e1 ) is the equilibrium location of the type-e1 household, the bid rent of the type-e2 household can be either equal or inferior to that of household e1 in x(e1 ). Given these assumptions, there are two possibilities (we denote ν (e1 ) and ν (e2 ) the equilibrium utilities of the two households): - If Ψe2 (x(e1 ), ν (e2 )) < Ψe1 (x(e1 ), ν (e1 )), then Ψe2 and Ψe1 do not intersect at all and Ψe2 is completely dominated by Ψe1 over all values of x. This is due to the our assumption that e2 < e1 and therefore Ψe1 has a stronger curvature than Ψe2 . - If Ψe2 (x(e1 ), ν (e2 )) = Ψe1 (x(e1 ), ν (e1 )), then Ψe2 and Ψe1 are are confounded 39 over [0, x(e1 )] and then Ψe2 is strictly dominated by Ψe1 for all values of x > x(e1 ). Second, observe that the two households’ bid rents must intersect in equilibrium because no household can outbid another household over the whole city (otherwise the outbid household would not be able to purchase a plot in equilibrium). This means that for any other tenure choice than πmax and location x > x(e1 ), household e2 is outbid by household e1 which contradicts the fact that household e2 is in group 2. Therefore e2 is larger than e1 . Now that we know that e1 < e2 , and that the type-e1 household chooses π ∗ = πmax in location x(e1 ), the bid rents Ψe1 and Ψe2 can only intersect in two ways: - If Ψe2 (x(e1 ), ν (e2 )) < Ψe1 (x(e1 ), ν (e1 )), which implies that the two bid rents intersect in x∗ > x(e1 ). Ψe2 dominates (eventually weakly) Ψe1 over [x∗ , +∞]. e2 is therefore located further away than e1 at equilibrium. - If Ψe2 (x(e1 ), ν (e2 )) = Ψe1 (x(e1 ), ν (e1 )), then in x(e1 ) < x(e2 ) which means that at equilibrium e2 locates further out than e1 . So in equilibrium, e2 is located further out than e1 : x(e1 ) < x(e2 ). A similar demonstration (not reported) would show that: ∀e2 ∈ G2 , e3 ∈ G3 : e2 < e3 and x(e2 ) < x(e3 ) Given these properties and the continuity of our problem, we can define four unique ¯ and x < x threshold values e < e ¯ which characterize groups 1, 2 and 3 such that: - Households with e smaller than e all choose maximum tenure security πmax , and are located in the central segment of the city defined by x < x. These are the households of the group 1 we previously defined. - Households with e between e and e ¯ all choose intermediate levels of tenure security π ∈ [πmin , πmax ], and are located in the middle segment of the city defined by x < x < x ¯. These households belong to group 2. 40 ¯ all choose minimal tenure security πmin , and - Households with e larger than e are located in the peripheral segment of the city defined by x > x ¯. These households belong to group 3. We are now left to show that within the central and peripheral segments, the location of each household cannot be determined precisely, but that in the intermediate area it can. First of all, all households located in the central or peripheral segments choose levels of tenure security equal to πmax or πmin . This means that their bid rents have the same slope and are all confounded over the area. It is therefore impossible to determine the exact location of each household since they will all be indifferent between the locations.22 We consider two households ea and eb within group 2 and assume that ea < eb . As previously stated, the two bid rents must intersect so that no household is outbid by another. Moreover, in order for households ea and eb to be located in the interme- diate zone of the city, their equilibrium bid-rent must dominate in at least one lo- cation of that zone. This means that in x(ea ), Ψea (x(ea ), ν (ea )) ≥ Ψeb (x(ea ), ν (eb )) and in x(eb ), Ψeb (x(eb ), ν (eb )) ≥ Ψea (x(eb ), ν (ea )). Given the typology of possible intersections between bidrents of ea < eb , given that neither household chooses πmax or πmin in equilibrium, it can only be that the intersection of both bid rents is a point, before which Ψea dominates strictly and after which Ψeb dominates strictly. Therefore x(ea ) < x(eb ), for any ea < eb . QED. C Aggregates and Surplus Utilities To illustrate the functioning of the model, it is useful to also discuss the shape of the utility distribution. Figure 9 below represents the ex ante utility (left panel) 22 Different households will have different levels of utility, but each household will be indifferent between all locations within the considered segment. 41 and ex post utilities (right panel) as a function of type. On the right panel, over [0, ep ], there are two ex post utilities which are conditional on the realization of each state of the nature. The higher curve is for households who invested in an urban plot and managed to keep it whereas the lower curve is for those who lost their plot. On [ep , 1], the curve represents the utility of households who did not purchase a plot in the city and remained in the rural area all along. On this segment, there is no distinction between the two states of the nature: both curves are flat and take the same constant value equal to the expected utility over the same segment shown on the left panel. While the ex ante utility is decreasing, the ex post utilities show an irregular pattern. The decrease in both ex post utilities for households in the formal sector (e < e) reflects the higher fees households with higher types need to pay to the administration. Interestingly, in this simulation, both ex post utilities of households with precarious tenure (e ∈ [e, e]) increase with the type because their purchase of lower tenure security dominates their cost disadvantage (which means that higher types make a smaller rather than a greater payment to the land administration). In the informal section of the city (e ∈ [e, ep ]), the ex post utility decreases with the type for households who keep their plot but increases with the type for household who lose it. This is due to the fact that on this segment of the land market, variations in utilities are explained by the comparison of rents and transport costs: rents (which are determined in the first stage) compensate for expected not ex post transport costs, hence a decrease in ex post utilities for higher types who manage to keep their plots located further away from the city center. 42 U U e e e e ep e e ep Figure 9: Ex ante (left panel) and ex post utilities of house- holds (right panel) Revenues The land administration’s revenue gathered from collected fees is given ep by I = 0 C (π ∗ (x(e), e), e)N de. The aggregate income of landown- ers is the sum of all rents paid by buyers (including multiple sales): e R = 0 p R(x(e))N de. The ex post welfare of households who bought an urban plot and are able to keep it in the second stage of the model is e Wurban,kept = 0 p π ∗ (x(e), e)[yu − x(e)t − R(x(e)) − C (π ∗ (x(e), e)), e)]N de. The ex post welfare of households who bought an urban plot but lose it to a multiple e sale is Wurban,lost = 0 p [1 − π ∗ (x(e), e)][yr − Ra − R(x(e)) − C (π ∗ (x(e), e)), e)]N de. The welfare of rural households who did not purchase a land plot in the city 1 is Wrural = ep (yr − Ra )N de.23 Finally, the ex ante welfare of households e purchasing a plot in the city is Wurban = 0 p [[1 − π ∗ (x(e), e)][yr − Ra − R(x(e)) − C (π ∗ (x(e), e)), e)] + π ∗ (x(e), e)[yu − x(e)t − R(x(e)) − C (π ∗ (x(e), e)), e)]]N de which is mathematically equivalent to the sum of Wurban,kept and Wurban,lost . Given the linearity (and a fortiori quasi-linearity) of the utility function, we can define a total surplus S = I + R + Wrural + Wurban (from ex ante definitions). The surplus is constant but its share among the different agents varies. In this respect, 23 To understand the formulas for Wurban,kept and Wurban,lost , observe that we are summing over e not x, hence the multiplicative terms π ∗ (x(e).e) and 1 − π ∗ (x(e).e). Also observe that there is no distinction between ex ante or ex post for Wrural . 43 our model is relevant to analyze the redistributional effects of land administration practices. D Comparative statics We are interested in the variation of city structure patterns with respect to the parameters of our model, namely the total population N , the trans- port cost t and the urban rural-wage differential yu − yr . We first show how ¯(e) and x(e), and then their intersections. these parameters affect the curves x(e), x The curves x(e) and x ¯(e) are defined by the equations : 1 ∂C x(e) = [yu − yr + Ra − (πmax , e)] t ∂π 1 ∂C x ¯(e) = [yu − yr + Ra − (πmin , e)] t ∂π The curve x(e) is determined by the following differential equation : x (e) = N π ∗ (x(e), e) x(0) = 0 A variation in N . We will now show that for N1 < N2 , the corresponding solutions to the differential equation x1 (e) and x2 (e) verify x1 (e) < x2 (e) for e > 0. We have x1 (0) = x2 (0) = 0, and x1 (0) = N1 π ∗ (0, 0) < x2 (0) = N2 π ∗ (0, 0) because π ∗ (x, e) does not depend on N . There is therefore a small neighborhood around 0 over which the function x2 (e) − x1 (e) is strictly positive. We can find ε > 0 such that x2 (e) − x1 (e) is strictly positive over ]0, ε[. 44 Let us note E the set of values e ≥ ε for which the two curves x1 and x2 intersect. E = {e ≥ ε | x1 (e) = x2 (e)} Let us suppose that E is a non-empty set. It therefore has an infimum e0 ≥ ε and because x1 (e) − x2 (e) is continuous, e0 belongs to E and x1 (e0 ) = x2 (e0 ). In e0 : x2 (e0 ) − x1 (e0 ) = (N2 − N1 )π ∗ (x1 (e0 ), e0 ) > 0. Therefore, there is a neighborhood to the left of e0 over which x2 (e) − x1 (e) is strictly negative. We can find η > 0 such that x2 (e) − x1 (e) is strictly negative over ]e0 − η, e0 [. Therefore, the function x2 (e) − x1 (e) is strictly positive over ]0, ε[ and strictly negative over ]e0 − η, e0 [ which implies that ε < e0 − η . By the theorem of intermediate values, there is a value e ∈ [ε, e0 − η ] for which x2 (e) − x1 (e) is equal to zero. This contradicts the definition of e0 . E is therefore an empty set and x2 (e) > x1 (e) for all e > 0. The equations which define x(e) and x ¯(e) do not depend on the parameter N and are decreasing functions of e: an increase in N therefore leads to an increase in x and x ¯. Figure 10 illustrates this property: an increase in ¯ and a decline in e and e N leads to an upward shift in x(e) (red curve) without changing x(e) and x(e). Therefore, for N2 > N1 : e2 < e1 and x2 > x1 e2 < e1 and x2 > x1 A variation in t. A decrease in transport costs t or an increase in the urban-rural wage differential yu − yr have the same effect on city structure since they both increase the gains from living in the city. Let us consider two different unit transport cost t1 > t2 , and x1 (e) and x2 (e) the solutions to the corresponding differential equations. 45 x xf x2 x1 x2 x1 e ee 21 e2 e1 ep2 ep1 Figure 10: Variations of x(e) with N . The functions x(e) and x ¯(e) are decreasing functions of t, such that x1 (e) < x2 (e) and x ¯1 (e) < x ¯2 (e) for all values of e for which all functions are not equal to zero. As x and e are defined by the intersection of x(e) and N πmax e we have the following inequalities: e1 < e2 , x1 < x2 and x1 (e) and x2 (e) are confounded over [0, e1 ]. Furthermore, there is a neighborhood of e1 over which x2 (e) > x1 (e) : ∀e ∈]e1 , e2 [, x2 (e) = N πmax > x1 (e). As with N , let’s note E the set of values e ≥ e1 + ε for which the two curves x1 and x2 intersect. E = {e ≥ e1 + ε | x1 (e) = x2 (e)} Let us suppose that E is a non-empty set. It therefore has an infimum e0 ≥ e1 + ε and because x1 (e) − x2 (e) is continuous, e0 belongs to E and x1 (e0 ) = x2 (e0 ). In e0 either x2 (e0 ) > x1 (e0 ) or x2 (e0 ) = x1 (e0 ) = N πmin because t2 < t1 and that π ∗ is defined by the intersection of yu − yr + Ra − xt and ∂C ∂π when the intersection exists. Furthermore, x1 (e0 ) < πmax as e0 > e1 . - If in e0 , x2 (e0 ) = x1 (e0 ) = πmin , then e0 = e ¯ 2 and x1 and x2 are confounded over [e¯2 , 1]. This leads to e ¯1 < e¯ e1 , e 2 . Over ]¯ ¯2 [, x2 (e) > x1 (e) which in turn leads to x2 (e) − x1 (e) < 0. The intermediate value theorem contradicts the definition of e0 . 46 - If in e0 , both derivatives are not equal to πmin then it must be that x2 (e0 ) > x1 (e0 ) as yu − yr + Ra − x2 (e0 )t2 > δ1 + Ra − x2 (e0 )t1 . By a similar argument as previously, this means that x2 (e)−x1 (e) is strictly negative in a neighborhood around e0 , which contradicts the definition of e0 due to the intermediate value theorem. We have therefore shown that for t1 > t2 , x2 (e) > x1 (e). ¯, but has an ambiguous A decrease in transport costs leads to an increase in x, e, x ¯. e effect on e ¯ is defined by the equation x e) = x(¯ ¯(¯ e). If we differentiate this equation with respect to t, we can show that the sign of ∂ ¯ e ∂t is not determined: ¯ ∂x ∂e ¯ ¯ ∂x ¯ ∂x ∂e ∂x e) + (¯ e) = (¯ e) + (¯ e) (¯ ∂t ∂e ∂t ∂t ∂e ∂t ¯ ∂x ∂e ¯ ∂x ∂x ¯ ∂x ( (¯e) − (¯ e)) = e) − (¯ e) (¯ ∂t ∂e ∂e ∂t ∂t These effects are illustrated on Figure 11. x xf2 xf1 x x2 1 x2 1 e e12 e e1e2 ep1 ep2 Figure 11: Variations of x(e) with t. The effect on the urban fringe. In the most general case, we cannot say what the effect of an increase in N , yu − yr or a decrease in t will have on the size of the city xf and the population who tries to migrate ep . 47 However, in the case in which (i) C (πmin , e) = C (πmin ), and (ii) there is an informal area to the city, the urban fringe is defined by the following equations : yu − yr Ra (1 − πmin ) + C (πmin ) xf = − t tπmin xf = x(ep ) An increase in N has no effect on the city size xf , but leads to a decrease in ep the last migrant to move to the city. This reflects the fact than an increase in N means that there are more well-connected individuals using up the city space, and that this is pushing the less well-connected to the outskirts where the gains from living in the urban area are smaller. An increase in yu − yr or a decrease in t leads to an increase in city size xf , and a decrease in ep . As the gains from living in the city center increase, well-connected households will be choosing more secure tenure and thereby taking up more of the urban space, hence pushing less well-connected households towards the outskirts. E Simulation of the cost function Specification of the cost function The key feature of our model is the pricing of property rights by the administration through the tenure cost function C (π, e). In our simulations, we use the following separable specification ˜(e) C (π, e) = c(π )f with 2 c(π ) = K (π 2 − πmin ) ˜(e) = f (em ) + θ[f (e) − f (em )] f f (e) = 1 + δe where K , θ, δ and em are parameters that can be adjusted to account for different practices of the land administration. 48 K is a multiplicative parameter that affects all households equally. If K is increased then the fees are increased in the same proportion for all households. On the contrary, f ˜(e) is a type-specific multiplicative term that accounts for the differential treatment of well-connected and poorly-connected house- holds. To understand how this works, first consider the case of θ = 1 so that f˜(e) = f (e) = 1 + δe. In this case, a type-0 household pays c(π ) to purchase a property right π , whereas a type-e household pays c(π )(1 + δe) > c(π ). Increasing δ will reinforce the advantage of well-connected households (those with a low e) relative to poorly-connected ones (those with a high e) who will pay a higher fee. Choosing θ < 1 operates a rotation of the f (.) function around a pivotal household of type em ∈ [0, 1]. If θ = 0 and em = 0 then all households are treated as the most-favored household (type e = 0). If θ = 0 and em = 0.5 then all households are treated as the median household in the distribution of types. The base case: a clientelistic land administration We can now turn to the presentation of the base case which we constructed assuming that the administration serves the interests of a small elite. Examples of such clientelism abound in many different countries (Van der Molen and Tuladhar, 2007; Deininger and Feder, 2009) and it makes sense to make this situation the initial stage or benchmark from which we will explore the desirability and feasibility of policies. The base case is obtained by maximizing the expected welfare of the best- connected household e = 024 with respect to K and θ, taking δ as fixed. We find that the well-connected households benefit from a land administration that treats households as unequally as possible. This is due to a pecuniary externality: the more expensive tenure security is made for higher types, the smaller their demand for tenure security and the more depressed their bid rents (because of the negative tenure insecurity premium which materializes in the form of a flatter bid-rent). In this context, it is thus easier for lower types to bid away higher types to the 24 Alternatively we could consider any other objective function, for instance to satisfy a fraction of the population e ∈ [0, ec ] representing the political clientele (where ec is the “last”” client). 49 periphery of the city without having to bid too much for central locations (prices are depressed throughout the city and in fine the land rent in x = 0 is low). In a sense, the land administration makes land available under advantageous conditions for the elite while protecting them from the competition of others on the formal segment of the market. As a matter of fact, household e = 0 would always be better off with a higher value of δ and with θ = 1. Assuming an exogenous value δ = 9 and considering yu = 111, 000, yr = 70, 000, Ra = 15, 000, t = 1, 000, πmax = 0.95, pimin = 0.45 and N = 50, we obtain K0 = 8, 948 and θ0 = 1.25 More affordable property rights It is often argued that excessive land administration fees push people into informality. To explore this argument within the framework or our model, we simulate a policy that would make fees smaller without making them less inequitable, i.e. keeping θ0 = 1). We simulate a case where K is decreased by 25 percent (K1 = 0.75K0 = 6711). There are three interesting impacts we can graphically highlight from those simulations: First, the effects on the demand for tenure security as a function of distance to the CBD are represented on Figure 12 below. The blue curve is the base case demand for tenure security whereas the red curve is demand under a lower K (when fees are more affordable to all). The simulation results in an increase in (and displacement of ) the spatial extent of the secure and precarious sections at the expense of the insecure zone.26 Second, making land administration fees more affordable results in an increase in land rents associated with tenure security improvement (see Figure 13 below). This comes from the combination of two mechanisms: (i) the shrinking of the insecure portion of the city, which mechanically pushes all prices up throughout the city, 25 With θ = 1, the value of em is irrelevant. 26 Because in our simulations C (e, πmin ) = 0, xf will not vary with K as long as there are three zones, in which case the city size remains the same but the insecure zone is smaller than before. With a more drastic reduction in K , the insecure zone will completely disappear and the overall city size will increase. Indexing by 0 the base case and 1 the policy simulation, we would have: x1 > xf 1 and xf 1 > xf 0 (simulation not shown). 50 Πx x x0 x1 x0 x1x f Figure 12: Tenure zones when property rights become more affordable (base case in blue; policy in red, with K1 = 0.75K0 ) and (ii) the capitalization in land prices of more secure tenures over the precarious and secure portions of the city. The greater the decrease in K the higher the equilibrium land prices (simulations not shown). Third, looking at figure 14 below, we see that the impact of lowering land administration fees has disparate effects on the expected utility. Lowering K results in a decrease in the expected utility of households in the “social vicinity” of the land administration (this is understandable since the base case is built by choosing the value of K that maximizes the expected utility of the household e = 0). However, lowering K results in an increase in the expected utility of households with relatively higher types who reside in the secure and precarious sections of the city. The reason for this is that, for the lowest type household, the increase in land rent outweighs the decrease in the cost of acquiring secure property rights (since they were paying a low fee anyway), resulting in an expected utility loss, while it is not the case for higher-type households.27 We find that these effects are magnified for even lower values of K (not shown). Observe that lowering K makes no difference for the expected utility of households in the rural sector and in the insecure section of the city (because all households in these 27 This occurs because land rents are shifted upwards additively and irrespective of the exact type of the households who reside in the secure section of the city, whereas fees are reduced multiplicatively, favoring households with relatively higher types. 51 Rx x x0 x1 x0 x1x f Figure 13: Land rents when property rights become more af- fordable (base case in blue; policy in red, with K1 = 0.75K0 ) areas have the same expected utility yr − Ra by virtue of C (πmin , e) = 0). We also simulated a policy making the menu of fees more equitable. We obtain an increase of the spatial extent of the precarious section of the city at the detriment of the secure and insecure sections and non-monotonic effects on land prices (see results in Appendix E). More equitable property rights It is often advocated to increase transparency in the land administration in order to avoid bribes and discretionary treatment. Measures include for instance the menu of fees posted on a wall. We simulate such a policy by making the land administration more equitable by changing the parameter θ.28 We explore the case of θ1 = 0.5, which makes acquiring property rights cheaper than under the base case for poorly-connected households of type e > em and more expensive for well-connected households with e < em . All curves for this policy are represented in green on the figures below. We have the three following comments: 28 Under θ = 1, which we considered up to now, the value of em was irrelevant. The value of em reflects the choice of authorities regarding which household should be the pivot in rotating e −e the fee schedule. We consider em = e0 + 0 2 0 = 0.31, where the pivot is the median type of households with semi-formal tenure. 52 u e e e0 e1 e0 e1ep1 ep0 Figure 14: Ex ante utilities when property rights become more affordable (base case in blue; policy in red, with K1 = 0.75K0 ) First, more equitable fees makes the precarious section of the city larger (at the expense of both the spatial extent of the secure and insecure sections of the city). This is because the highest level of tenure security becomes less affordable for some households who could afford it under the base case and because precarious tenure becomes more affordable for households who would optimally choose insecure tenure under the base case (see Figure 15). The tenure security of poorly-connected households is thus improved at the expense of some well-connected households. Second, effects on the land rent curve are nuanced, as illustrated by Figure 16 below. At the periphery of the city, some locations held under insecure tenure under the base case are now held under precarious tenure, which raises the price of land for these plots and exerts an upward pressure on land rents in more central locations (due to competition for land). But because of increased fees, households in the precarious section with type e < em and a fraction of households who formerly had secure tenure (who a fortiori have a type e < em ) now demand lower tenure security. There is thus a countervailing force towards the decrease in land prices for central locations. In our simulation, the latter effect dominates the former throughout the secure tenure section and on part of the precarious tenure 53 Πx x x1 x0 x0 x1 xf Figure 15: Demands for tenure security in each location fol- lowing a reduction in θ (θ1 = 0.5) (base case in blue; policy in green) section of the city, resulting in lower land prices there than under the base case. Rx x x1 x0 x0 x1 xf Figure 16: Land market prices following a reduction in θ (θ1 = 0.5) (base case in blue; policy in green) Third, looking at the expected utility of households (Figure 17), there are nuanced effects as well. Households who have switched from insecure to precarious tenure (in the middle of the distribution) experience an increase in expected utility, wheras household with lower types face a decrease in their expected utility due to an increase in tenure costs (which may result in the choice of a lower level of tenure security) which is not completely compensated by the decrease in land 54 rents in more central locations. u e e e1 e0 e0 e1 eep1 p0 Figure 17: Ex-ante utility following a reduction in θ (θ1 = 0.5) (base case in blue; policy in green) 55