Gray Zones: Slums and Urban Structure in Developing Countries

Gray Zones: Slums and Urban Structure in Developing Countries Tiago Cavalcanti University of Cambridge and Sao Paulo School of Economics - FGV tvdvc2@cam.ac.uk Daniel Da Mata IPEA daniel.damata@ipea.gov.br Marcelo Rodrigues dos Santos Insper marcelors2@insper.edu.br Version: September 1, 2017. Preliminary and Incomplete. Abstract We construct an equilibrium model of a city with heterogeneous agents to explain why in urban areas slums vary in location, size, and quality. We implement counterfactual experiments to assess the role of regulation and transportation policies. We show that policies can have unintended consequences. Transportation policies decrease the number of slums close to the business district as improvements in road networks and public transportation are capitalized into housing prices. However, such policies can substantially increase slums in areas away from the business center district due to immigration of poor households from rural areas. Keywords: Slums; Rural-Urban Migration; City Growth; Regulation; Transportation Policies. J.E.L. codes: O18, O15, R52. We have benefited from discussions with Cezar Santos. All remaining errors are of our responsibility. 1

1 Introduction The goal of this paper is to understand how public policies affect the formation and location of slums, a major phenomenon in large cities in developing countries. There is no consensual definition of slums, but according to the United Nations a slum household is characterized by the lack of at least one of the following conditions: access to improved water and sanitation; sufficient living area; durability of housing; and security of land. Evidence from the United Nations (2013) show that slum prevalence is highest in Sub- Saharan Africa where about 62 percent of the urban population live in slums but it is also present in a large scale in Western Asia (25 percent), South Asia (35 percent), and Latin America and the Caribbean (24 percent). Slums are also present at a smaller scale in cities in the developed world. For instance, some cities in the United States have also a large prevalence of slums, such as Hidalgo County in Texas, in which about 6.6 percent of the population live in the colonias with no proper water supply and secured land. Even though slum dwellers generally present worse social outcomes, there is a large variability in the number of slum dwellers and quality of life for different slums between cities and within a city, i.e., not all slums are the same. In other words, slums vary in the extensive margin, intensive margin, and location. Indeed, within cities in developing countries, slums can be located close to the city center or farther away from labor market opportunities. Besides, in the same city, slum areas can include small units made of wood and waste materials, or even places with several floors of informal construction. Despite the widespread presence of slums across cities in developing country, not much is written and known about how public policies affect their prevalence and location. This article aims to contribute to the literature and policy discussions on slum formation and location. We introduce a spatial general equilibrium model of a city to understand the formation and location of slums in developing countries, and to perform ex-ante evaluation of the impacts of policy interventions on slums. In our model, households who are heterogeneous on their labor productivity have to choose whether to migrate to the city, where to locate within the (circular) city s boundaries, and to decide on the type of housing tenure (formal or informal housing). In our model, there is a simple trade-off from choosing informal housing over formal housing. Households choosing informal housing units do not benefit from public goods and must protect the informal plot from property theft or eviction, but they dodge complying with building regulations and paying property tax. There is an opportunity cost to protect the informal plot as households must forgone labor income to spend part of their time protecting the plot. 2

For each location on the city space the trade-off between formal and informal housing generates two income thresholds which separate formal and informal housing agents. Firstly, formal housing means compliance with a myriad of building constraints, such as minimum lot size and height restriction. Therefore, only households with high enough level of income are able to live in the formal housing sector. Secondly, households in the informal housing sector need to protect their plot and end up forgoing labor income. As a result, high income households do not choose informal housing. Given that we take into account rural-urban migration and the decision on where to live within a city, the model is able to explain important features of urbanization in developing countries. Richer households tend to live in formal housing units, near the city center, and surrounded by a greater availability of public goods. Informal housing agents tend to live in urban slums in the outskirts of the city, and they only live close to the center when they can occupy marginal unauthorized land. Not all slum units are the same in our model because of the intensive margin of housing space consumption, which varies with distance to the city distance and institutional characteristics of the city. Formal housing rents are higher than informal housing ones (cf., Jimenez, 1984; Friedman, Jimenez, and Mayo, 1988). We also show that both formal and informal housing rents decrease as distance from the city center increases. Even though the model is able to account for general patters of urbanization, it offers flexibility to explain the location of slums in different cities. For instance, while Mumbai and Rio de Janeiro have their largest slums in centrally located area, in São Paulo most of slum formation takes places in areas farther from the center. We implement policy simulations to assess the role of changing regulation and transportation policies. We show that welfare-boost policies can have unintended consequences as they may increase urban land values and change both tenure and location decisions. In the simulations, we concentrate the effects of infrastructure policies because our model features the formation and location of slums. Transportation policies decrease the number of slums close to the business district as improvements in road networks and public transportation are capitalized into housing prices. However, such policies can increase substantially slums in areas away from the business center district due to immigration of poor households from rural areas. Paragraph on optimal policy. As we cope with the location of households within a city, our model is related to the standard Alonso-Muth-Mills model in urban economics (cf., Alonso, 1964; Muth, 1969; Mills, 1967). While there are a number extensions of the AMM model to developed coun- 3

tries (see Brueckner, Thisse, and Zenou (1999)), we contribute by explaining spatial patters in developing countries. We should stress up that our model is not an extension of the AMM model, though. We are specially connected to Kopecky and Suen (2010) and Baum-Snow (2007) as they are both extensions of the AMM-type of model and explore its quantitative implications. The main difference from previous works using the AMMtype of model is that we introduce the household choice between living either in formal or informal housing sector. The literature also shows that transportation impacts location decisions. Glaeser, Kahn, and Rappaport (2008) indicate that better public transportation attracts the poor to live in central cities. In the present model, transportation costs change households disposable income and housing prices in general equilibrium, and thus alter their location choice. Precisely, lower transportation costs reduce slum formation in the city center and induce slum formation in the outskirts because transportation policies attract migrants willing to live in a more attractive city. This paper is connected to the literature on urban squatting (cf., Jimenez, 1985; Field, 2007; Brueckner and Selod, 2009; Brueckner, 2013; Cavalcanti, Da Mata, and Santos, 2016). We differ from Cavalcanti, Da Mata, and Santos (2016) by introducing location decisions as well as rural-urban migration in a slum formation model. These additions provide new properties to the model and allow us to study the effect of new policies on slum formation. Finally, this paper also connects with articles on the ex-post evaluation of slum upgrading interventions (e.g., Lanjouw and Levy (2002), Field (2005), Field (2007), Di Tella, Galiani, and Schargrodsky (2007) and Galiani and Schargrodsky (2010)). Section 2 presents the model and Section 3 discusses the steps taken to calibrate the model and estimate its parameters. Section 4 shows the quantitative analysis to measure how policies affect the location and number of slums. Section 5 contains concluding remarks. 2 The Model There is a mass P of individuals who could live in the city. The city has n = 1,..., N locations that differ in their distance, d n D = {d 1,..., d N }, to the Central Business District (CBD) and in the area available for housing, L n L = {L 1,..., L N }. All households living in location n have the same road network and public transportation so that they all spend the same commuting cost to the central business district. We interpret this as a circular city with a fixed radius S such that d n = N n S, n = 1,..., N, and the total land area of the city 4

is S = πs 2. The area at location n is S n = πs2 N 2 (2n 1), but only L n = σ n S n with σ n [0, 1] is available for housing. Households are ex-ante distinguished by their labor productivity and they have the option to choose two types of housing tenure, i.e., formal or informal. The inclusion of informal housing allows the model to explain important features of cities in developing countries. 2.1 The Environment Households. There is a continuum of households of measure P, each of whom with labor productivity λ who can live in the city or in a rural area. The distribution of labor productivity in this economy follows a Pareto cumulative distribution Υ(λ) with scale parameter λ 0 > 0 and shape parameter ɛ > 1 such that Υ(λ) = 1 ( λ0 λ ) ɛ. Each household has one worker and occupies one housing unit, so that the number of people equals both the number of households and housing units. There is a value V of living in a rural area and working in the agricultural sector. We assume that this value is independent of the productivity level λ. Therefore, individuals will live in this city only if it is optimal to do so. Households who decide to live in the city have to choose their location in the city and their housing tenure type: formal or informal housing. They obtain utility over non-housing consumption (a homogeneous good c), leisure (l), and housing services (s j ), where j equals F if formal housing and I if informal housing. The utility function is specified as U(c, s j ) = c α c s α h j l 1 α c α h, α c, α h (0, 1), α c + α h < 1, and j = F, I. Housing service flow (s j ) entails housing space (h), public goods surrounding the housing unit (G), and a parameter θ such that: Gh s j = θgh if formal housing (j=f), if informal housing (j=i), where θ (0, 1). The expression θg means that informal housing agents do not reap all the benefits from public goods provision within the city boundaries. The parameter θ may be seen as a congestion cost due to living in high density neighborhood (as is typically the case of slums). Household s labor income is wλ, where w is the wage rate per efficiency units of labor, and λ is the labor productivity of the household. Each household has one unit of pro- 5

ductive time and face a cost η(d n ) = η 0 d η 1 n in commuting time, where d n is the location of the house of the household and η 0 and η 1 are parameters. There is also a transportation cost A household allocates her income net of commuting cost wλ(1 η(d n )) between non-housing consumption (c), and housing services (s). The two types of housing entail different costs. Only households in formal housing pay a property tax τ p and a lump-sum fee φ associated with the cost of some public goods. 1 A formal housing unit must also comply with specific building-related requirements: the minimum lot size (MLS). This is a catch up parameter summarizing several building regulations related to formal housing, such as building standards, height restrictions, drainage systems, among others. The MLS is represented by parameter h, which is exogenously set by the local government. 2 If a household decides to live in the informal sector, then they can evade property taxes, τ p and lump-sum fee φ, but she incurs protection costs Ψ(d n, h) = ψ(d n )h, where ψ(d n ) = ψ 0 d ψ 1 n with ψ 0 > 0 and ψ 1 > 0. Therefore protection costs are increasing with the size of the property h and the closer the house is to the central business district. Informal housing lacks property rights and is associated with insecurity due to exogenous risks of eviction and demolitions by the government. Therefore, if an agent chooses to dwell illegally, she will spend 1 ηd n Ψ(d n, h) of her time working, i.e., informal housing agents allocate part of their time to protect their informal plot. This is a reduced form approach to represent the cost of informality which we assume to be decreasing with the distance to the business district. Therefore, on the border of the city the protection cost to live in a slum is lower than in locations closer to the business district. The implicit assumption is that if households living in informal housing do not spend time or resource in protecting their land, then they will be evicted. This implies that spending such time and resource are incentive compatible. Parameter θ and function Ψ(d n, h) may represent other costs of informal housing such as the adverse effect of lack of job networks for slum dwellers. Specifically, θ can be an utility cost associated with stigma to live in a slum and Ψ(d n, h) is associated with any time spent due to living at a slum. Anecdotal evidence on fires in slums supports the idea that slum dwellers spend extra time on (re)constructions after fire. In some cases, the same slum is destroyed by fire more than once in a single year: this was the unfortunate case of Favela Moinho in São Paulo in 2012. 3 In addition, evidence shows 1 We therefore assume that slum dwellers escape from paying for property taxes and the cost of some public goods such as street lights. Caju s slum data supports, for instance, the assumptions of no-payment of property taxes by slum dwellers (94.58% of the households in Caju slum do not pay property taxes). 2 The minimum lot size is exogenous in the model, i.e., we assume that the city and its households take the MLS parameter as given. There are cases where building regulations are nationally mandated. One example is the national law in Brazil which stipulates the minimum lot size of 125m 2. 3 http://g1.globo.com/jornal-nacional/noticia/2012/09/incendio-em-favela-de-sp-deixa-uma-pessoa- 6

that after a fire the slum dwellers also suffer from robbery from opportunistic thefts that try to enter houses taking advantage of the fire despair. 4 In sum, a household faces the following trade-off: if she chooses to live in an informal settlement she avoids paying property taxes (τ p ), formal housing lump-sum fees φ, and complying with the minimum lot size regulations (h). However, informal housing is insecure and therefore she incurs a utility discount per housing space (θ) and an opportunity cost related to informal housing (Ψ(d n, h)). As a result, the informal housing sector is related to the general notion of a slum, where there is insecure land tenure (represented by the opportunity cost Ψ(d n, h)), low infrastructure provision (represented by θ), noncompliance with building regulations (represented by h), and evasion of taxes (τ p ) and other costs associated with public goods provision (φ). Consumption Goods. The homogeneous non-housing consumption good is the economy s numeraire. Its production takes place in the central business district in a competitive environment where all firms have a constant returns-to-scale production function Q(N, K) = BN υ K 1 υ, where υ (0, 1) is the labor share in production, N represents units of labor efficiency, K represents units of capital and B is a total factor productivity parameter of the city. Labor units are supplied by households at the wage rate w. Capital, K, is an elastically supplied factor with rental price r determined outside the city - the city is small relative to the country, which might be integrated to the international capital market. The optimization problem of the representative firm operating in this sector is represented by: π = max N,K {BNυ K 1 υ wn rk}. The first-order condition for K is r = (1 υ)b( N K )υ, while the first-order condition for N is given by w = υb( K N )1 υ. Combining both conditions generates the labor demand of the city: ( (1 υ)b w = υb r ) 1 υ υ. (1) Equation (1) shows that labor demand is perfectly elastic because the rental rate of capital r is exogenous to the city point of view. As a result, the wage rate w in the model is determined solely by the labor demand equation. Observe that in equilibrium profits are zero and firm ownership is unimportant. morta-e-300-desabrigadas.html 4 http://noticias.bol.uol.com.br/brasil/2012/11/27/incendio-em-favela-de-manaus-deixa-545-familiasdesabrigadas.jhtm 7

Housing. Housing space is produced using land and capital. Land in each location L n is available in fixed supply, while capital is elastically supplied with price r. Land belongs to a group of absentee landlords who spend land rents outside the city. Land can also be allocate to agriculture production and the rental rate of land for agricultural production is r a and it is determined outside the city. Developers also live outside the city. The production function of housing space is H j (L j (d n ), K j (d n )) = A j L j (d n ) γ K j (d n ) β, where j {F, I}, γ, β (0, 1) and γ + β < 1. The rental price of land (p L (d n )) is the same at every unit of land in location n. Developers in each location n have the following optimization problem: Π j (d n ) = max {p j(d n )A j L j (d n ) γ K j (d n ) β p L (d n )L j (d n ) rk j (d n )}, L j (d n ),K j (d n ) where p j (d n ) is the housing price of type j {F, I} at location n. The first-order conditions for this problem are: p L (d n ) = γp j (d n )A j L j (d n ) γ 1 K j (d n ) β r a, (2) r = βp j (d n )A j L j (d n ) γ K j (d n ) β 1, (3) for each j {F, I}. Using the first-order conditions for this problem we can define how land in location n is allocated for formal and informal housing, respectively: In addition, we have that: p L (d n ) = γ L F (d n ) L n = L I (d n ) L n = ( Aj p j (d n ) L j (d n ) 1 γ β ( ) 1 pf (d n )A F 1 γ β p I (d n )A I ( ) 1 1 + pf (d n )A F 1 γ β p I (d n )A I 1 ( ) 1 1 + pf (d n )A F 1 γ β p I (d n )A I, (4). (5) ) 1 1 β ( β r ) β 1 β ra, j {F, I}. (6) 8

Housing supply is ( ( ) γ γ ( ) ) 1 β β 1 γ β H j (p L (d n ), r, p j (d n )) = A j p j (d n ) γ+β, j {F, I}. (7) p L (d n ) r Government and Land Law Enforcement. Let e(g) be local government expenditures and assume that e(g) is linear in the public good G, i.e., e(g) = G. The government finances expenditures through a property tax rate (τ p ) and a lump-sum fee φ paid only by formal housing residents. Define E F (d n ; o) as the measure of households who live in formal housing units in location n. We will define E F (d n ; o) and the housing demand function h F (λ, d n ; o) precisely shortly. Government runs a balanced budget and therefore G = N ˆ ( τp R F (d n )h F (λ, d n ; o) + φ ) Υ(dλ), n=1 E F (d n ;o) where R F (d n ) is the housing rent at location n and h F (λ, d n ; o) is the housing size chosen by each formal housing household at location n. Property tax τ p and lump-sum fees φ are government parameters. Note that the local government spending G enters the utility function of all households, but there is a discount in utility, θ (0, 1), for informal housing agents. It has been said that informal housing agents do not pay property taxes. But what is the reaction of the government? Land law enforcement is exogenous in the model. For simplicity, we assume that government spends on public goods and guarantees the security of tenure for those household that have opted for living in formal housing. Then, only households in formal housing can reap all the benefits from public expenditure. We also implicitly assume that there is a collective threat from all time spent by informal housing agents to protect their plots such that the government does not carry out evictions 5. 5 The assumption that the government is capable of protecting the plots of those who stay in the formal housing market (and pay property taxes) is central to the present analysis. How come the government is able to enforce property tax collection, but does not prohibit squatting? Squatting in developing countries was accompanied by huge rural-urban migration. For instance, in Brazil the urban population was 80 million in 1980 and 160 million in 2010. In 30 years, the urban population growth in Brazil was equivalent to the population size of Germany or the combined population of Spain and Canada. In this scenario, the government may not be able to prevent every single land invasion, especially when there is vacant land. At the same time, the government cannot guarantee public good provision and land security for all new residents and, therefore, those who pay property taxes may be the only ones benefiting from government actions. There is evidence of under-service of migrants in Brazil (cf., Feler and Henderson, 2011). Government protection (of plots) could induce voluntary payment of property taxes by households. 9

2.2 Decision Problems and Properties of the Model We define the problem of the household into three steps. First, for a given location n and housing tenure j {F, I} we determine how households choose housing services and consumption to maximize utility subject to the budget constraint. Then, we define the housing tenure choice for each location and finally we determine the location choice. Formal housing problem. Define the vector of prices, policies and institutions by o = (w, r, r a, R F (d n ), R I (d n ), τ p, φ, h, G, η(d n ), ψ(d n ), V, P). Taking w (average city wage) and R F (d n ) (housing rent) as given, households who have chosen to live in formal housing at location n solve the following problem: V F (λ, d n ; o) = max c F,h F [c α F (Gh F) 1 α ], (8) subject to c F + (1 + τ p )R F (d n )h + φ wλ(1 η(d n )), (9) h F h 0, (10) c F 0. (11) The budget constraint (9) states that expenditures on consumption and housing cannot exceed disposable income. Equation (10) means that there is a minimum housing space that stems from minimum lot size (MLS) regulations from a zoning constraint in the city 6. When equation (10) is not binding, the optimal choice of consumption and housing for φ formal households with λ w(1 η(d n )) = λ nb(d n, o) are determined by: h F (λ, d n ; o) = c F (λ, d n ; o) = α (wλ(1 η(d n )) φ), (12) (1 α) (1 + τ p )R F (d n ) (wλ(1 η(d n)) φ). (13) Substituting these two equations into the utility function yields the following value function: V F nb (λ, d n; o) = α α (1 α) 1 α G 1 α (wλ(1 η(d n )) φ) ( (1 + τp )R F (d n ) ) 1 α. (14) 6 Recall that the minimum space constraint given by Equation (10) can incorporate any regulation constraints that affect the density of the housing unit such as height controls and minimum setbacks. 10

When equation (10) is binding, we have that c F (λ, d n ; o) = wλ(1 η(d n )) φ (1 + τ p )R F (d n )h, (15) h F (λ, d n ; o) = h. (16) In this case, the indirect utility function of the household is given by: V F b (λ, d n; o) = [wλ(1 η(d n )) φ (1 + τ p )R F (d n )h] α h 1 α G 1 α. (17) Constraint (10) binds only for sufficient small values of λ. Moreover, there is a minimum value of λ, which is equal to λ(d n ; o) = (1+τ p)r H (d n )h+φ w(1 η(d n λ )) nb (d n, o) 0 such that for any λ < λ(d n ; o) the problem of the household living in formal housing is not well defined. Then, for any λ > λ(d n ; o) we have that: Condition (18) defines (1 α) h F (λ, d n ; o) = max{ (1 + τ p )R F (d n ) (wλ(1 η(d n)) φ), h}. (18) λ MLS (d n ; o) = (1 + τ p )R F (d n )h (1 α) (w(1 η(d n )) φ), (19) such that if λ λ MLS (d n ; o), then V F (λ, d n ; o) = V F nb (λ, d n; o) and if λ λ MLS (d n ; o), then V F (λ, d n ; o) = V F b (λ, d n; o). In addition, we require that the value of living in a formal house in any location n is higher than the value of moving to a rural area, which gives reservation utility V and therefore V F (λ, d n ; o) V. This condition implies that there is a λ F V (d n; o), such that if λ < λ F V (d n; o), then the household prefers to live in the rural area. We can easily show that for any V > 0, then λ F V (d n; o) > λ(d n ; o) and λ F V=0 (d n; o) = λ(d n ; o). Moreover, λ F V (d n; o) = ( ( 1 V w(1 η(d n )) 1 w(1 η(d n )) ) 1 ) α + φ + (1 + τ (hg) 1 α p )R F (d n )h ( ) V((1+τ p )R F (d n )) 1 α + φ α α (1 α) 1 α G 1 α if λ F V < λmls, if λ F V > λmls. The problem of a household choosing formal housing is summarized in the Lemma 1. Lemma 1. For each vector o and λ [ λ F V (d n; o), ), we have that V F : [ λ F V (d n; o), ) D R and h F : [ λ F V (d n; o), ) D [h, ) such that: 11

i. For each location n, there exists a productivity level λ MLS (d n ; o) given by (19) such that if λ [min{ λ F V (d n; o), λ MLS (d n ; o)}, λ MLS (d n ; o)) then h F (λ, d n ; o) = h ; and if λ [max{ λ F V (d n; o), λ MLS (d n ; o)}, ) then h F λ > 0. iii. V F (λ, d n ; o) is continuous and strictly increasing in λ. Moreover, V F (λ, d n ; o) is strictly concave in λ [min{ λ F V (d n; o), λ MLS (d n ; o)}, λ MLS (d n ; o)), and it is linear in λ [max{ λ F V (d n; o), λ MLS (d n ; o)}, ). Finally, lim λ + λ F V (d n;o) VF (λ, d n ; o) = V and when h > 0, then lim λ + λ F V=0 (d n;o) V F (λ,d n ;o) λ =. Proof. See Appendix B. Q.E.D. The black solid line in Figure 1 displays the value function V F (λ, d n ; o) for a given location n. Value functions V F nb (λ, d n; o) and V F b (λ, d n; o) are also displayed in this figure. V F nb (λ, d n; o) is represented by the dashed black line, while V F b (λ, d n; o) is represented by the dotted black line. Informal housing problem. Taking w (average city wage) and R I (d n ) (housing rent) as given, households who have chosen to live in informal housing at location n solve the following problem: subject to V I (λ, d n ; o) = max c I,h I [c α I (θgh I) 1 α ], (20) c I + R I (d n )h wλ(1 η(d n ) Ψ(d n )), (21) Ψ(λ, d n ) = ψ(d n )h I, (22) c I 0, h I 0. (23) This problem leads to the following consumption and housing demand equations: c I (λ, d n ; o) = αwλ(1 η(d n )), (24) λw(1 η(d n )) h I (λ, d n ; o) = (1 α) R I (d n ) + ψ(d n )wλ. (25) The indirect utility for households living in informal housing is given by: V I (λ, d n ; o) = α α (1 α) 1 α (θg) 1 α wλ(1 η(d n )). (26) (R I (d n ) + ψ(d n )wλ) 1 α 12

We also require that living in a slum should be incentive compatible and should give higher utility than living in a rural area such that V I (λ, d n ; o) has to be greater that V. This defines a λ I V (d n; o) such that for any λ > λ I V (d n; o), then the household prefers to live in the informal sector than in the rural area. Proposition 2 summarizes the decision to live in the informal house. Lemma 2. For each vector o and λ [ λ I V (d n; o), ), we have that V I : [ λ I V (d n; o), ) D R and h I : [ λ I V (d n; o), ) D R, such that: i. Informal housing is increasing in λ; ii. V I (λ, d n ; o) is continuous, differentiable, strictly increasing, and strictly concave in λ [ λ I V (d n; o), ). In addition, lim λ + λ I V=0 (d n;o) V I (λ, d n ; o) = 0, and lim λ V I (λ,d n ;o) λ = 0. Proof. See Appendix B. Q.E.D. Value function V I (λ, d n ; o) is displayed by the solid gray line in Figure 1. Income elasticities of formal and informal housing consumption will be important to determine housing choices. Proposition 1. For any location n, let V 0 be such that λ F V (d n; o) < λ MLS (d n ; o). If λ < λ MLS (d n ; o)), then formal housing has income elasticity of zero; and if λ λ MLS (d n ; o), then formal housing has income elasticity of greater than one. For any λ > λ I V (d n; o) Informal housing has income elasticity of less than one. Proof. See Appendix B. Q.E.D. The result of proposition 1 relates with the literature on the lack of incentives to invest in informal and illegal housing units (cf., Kapoor and le Blanc, 2008). For instance, Field (2005) points out that housing renovation increased after a land titling program in Peru and Galiani and Schargrodsky (2010) show that title increased housing investment in Buenos Aires. Housing choice. Let λ V (d n; o) = min{ λ F V (d n; o), λ I V (d n; o)}. Given location n, households will live in formal housing units if V F (λ, d n ; o) > V I (λ, d n ; o). The housing type decision can be described by value function V(λ, d n ; o) and policy function Ω(λ, d n ; o) where V(λ, d n ; o) = max {Ω(λ, d n; o)v F (λ, d n ; o) + (1 Ω(λ, d n ; o))v I (λ, d n ; o)}, (27) Ω(λ,d n ;o) {0,1} 13

for λ [ λ V (d n; o), ). Let s assume h = 0. The following lemma characterizes the housing choice in location n for a given labor ability in a partial equilibrium environment in which prices are exogenously given. Lemma 3. For each location n and each vector o, there exists a unique value function V(λ, d n ; o) and threshold productivity λ(d n ; o) > 0, such that: i. V(λ, d n ; o) is continuous, strictly increasing and concave in λ. ii. If V is sufficiently small such that λ V (d n; o) < λ(d n ; o), then for any λ λ(d n ; o) we have that Ω(λ, d n ; o) = 1, and for any λ [ λ V (d n; o), λ(d n ; o)) we have that Ω(λ, d n ; o) = 0 iii. If V is too large such that λ V (d n; o) > λ(d n ; o), then Ω(λ, d n ; o) = 1 for any λ [ λ(d n ; o), ). In addition: iv. λ(d n ;o) θ > 0; λ(d n ;o) τ p > 0; λ(d n ;o) ψ(d n ) < 0; λ(d n ;o) φ 0; v. λ(d n ;o) w < 0; λ(d n ;o) R F (d n ) > 0; and λ(d n ;o) R I (d n ) < 0. Proof. See Appendix B. Q.E.D. that When h = 0, then inequality V F (λ, d n ; o) V I (λ, d n ; o) 0 for any location n implies ( ) λw(1 η(d n )) φ (1 + θ 1 α λw(1 η(d n )) τp )R F (d n ) 1 α, (28) (R I (d n ) + ψ(d n )wλ) which defines a unique labor productivity threshold, λ(d n ; o) > 0, such that households are indifferent between living formally or informally. In addition, if V is sufficiently small 7 such that λ V (d n; o) < λ(d n ; o) then for any λ between these two threshold productivity levels, we have that households choose to live informally in location n; while if λ [ λ(d n ; o), ) then households prefer to live in formal housing. For given prices, in each location n, housing informality rises with formalization costs (φ) and with the property tax rate τ p ; and it reduces with relative more provision of public goods in slums (higher θ) an with lower protection costs ψ(d n ). Finally, in a given location n, informality reduces when the average income rises (w increases) and when the rental price for informal housing, R I (d n ), rises; while the presence of slums is positively related with the rental 7 Notice that λ V (d n; o) is increasing in V and lim V 0 λ V (d n; o) = 0. 14

Figure 1: Slums. Left graph: Non-binding housing regulations. Right graph: Binding housing regulations. price of formal housing, R F (d n ). When V is too large such that λ V (d n; o) > λ(d n ; o), then there will be no informality in location n. When h > 0, then if the productivity level λ MLS (d n ; o) associated with the minimum lot size in formal housing (from Lemma 1) is lower than λ(d n ; o), then housing regulations do not interfere with the housing tenure mode decision of households and Lemma 1 still applies. This is depicted in the left graph of Figure 1. Households are partitioned into three measures: Those with a small productivity level who prefer to live in the rural area (or other location) than in this location; households with income between λ V (d n; o) and λ(d n ; o) who will live in informal housing; and those with productivity above λ(d n ; o) who prefer to live in formal housing. When h > 0 is sufficiently large such that λ MLS (d n ; o) > λ(d n ; o) as defined by equation (28), then regulations might bind for the decision of the tenure type. This is shown in the right graph of Figure 1 and households are partitioned into five measures. Despite the three previous measures there are two more set of households due to binding housing regulations. In this case, since V F b (λ, d n; o) < V F nb (λ, d n; o) for any λ = λ MLS (d n ; o), then there will be some households that will live informally because they cannot comply with the minimum lot size, even though they would live in the formal housing sector if such regulations were not in place. Those are the households with informal housing due to regulations. In this case, a more restricted housing regulation (higher value for h) implies a large share of infor- 15

mal housing. 8 Another measure of households would prefer to live formally even when regulations are binding. Those are households with constrained formal housing. Location choice. Given the vector o containing prices, policies and institutions, a household with income λ will choose location n to live in order to maximize V(d n ; o) = max d n D {V(λ, d n; o)}. (29) This problem defines the optimal location policy function d n (λ; o) describing the decision of household to where to live. Notice that this decision depends on the following tradeoff. In one hand, households prefer to live close to the CBD because commuting costs (1 η(d n (λ; o))) are lower. In addition, informal housing suffers high protection costs when close to the CBD, since ψ(d n (λ; o)) is higher for smaller distances to the CBD. If all households decide to live close to the CBD, then house prices at the CBD will rise. This would imply that some households might decide then to move away from the CBD in order to enjoy better housing at lower price. Therefore, the location choice depends on general equilibrium effects and requires us to define the equilibrium of this economy. Before that let s determine the measure of households who choose to live in location n in a formal housing mode by: E F (d n ; o) = {(λ [ λ V (d n; o), ) : Ω(λ, d n ; o)d n (λ; o) 1}, (30) such that E F (o) = N n E F (d n ; o) corresponds to the measure of all households living in formal housing. Let (E F ) c (d n ; o) and (E F ) c (o) denote the complement set of E F (d n ; o) and E F (o), respectively. 2.3 Equilibrium In order to complete the analysis of the model, it remains to define the competitive equilibrium. The households optimal behavior was previously described in detail above as well as the government budget constraint. It remains, therefore, to characterize the market equilibrium conditions. There are 5N + 2 prices in this economy. They are: the wage rate, w, the interest rate, r, land price in each location, {p L (d n )} n=0 N, formal and informal house price in each location, {p F (d N ), p I (d n )} n=0 N, and formal and informal rental house price 8 Use the condition V F b (λ, d n; o) V I (λ, d n ; o) = 0 to define λ(d n ; o) and use the implicit function theorem to show that λ(d n ;o) h > 0. 16

in each location, {R F (d N ), R I (d n )} n=0 N. We are assuming that the rental price of capital is determined outside the city boundaries. Therefore, given r and Equation (1) the wage rate w is also determined. Equilibrium in the asset market with no uncertainty means that the housing price equals the discounted value of housing rents: p F (d n ) = R F (d n )/r and p I (d n ) = R I (d n )/r. Equations (2), (3) and (6) for j {F, I} imply that given r and once we determine p j (d n ) (or R j (d n ) by arbitrage), then land price p L (d n ) will be determined. Therefore, we need just 2N market clearing conditions to define rental prices R j (d n ), house prices p j (d n ) and therefore the land price, p L (d n ). Housing market equilibrium. Total housing demand in the informal sector in location n is ˆ λ(d n HI d,o) (R I(d n ); o, P) = P h I (λ, d n ; o)dυ(λ), λ V (d n,o) where λ(d n ; o) is the unique productivity cutoff that partitions households into informal and formal tenures in location n. In the formal sector, total housing demand in location n is ˆ HF d(r F(d n ); o, P) = P h F (λ, d n ; o)dυ(λ). λ(d n ;o) Recall that housing supply is given by Equation (7) and p j (d n ) = R j (d n )/r. The housing market clearing condition at location n for housing type j {F, I} is: H d j (R F(d n ); o, P) = H j (p L (d n ), r, R j (d n )/r), (31) where the left-hand side (LHS) represents the housing demand and the right-hand side (RHS) is total housing supply at location n for tenure mode j. We formally define the spatial equilibrium for this city. Definition 1. A spatial equilibrium for this city is a vector of prices, policies and institutions o = (w, r, r a, R F (d n ), R I (d n ), τ p, φ, h, G, ψ(d n ), V, P) for each location n, such that: (i) the interest rate r is given and profit maximization in the goods market implies that w is given by (1); (ii) no-arbitrage in the housing market such that p j (d n ) = R j(d n ) r for j {F, I}; (iii) developers take their technology and o as given to maximize profits such that land allocated for formal and informal housing in each location n are given by (4) and (5), the rental price of land, p L (d n ) r a, in each location n is given by (6), and housing supply of mode j {F, I} in each location n is determined by (7); (iv) given the vector o households choose their housing location d n (λ; o), housing mode Ω(λ, d n ; o), housing services h j (λ, d n ; o), and consumption c j (λ, d n ; o) in order to maximize utility subject to their budget constraint; (v) the housing market is in equilibrium for 17

Figure 2: Slums. Left Photo: Rio de Janeiro, Brazil. Right graph: São Paulo, Brazil. Concentric Areas: Blue - Business Center District. Black shades are areas with slums each housing mode j and location n - Equation (31) is satisfied; and (vi) the city government runs a balance budget - Equation (8) is satisfied. 3 Calibration and Estimation The model formalizes the mechanisms through which income, land use regulations, and other variables affect housing tenure choice. Now we want to determine the magnitude of those factors on slum formation. Since our model is one of a city, the parameters of the model are set to match relevant features of selected cities in 2010. We look at two cities in Brazil: São Paulo and Rio de Janeiro. We verify the model s predictions and test these predictions using the data. Once the model is estimated to fit the data, we perform ex-ante policy evaluations (see Section 4). We need to calibrate 20 parameters generated by the structure of the model and represented by the following vector of parameters: ξ = {λ 0, ɛ, α, γ, β, τ p, P, S, υ, r, B, φ, η, ψ 0, ψ 1, θ, h, A F, A I, V}. The values of the parameters of vector ξ were either selected from existing observations ( external calibration ) or determined by a distance minimization procedure ( internal calibration or estimation). Let ξ ext be the partition of vector ξ containing eleven parameters to be (externally) calibrated and ξ int be the partition of ξ with the additional nine unknown parameters to be estimated. 18

External Calibration. The vector to be determined by external calibration is given by ξ ext = {λ 0, ɛ, α, γ, β, τ p, P, S, υ, r, B}. Due to the shortage of empirical literature on slums, in particular applied to cities in Brazil, we had to select parameters directly from datasets to use in the calibration. The parameters of the external calibration are: two parameters for the distribution of labor productivity (λ 0 and ɛ), weight of housing consumption (α), weight of land and capital in housing production (γ and β, respectively) 9, property tax rate (τ p ), total population (P), city radius (S), labor share in the production of the consumption good (υ), real interest rate (r), and city-level productivity (B). As we will calibrate the model for two cities in Brazil, most of the data sources are similar in the two cases and come from Brazil s Official Bureau of Statistics (IBGE). One key aspect of the numerical exercise is the distribution of labor productivity (Υ(λ)). In the calibration exercise, the distribution of abilities Υ(λ) is set to be a Pareto distribution in order to fit the empirical income distribution. We adjust the two parameters of the income distribution to match moments from the income distribution of São Paulo and Rio de Janeiro in 2010, i.e., the artificial income data is set to match moments of the empirical income data. We use two income data moments from IBGE s 2010 Brazilian Population Census (see Appendix A) for this exercise: the per capita income of the poorest quintile of the population (proxy for λ 0 ) and the Gini index (proxy for ɛ). The estimated Pareto distribution provides the income distribution for the calibration exercise. The weight of housing consumption in the utility function (α) is set at 0.21 to São Paulo and at 0.23 to Rio de Janeiro. These values stem from IBGE s Household Budget Survey, the main data source on the distribution of spending of Brazilian households. 10 As for housing production, we set the weight of land input (γ) to be equal to 0.25. The parameter γ can be rewritten as the share of land out of total output given the Cobb- Douglas production function of the developers. Construction companies listed on the São Paulo Stock Exchange display information on the price of land compared to the their total sales (both by construction unit and the total values). Production technology of developers varies because several factors influence the ratio of land prices to total sales, such as whether the land was bought directly or whether the company traded residential units for 9 Recall that housing supply price-elasticity comes from γ and β. 10 The Brazilian values are somewhat higher than those from other studies (cf., Davis and Heathcote, 2005; Davis and Ortalo-Magne, 2011). For instance, Lebergott (1996) reports that historically the value has been 0.14 for the US. 19

land. However, data from construction companies in São Paulo indicate that the target value for land price lies between 25% to 35% of the total output when it comes to residential units. Additionally, we set the weight of capital input (β) to be equal to 0.25 using data from IBGE s SINAPI (National Survey of Costs in Construction Sector). The property tax rate (τ p ) equals 1% of the housing price in São Paulo and 1.2% in Rio de Janeiro. 11 Note that τ p multiplies housing rents (R F ) for formal households in the model, so it is necessary to calculate the effective property tax as of housing rents values (instead of housing prices). For instance, a tax of 1% on housing prices is equivalent of a tax of 25% on rents, for a given interest rate of 4% per year. Parameters P and S are set to one. We also need to stipulate the interest rate r, for which we use the national interest rate. Based on a 30 years average for Brazil, the interest rate is set to be 4% per annum in real terms. Parameter υ is the labor share in the production of the consumption good. We set it equal to 0.6, which is consistent with the estimates provided by Gollin (2002). Given r and υ, we choose productivity factor B such that the wage rate is equal to 1. Internal Calibration. Due to the paucity of data, the additional nine parameters from vector ξ were found by minimizing the distance between the model and the data moments. Therefore, we find the nine parameters focusing on selected data moments (along the lines suggested by Hansen (1982)) instead of focusing on the entire distribution of the observed variables. Partition ξ int is represented by ξ int = {φ, η, ψ 0, ψ 1, θ, h, A F, A I, V}. The parameters of the internal calibration are: formal housing lump-sum fees φ, a parameter for transportation costs (η), two parameters for the protection costs function (ψ 0 and ψ 1 ), informality disutility θ, the minimum housing space h, housing production parameters A F and A I, and reservation utility V. 12 Internal calibration consists of estimating the parameters from ξ int such that the model would match key statistics of the São Paulo s and Rio de Janeiro s urbanization process. 11 It is useful to note that a municipality can adopt a single property tax rate or several rates depending on the housing value, housing location, etc. For instance, two-thirds of the 365 Brazilian municipalities studied by Carvalho Jr. (2008) adopted a single rate. In São Paulo, property tax varies between 0.8% and 1.6% of the market value (housing price) of the housing unit. Although there is some variability, the median and mode rates for residential units were 1%. 12 Notice that it is difficult to obtain a measure of minimum housing space from the existing legislation, so the parameter h is obtained by the internal calibration. 20

We aim to match ten predictions of the model to their data counterparts. Tenure choice and housing rents are used as moments in the minimization procedure. Recall that the model is able to generate information on housing tenure choices (so we have the fraction of population in slums) and on housing rents (so we can construct the housing rent ratio between formal and informal housing units). We split each city into five concentric circles, such that we have ten moments, i.e., the fraction of population in slums and housing rent ratio in each circle. In order to create the slums share, we verify the decision rule regarding housing tenure (represented by Ω(λ, d n ; o) in the model) for each agent and then aggregate it to create the slums share value. After verifying the optimal choices of each agent, it is also possible to retrieve information on housing rents. We use a Method of Moments or Minimum Distance procedure to find the unknown parameters. The distance minimization procedure consists of finding the vector ˆξ int that minimizes the distance between the model s prediction and the data. By carrying out this procedure, we aim to obtain parameter values which most closely reproduce key features of the data. The objective is to minimize the quadratic loss function of deviations of predicted moments from their empirical counterpart (weighted sum of squared errors of model moments and data moments). The model is overidentified, since the number of parameters in ξ int is nine and the number of moments is ten. The nested fixed point algorithm used to compute the parameters has two loops: the inner loop computes the housing market equilibrium for all five concentric circles given each parameter value, while the outer loop searches for optimal parameter values. Let L(ξ) be the quadratic loss function to be minimized, M d be the vector of data moments, and M m (ξ) be the correspondent vector of model moments. Formally, we want to find a ˆξ int such that 1 ˆξ int = arg min L(ξ) = arg min ξ int ξ int N [M m(ξ) M d ] 1 W N N [M m(ξ) M d ]. (32) where M m (ξ) M d is the orthogonality condition and W N is a positive semi-definite weighting matrix. The matrix W N converges in probability to W. It follows that ˆξ int is a consistent and asymptotically normal estimator of ξ int. In the analysis, we use an optimal weighting matrix W, which is given by the inverse of the variance-covariance matrix of the data moments 13 (S), i.e., W = S 1. 13 There are 10 moments, so the orthogonality condition is a 10x1 matrix and the weighting matrix is a 10x10 matrix. All five data moments are calculated from São Paulo s microdata. We computed datamoments variance-covariance matrix S directly from city-level variance-covariance information. The offdiagonal elements of the matrix S also come from the city-level information. 21

Given the structure of the model, we had to consider the following constraints in the estimation procedure: ψ 0 > 0: protection costs must be positive. 0 < θ < 1: there must be a disutility from living in an informal housing unit. h > 0: enforcement of MLS regulations. Notice that there is no formal proof that the parameters are identified from the chosen set of moments. However, the chosen moments are informative about the parameters to be estimated. All moments are important for the values of all parameters, but each parameter has more influence on a subset of the chosen moments. Specifically, minimum housing space (h) highly influences slum share (recall Figure??), while protection costs (ψ 0 ) and informality disutility (θ) impact more on housing space consumption for informal housing (and thus the distribution of housing rent ratio). Table 1 shows the value of the calibrated and estimated parameters. The internallyestimated parameters have the expected sign. The only parameter which its magnitude has a direct economic interpretation is ψ 0. Parameter θ is a utility-shifter and h is related to the concept of housing services (which enters the utility function as well) and a catch up of all variables which denoting housing regulations in the formal sector. Field (2007) estimates that in Peru no legal claim to property is associated with a reduction of 14% in total household working hours. Our estimate for São Paulo and Rio de Janeiro is that, for each housing space unit, the cost of informality is about XX% of disposable income. 4 Quantitative Analysis In this section, we use our model to predict the impact of hypothetical policies on the location and formation of slums. We focus on hypothetical policies that makes formal housing market more accessible to the urban poor and improvements in road networks and public transportation. By and large, policies towards slums are classified under the umbrella of slum upgrading interventions. Upgrading projects include a wide range of interventions. Typical interventions include a bundle of land titling, provision of basic infrastructure (e.g., water, electricity, and public lighting), construction of facilities (schools, health posts), and home improvement. In the simulations, we focus on a somewhat broader set of interventions that we split into two categories: apart from slum upgrading interventions, we also study the effects of policies that decrease barriers to formalization. The simulated policies have significant impact on the city so as to interfere with housing prices 22