metadata: authors: 0: author: Pfeiffer,Basile Fabrice
1: author: Rabe,Claus
2: author: Selod,Harris
3: author: Viguie,Vincent
count: South Africa
docna: 0: docna: Assessing Urban Policies Using a Simulation
            Model with Formal and Informal Housing : Application to Cape
            Town, South Africa
docty: Policy Research Working Paper
subtopic: Municipal Management and Reform,Urban Housing and Land Settlements,Urban Housing,Urban Governance and Management,Transport Services,Urban Economic Development,City to City Alliances,Urban Economics,National Urban Development Policies & Strategies,Urban Communities,Regional Urban Development,Labor Markets
teratopic: Social Protections and Labor,Public Sector Development,Urban Development,Transport,Communities and Human Settlements
abstracts: cdata!: Building on a two-dimensional discrete
            version of the standard urban economics land-use model, this
            paper presents a tractable urban land-use simulation model
            that is adapted to developing country cities, where formal
            and informal housing submarkets coexist. The dynamic
            closed-city framework simulates developers'
            construction decisions and heterogeneous households'
            housing and location choices at a distance from various
            employment subcenters, while accounting at the same time for
            land-use regulations, natural constraints, exogenous
            amenities, and dynamic scenarios of urban population growth
            and of State-driven subsidized housing. Designed and
            calibrated for Cape Town, the model is used to assess the
            impact of an urban growth boundary and of changes in the
            scale of subsidized housing schemes, informing a discussion
            of the potential trade-offs in policy objectives and of
            policy effectiveness.
geo_reg_key: 517195,517191
repnme: repnme: Assessing Urban Policies Using a Simulation
            Model with Formal and Informal Housing : Application to Cape
            Town, South Africa
display_title: Assessing Urban Policies Using a
            Simulation Model with Formal and Informal Housing :
            Application to Cape Town, South Africa
listing_relative_url: /research/2019/06/31206107/assessing-urban-policies-using-simulation-model-formal-informal-housing-application-cape-town-south-africa
topicv3: Transport m1326661 135,Urban Development m1327371 699,Land and Housing m1327648 817,Population, Aging and Health m1327736 640,Jobs m1503931 2670
publishtoextweb_dt: 2019-06-27T00:00:00Z
historic_topic: Social Protections and Labor,Public Sector Development,Urban Development,Transport,Communities and Human Settlements
no_of_pages: 44
author_keys: 0: author_key: 1459461
1: author_key: 1578897
2: author_key: 750638
3: author_key: 957932
model_extractions: Policy Research Working Paper 8921 Assessing Urban Policies Using a Simulation Model with Formal and Informal Housing Application to Cape Town, South Africa _Basile Pfeiffer Claus Rabe Harris Selod Vincent Viguié_ Development Economics Development Research Group June 2019
Policy Research Working Paper 8921

# **Abstract**

Building on a two-dimensional discrete version of the standard urban economics land-use model, this paper presents a tractable urban land-use simulation model that is adapted to developing country cities, where formal and informal housing submarkets coexist. The dynamic closed-city framework simulates developers’ construction decisions and heterogeneous households’ housing and location choices at a distance from various employment subcenters, while accounting at the same time for land-use regulations, natural constraints, exogenous amenities, and dynamic scenarios of urban population growth and of State-driven subsidized housing. Designed and calibrated for Cape Town, the model is used to assess the impact of an urban growth boundary and of changes in the scale of subsidized housing schemes, informing a discussion of the potential trade-offs in policy objectives and of policy effectiveness.

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

_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
# Assessing Urban Policies Using a Simulation Model with Formal and Informal Housing: Application to Cape Town, South Africa<sup>_∗_</sup>

Basile Pfeiffer<sup>a</sup> , Claus Rabe<sup>b</sup> , Harris Selod<sup>c</sup> , Vincent Viguié<sup>a</sup>

> _aCIRED, Nogent-sur-Marne, France_

> _bIndependent Consultant, Cape Town, South Africa_

> _cThe World Bank, Washington D.C., USA_

**Keywords** : LUTI model, urban planning, informal settlements, backyarding, Cape Town **JEL classification** : R14, R31, R52

> _∗_ We thank the City of Cape Town for its support and facilitation of access to the data used in this paper. We are especially grateful to Peter Ahmad, Hugh Cole, Jaco Petzer, Eloise Rousseau, John Spotten and participants to the City of Cape Town’s workshop “Economics of Human Settlements” (August 27, 2018) where preliminary simulations were presented. We are also indebted to Roger Behrens, Edward Beukes, Jan Brueckner, Rob McGaffin, Danie du Plessis, Hazvinei Tamuka Moyo, Ivan Turok, Tony Yezer and Mark Zuidgeest for useful discussions and remarks, as well as to participants to the World Bank/GWU/IGC Urbanization and Poverty Reduction Research Conference (September 7, 2018), the World Bank’s seminar “Backyarding in South Africa” (October 18, 2018) for valuable feedback. Funding from DFID under the World Bank’s Strategic Research Program is gratefully acknowledged.
# **1. Introduction**

Urban planning requires ex-ante assessments of potential policy impacts. These impacts need to be assessed at the scale of a whole city, with the understanding that a city forms a system and that policies can have systemic consequences. For instance, urban planners may try to anticipate how land markets could respond to a major transport infrastructure investment, potentially modifying the spatial organization of a city and its footprint over the long term. Doing so requires an understanding of the market forces that drive city structure, including housing construction decisions, household location and transport mode choice, subject to physical constraints and zoning. To answer such questions, urban economists have come up with sophisticated spatial simulation tools which are grounded in urban economic theory and which can be used to support urban planning at the local government level. Land use and transport integrated (LUTI) simulation models which simultaneously predict land use and transport decisions have been designed for a few major cities in the world, and various modeling approaches have emerged, including TRANUS (de la Barra, 1989), UrbanSim (Wadell, 2000) and RELU-TRAN (Anas and Liu, 2007) among others (see Acheampong and Silva, 2015, for a full literature review of such models). Although these simulation models have mostly been applied to cities in the United States and Europe, they are increasingly being applied to metropolitan areas elsewhere (see for instance the recent applications of the RELU-TRAN model to Beijing (Anas and Timilsina, 2015) or Cairo (Anas et al., 2017)). In the case of South Africa, a local version of UrbanSim was developed by the Centre for Scientific and Industrial Research (CSIR) for the East Rand area, Durban and Nelson Mandela Bay (see Wray and Cheruiyot, 2015, for a survey of land use modeling in South Africa).<sup>1</sup> Although the development of LUTI models for developing countries is facilitated by improved data availability, it also faces at least three major challenges: First, these models are often complex (involving hundreds of equations) and computationally intensive, which can make them cumbersome to design, difficult to operate, and intractable. Second, because of their complexity and the skills required to operate them, these models may be out of reach and unaffordable to local authorities with limited fiscal resources. Finally, the existing models overlook a key feature of land markets in developing countries: the presence of _a large informal housing sector which coexists and interacts with the formal housing sector_ (Durand-Lasserve and Selod, 2007, Napier et al. 2013). This gap precludes analyses of informal housing (which hosts the poor and often a large fraction of the middle class) and of how and the degree to which informal housing affects the whole system. More importantly, because, for many cities, housing informality is far from being a marginal phenomenon, it is possible and even likely that the predictions from models that lack an informal housing sector would not hold if the informal sector were accounted for.

In the face of these challenges, simple urban simulation models that are based in urban economic theory emerge as a less costly alternative to previously developed models (see Arnott, 2012). They are useful for urban planning as they can account for broad patterns of urban development over the long term, while remaining tractable so that users understand the forces at play. The

> 1The South African initiatives include a simple cellular-automata model (Dyna-CLUE) that was developed for Johannesburg to investigate land-use conversion processes (Le Roux and Augustijn, 2017). A conceptual framework for an agent-based model of slum evolution was introduced in Shoko and Smit (2013).

2
NEDUM model (Viguié et al. 2014) for instance, which was initially developed for the Paris metropolitan area, is an example of such a model.<sup>2</sup> It directly applies a discrete two-dimensional version of the standard urban monocentric model (see Fujita, 1989) on a grid of pixels, accounting for zoning and land availability constraints defined at the pixel level.

In the present paper, we address all three challenges with the proposal of a LUTI model for Cape Town that builds on a polycentric version of the NEDUM model. To account for the key features of Cape Town, a highly unequal city with a large informal housing sector and a history of subsidized housing provision, we also introduce heterogeneous income groups in the model, as well as informal housing situations that coexist with market and state-driven formal housing.

Our modeling approach builds on a handful of theoretical papers that previously adapted the standard urban land-use model to South African cities (Brueckner, 1996, and Selod and Zenou, 2003) or that proposed ways to model a spatial equilibrium in the presence of interacting formal and informal land and housing markets (see Brueckner and Selod, 2009, Cai, Selod and Steinbuks, 2018, and Selod and Tobin, 2018). The important feature common to all these models is that households can make constrained choices whether to occupy land formally or informally so that an equilibrium relation emerges between the price and extent of formal housing and the size of the informal housing sector.<sup>3</sup> In our framework, we consider two types of land and housing informality: informal settlements in predetermined locations (which is akin to squatting as in Brueckner and Selod, 2009) and a rental market for backyard structures erected by owners of state-driven subsidized housing as recently modeled by Brueckner, Rabe and Selod (2018). We integrate these elements in a closed-city model with exogenous population growth and simulate developers’ construction decisions as well as the housing and location choices of households from different income groups at a distance from several employment subcenters (while accounting for state-driven subsidized housing programs, natural constraints, amenities, zoning, transport options, and the costs associated with each transport mode). To our knowledge, our framework is the first two-dimensional urban economics spatial simulation tool to model the internal residential structure of a city with endogenously determined informal housing.<sup>4</sup> As a proof of concept, we conduct ‘what-if’ evaluations of policy scenarios, investigating the spatial consequences of policies relevant to the city of Cape Town. We first simulate the impact of an urban growth boundary adopted by Cape Town’s metropolitan planning authority to limit sprawl. The second policy we simulate is the continuation of the ongoing subsidized housing program at varying rates of implementation, asking ourselves whether or not this will be sufficient to significantly reduce housing informality in the city.<sup>5</sup> Our simulations show the long term spatial effects and trade-offs in terms

> 2The acronym NEDUM stands for Non-Equilibrium Dynamic Urban Model, where the term “non-equilibrium” refers to the adjustment process between any two periods.

> 3Note that housing informality was first modeled by Jimenez (1984 and 1985) in a partial equilibrium setting. For a review of these models, see Brueckner and Lall (2015).

> 4For a calibrated simulation model with housing informality but no internal city structure see Cavalcanti et al. (2019). For a city-system model of slums, see Alves (2018). For an agent-based model of slums, see Patel et al. (2012). For a dynamic simulation of slums with exogeneous price variations, see Henderson et al. (2018). For a monocentric version of the NEDUM model in a developing country but with no informal housing, see Avner et al. (2017). For city-structure simulations with both endogenous job and residence locations but without explicit consideration of informal land uses, see Ahlfedlt et al. (2015) and Tsivanidis (2018).

> 5To the extent that insecure tenure and associated slum conditions have been shown to entail a range of harmful

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of footprint reduction versus housing affordability.

In Section 2 below, we briefly present relevant Cape Town stylized facts regarding job locations (polycentrism), the high level of income inequality that prevails in the city, and the different housing submarkets that coexist. Section 3 then details the theoretical model. Section 4 presents the data and calibration. Section 5 describes a benchmark simulation and the effects of the two sets of policies. Section 6 briefly concludes.

# **2. The Cape Town context**

Cape Town is a sprawling middle-income city of 4.2 million residents, with an ethnically diverse population (46% Black Africans, 40% “Coloured” (Mixed Descent), 13% Whites and 1% Indians/Asians). The city faces a population growth of 2.4% annually, fueled by in-migration from South African rural areas and other African countries. It has inherited high levels of poverty and acute income inequality (which is highly correlated with race) from past Apartheid policies: As a result, the Gini index for Cape Town is among the highest in the world, reaching .62 in 2017.

This high level of inequality is associated with a fragmented housing market, consisting of four main segments: (i) privately developed formal housing (which houses 52% of residents in 2016), (ii) State-subsidized formal housing (29%), (iii) informal structures in informal settlements (9%), and (iv) structures erected illegally in the backyards of formal housing, mainly State-subsidized housing (8%) (source: Statistics South Africa). Informal settlements first appeared in Cape Town with the rapid urbanization of Black Africans that was stimulated by the labor demands of the wartime economy in the late 1940s (Wilkinson, 2000). They re-appeared at scale during the 1970s as a result of inadequate affordable housing provision coupled with the relaxation of Apartheidera "influx control laws" (which sought to limit internal in-migration of rural Black Africans to cities). As in the rest of South Africa, housing in informal settlements in Cape Town is characterized by small one-story structures made of corrugated iron sheeting and packed at relatively high densities on peripheral, publicly-owned land originally reserved for future roads, social facilities or public housing. The same type of housing can be encountered as backyard structures although these backyard structures may also be made of brick and mortar. Backyarding was historically a non-transactional, kin-based arrangement first associated with Council housing rolled out for households of Mixed Descent in the 1950s and 1960s. Its proliferation as a housing market accelerated in earnest, alongside informal settlements, in the wake of large-scale state-driven housing programs from the 1980s onwards, such as the Reconstruction and Development Programme in 1994 (’RDP’) and, more recently, with the Breaking New Ground (’BNG’) program (Lemanski, 2009). Both programs aimed to allocate individual, free-standing dwellings by means of full capital subsidy to eligible households (Landman and Napier, 2010). These capital subsidies are effects (including e.g., removing workers from the labor force, reducing education and health outcomes, and more generally, a loss in efficiency from the misallocation of land), reducing the level of housing informality in a city can be a justified policy objective. Note, however, that completely eradicating informality may not necessarily be desirable in a second best setting wher formalization is very costly as moderate levels of slums may provide the poor access to urban economic opportunities in excess of the negative externalities they generate (see Cai et al., 2018, who derive this result by contrasting _laissez-faire_ and social planner equilibria in a spatial dynamic stochastic setting).

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other non-residential uses. In addition, each location has an exogenous quantity _A_ ( **x** ) of natural and historical amenities.

_Job centers, commuting and net income._ The city is inhabited by _N_ households (closed city assumption) divided into 4 skill/income groups (indexed by _i_ ). Each group has an exogenous number of households _Ni_ and each household has one worker and other family members. There are _C_ employment locations in the city, indexed by _c_ = 1 _,...,C_ .<sup>7</sup> If considering employment in _c_ , a worker of group _i_ could earn a wage _wic_ and would have expected income _yic_ = χ _iwic_ , where χ _i_ is the exogenous employment rate prevailing in group _i_ .<sup>8</sup> There are _M_ possible modes of transportation in the city, denoted by _m_ . For each residential location **x** , job center _c_ , income group _i_ , worker _j_ and mode _m_ , the expected commuting cost is: _tm j_ ( **x** _, c, wic_ ) = χ _i_ (τ _m_ ( **x** _, c_ )+ δ _m_ ( **x** _, c_ ) _wic_ ) + ε _m_ **x** _cij_ , where χ _i_ τ _m_ is the expected monetary cost of using transport mode _m_ to travel from _c_ to _x_ (accounting for the expected frequency of commuting),<sup>9</sup> χ _i_ δ _mwic_ is the expected cost associated with the time spent commuting, assumed to be proportional to the wage _wic_ (opportunity cost of time), and ε _m_ **x** _cij_ is a random term that follows a Gumbel minimum distribution of mean 0 and scale parameter<sup>1</sup> _/_ λ.

Commuters choose the mode that minimizes their transport cost. By property of the Gumbel distribution, we can thus write the commuting cost between **x** and _c_ as:

> 7Note that our framework can account both for polycentric (if _C_ >1) and monocentric cases (if _C_ =1).

> 8This approach can account for stark variations in employment and in wages across skill groups (with low skill workers earning lower wages and being more unemployed than the high skill workers), as well as moderate variations in wages within groups (which entirely stems from differences in labor remuneration across employment centers). For simplicity, and in spite of within-group income heterogeneity, in the rest of the text, we refer to groups i=1,...,4 as “income groups”, with i=1 the poorest, and i=4 the richest.

> 9Note that, with workers potentially cycling in and out of employment, the employment rate also gives the fraction of time spent employed and thus the frequency of commuting.

> 10Note that although our modeling differs from the random-utility approach (see Anas and Liu, 2007) and from the match-productivity approach (see Ahlfeldt et al., 2015; Tsivanidis, 2018), all yield similar types of gravity equation such as (2).

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We denote _y_ ̃ _i_ ( **x** ) the expected income (over all possible employment centers) net of commuting costs for residents of group _i_ living in location **x** , that is: We can calculate _y_ ̃ _i_ using (1) and (2), which gives: From equation (2), we can derive the expected number of residents of income group _i_ choosing to work in _c_ , denoted _Wic_ , providing that we know the number of residents of income group _i_ with their residence in **x** , denoted _Ni_ ( **x** ), in all _x_ . We have: _Housing types._ There are potentially four types of housing, generically denoted _h_ . The four categories include _h_ = _FP_ (“formal private”) for housing formally provided by the private sector, _h_ = _FS_ (“formal subsidized”) for housing delivered under a subsidized-housing program such as the RDP or BNG, and two types of informal housing: _h_ = _IS_ (“informal settlement”) for housing in an informal settlement, and _h_ = _IB_ (“informal backyard”) for housing in a backyard structure of a plot that was initially delivered under a subsidized-housing program.<sup>11</sup> As will be detailed below, subsidized housing is accessed outside a market-determined price (see below) but a market exists for formal private housing ( _h_ = _FP_ ) as well as for informal backyard structures ( _h_ = _IB_ ) and for informal settlement structures ( _h_ = _IS_ ).<sup>12</sup> In line with empirical observations, we assume that the set of housing options varies across income groups, with only the lowest income groups considering the possibility of informal housing. In our framework, individuals from income group 1 (the poorest) are eligible for public housing. Only a fraction of individuals from this income group, however, will benefit from the relatively

> 11Note that no central, authoritative registry of public housing is available in Cape Town documenting all housing delivered under the succession of government programs that were implemented since the 1920s (Wilkinson, 2000). Earlier public housing varies greatly in terms of typology, tenure arrangement and quality, and some of it has subsequently re-entered the formal housing market. For the purpose of this model, a series of explicit neighborhood, zoning, and physical attributes were used to delineate public housing characteristic of the RDP and BNG housing programs from overall housing stock in existence today.

> 12Although local surveys suggest that a significant proportion of beneficiaries resell the properties that were initially allocated to them under subsidized-housing programs (Tissington et al., 2013), for simplicity, we do not model this secondary market. Because the sales likely remain within the same income group, this has no impact on income sorting in the model.

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small amount of housing provided under the public housing program. The other individuals in this income group will be rationed out and may decide to live in informal settlements, in other people’s backyards, or in formal private housing. Individuals from income group 2 (the second poorest group) face the same housing choices as individuals from income group 1 but are not eligible for public housing which only targets the poorest individuals. Finally, income groups 3 and 4 (the richest groups) may only be housed in formal private housing.<sup>13</sup> These assumptions are summarized in Table A.2 in Appendix A.

_Utilities._ The type and quantity of housing consumed affect household utility. In our model, households derive utility by consuming a composite good _z_ , housing quantity _q_ , and facing amenities _A_ and a housing type externality _B_<sup>_h_</sup> , where _h_ = _FP, FS, IS, IB_ . Assuming a Stone-Geary specification–which imposes a minimum housing size consumption–household utility is expressed as: where _q_ 0 _>_ 0 is the minimum need for housing quantity, α + β = 1, and _B_<sup>_FP_</sup> = _B_<sup>_FS_</sup> = 1 and _B_<sup>_IS_</sup> and _B_<sup>_IB_</sup> _<_ 1.

Because _B_<sup>_h_</sup> is a multiplicative term in the utility function, the condition _B_<sup>_FP_</sup> = _B_<sup>_FS_</sup> = 1 means that there is no externality associated with formal housing, whereas _B_<sup>_IS_</sup> and _B_<sup>_IB_</sup> _<_ 1 capture the negative externalities associated with informal housing (see Galiani et al., 2018). It can easily be shown that the Stone-Geary specification also implies that the rich will value more than the poor residing in locations with better amenities (as in Brueckner et al., 1999).

Having laid out these general assumptions, we can now say a few words about the functioning of the model: Housing will be provided exogenously by the government in the form of limited subsidized housing in areas of the city zoned for such developments, and endogenously by competitive developers (for formal private housing), by illegitimate absentee “landowners” (for informal settlements) and by beneficiaries of subsidized-housing (for backyard structures). Those among low-income households who are not granted subsidized housing (a fraction of income group 1 households and all of income group 2 households) will compete for locations within and across the different market segments (formal private, informal settlements and backyards), with housing being allocated to the highest bidder in each market segment.<sup>14</sup> In the subsection below, we begin by deriving the demand and supply for the different housing types before presenting the equilibrium in the subsection that follows.

> 13Although income groups 3 and 4 face similar housing choices, we distinguish between these two groups in order to better account in our simulations for income heterogeneity and spatial sorting along income lines.

> 14Observe that although there is competition for land within each market segment, there is no _direct_ competition for land across market segments in the sense that households choosing to reside in one type of housing do not need to outbid households choosing to reside in another type of housing. This stems from the fact that the locations of informal settlements and of subsidized-housing programs–where backyarding occurs–are exogenously given. Households from income groups 1 and 2 can nevertheless decide in which market segment to demand land, increasing or decreasing the demand for land in the different market segments accordingly.

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# _3.2. Housing markets_

# _3.2.1. Housing supply_

In each cell **x** , the total quantity of available land (free of constraints) is exogenously given by _L_ ( **x** ). This amount is further broken down into land available for each primary housing type, denoted _Lh_ ( **x** ) for _h_ = _FP, FS, IS_ . In our framework, because informal settlement locations vary little over time and because the quantity of land allocated to subsidized housing is a policy decision, _LFS_ ( **x** ) and _LIS_ ( **x** ) are exogenously given. This implies that the quantity of land available for private formal development is also exogenous and given by the residual: As will be detailed below, the fraction of subsidized-housing land allocated to backyarding will be endogenously determined.

The number of individuals residing in each housing type, _Nh_ ( **x** ) for _h_ = _FP, FS, IS, IB_ and the overall number of individuals residing in each cell, _N_ ( **x** ) (= ∑ _h Nh_ ( **x** )) are also endogenous quantities.

Below, we derive the supply of each housing type in a given location **x** .

_Formal private housing._ Let us start with presenting the supply of formal housing by competitive developers. In a location **x** , a developer will purchase land at a price _P_ ( **x** ) from absentee landlords and will combine land with capital to produce housing, before renting out housing to individuals at a price _RFP_ ( **x** ). Note that both prices _P_ ( **x** ) and _RFP_ ( **x** ) will be determined in equilibrium but for the time being, we consider them as given and express housing supply conditional on these prices. As standard in the developer model (see Fujita, 1989), the housing surface built, _SFP_ , is given by a production function with constant returns to scale: _SFP_ ( _K, L_ ) = κ _L_<sup>_a_</sup> _K_<sup>1</sup><sup>_−a_</sup> , where 0 _< a <_ 1 is the land elasticity of housing production, _L_ is the land surface occupied by the building, _K_ is the capital used for development, and κ is a scale parameter.<sup>15</sup> We express the quantity of housing produced per unit of land as: where _k_ = _K/L_ is the capital per unit of land.

For a developer, the profit per land unit in location **x** is thus: where ρ is capital depreciation and δ is the cost of capital. Profit maximization with respect to capital per unit of land yields the solution:

> 15The literature is split regarding the specification to use for housing production functions. The practices in the US has long been to use a CES specification with an elasticity of substitution lower than 1, implying that the ratio of capital to land value decreases with distance to the city center (see Larson and Yezer, 2015). Recent papers for the US and France, however, have concluded that a Cobb-Douglas function (implying an elasticity of substitution equal to 1) is a good approximation (see Epple et al. 2010, Combes et al. 2016).

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and Note that Eq. (6) expresses the supplied housing quantity per unit of land in location **x** as a function of the market-determined rent for formal private housing in that location.<sup>16</sup> In location **x** , the total quantity of supplied formal private housing will be _SFP_ ( **x** ) = _sFP_ ( **x** ) _LFP_ ( **x** ).<sup>17</sup> _Formal subsidized housing._ Let us now turn to the supply of subsidized housing (RDP/BNG programs). For simplicity, in our framework, subsidized housing is exogenously supplied for free to a limited number of individuals among income group 1 (the low-income group).<sup>18</sup> Each plot in the single-family subsidized housing scheme is of fixed size _qFS_ , including a backyard of fixed size _Y_ . As we will see below, occupants of subsidized housing may decide to rent out a fraction _μ_ ( **x** ) _<_ 1 of their backyard, so that the remaining quantity of housing that they end up consuming is _qFS − μ_ ( **x** ) _Y_ .

_Informal housing in backyards._ We adopt here a simplified version of the “backyarding model” recently proposed by Brueckner et al. (2018). In our setting, some individuals from income group 1 will be granted subsidized housing for free. The other individuals from income groups 1 and 2 may decide to reside informally in backyard structures, paying a rent _RIB_ ( **x** ) per unit of housing (to be determined in equilibrium) to beneficiaries of subsidized housing.

In each location **x** , the fraction of backyard space rented out _μ_ ( **x** ) is chosen to maximize the utility of subsidized housing beneficiaries, under the budget constraint:<sup>19</sup>

> 16Using the zero profit condition, the price of land paid by developers to absentee landlords is also a function of the price of housing sold by developers to individuals, with _P_ ( **x** ) = _a_ (1 _− a_ ) <u>1</u> _−a_ _<u>a</u>_ � λ _R_ ρ _<u>FP</u>_ +δ( _x_ <u>)</u> � <u>1</u> _<u>a</u>_ .

> 17We abstract from modeling the construction and funding of infrastructure networks (water, electricity, transport) to support spatial urban expansion. Infrastructure network expansion costs could be considered in the model as additional costs borne by private developers through impact fees, or as a cost collectively funded by city residents under a property tax.

> 18Subsidized housing could be provided at a non-zero price without significantly altering the results of the model.

> 19  ̃ Observe that all subsidized housing beneficiaries belong to income group 1, hence the notation _y_ 1( **x** ) to denote income net of commuting costs.

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Note that _μ_ ( **x** ) increases with _RIB_ ( **x** ), which replicates a result from Brueckner et al. (2018): under well-behaved properties of the utility function, all things else equal, a higher rental price for backyard structures will increase the supply of backyard housing.<sup>20</sup> The quantity of backyard housing in location **x** will thus be _SIB_ ( **x** ) = _μ_ ( **x** ) _Y N_ ( **x** ), where _NFS_ ( **x** ) is the exogenous number of subsidized plots in location **x** .

_Informal settlements._ Zones where informal settlements occur are exogenously determined in the model (accounting for historic locations) so that the maximum supply of informal settlement land in a location **x** is _LIS_ ( **x** ). Individuals residing in informal settlements pay a rent _RIS_ ( **x** ), even though this payment is not made to the legitimate owner of the land (see Brueckner and Selod, 2009, for a description of squatting arrangements and associated payments). In our setting, the rent extracts informal settlers’ willingness to pay for living in an informal settlement given the negative externality and the fixed size of informal structures _qI_ . For simplification, we assume that it does not cost anything to build an informal structure and that no capital investment is required (as informal structures only have one floor, i.e. a floor-area ratio of 1) so that it is not necessary to model the building decisions of illegitimate absentee “owners” of informal settlement. This implies that in location **x** , given the quantity of land _LIS_ ( **x** ) available for informal settlers, there can be at most _LIS_ ( **x** ) _/qI_ informal settlement structures.

# _3.2.2. Housing demand_

Before deriving the demand for the different housing types, note that the budget constraint of a household of income group _i_ , and residing in location **x** , under housing type _h_ can be written as: where 1 _{h_ = _FS}_ is the indicator function equal to 1 for occupants of subsidized housing (as these households have rental income _μ_ ( **x** ) _Y RIB_ ( **x** )) and equal to 0 for everyone else, and _Rh_ is the rent per unit of housing of type _h_ (with _RFS_ = 0).

Below, we derive the demand for housing conditional on location **x** and on each housing type _h_ , starting with formal private housing.

_Formal private housing._ For a given location **x** , an urban resident will demand a quantity of housing that maximizes utility (4) under constraint (8) and the minimum dwelling size condition _qFP ≥ qmin_ .<sup>21</sup> This yields the following first-order conditions:

> 20With a general utility function, the effect of land rents on backyard space rental is ambiguous because the increment in income associated with higher rents (which tends to decrease the rental of backyard space) plays in the opposite direction of the substitution effect associated with a greater opportunity cost of own yard space consumption (which tends to increase the rental of backyard space). In theory, the supply of backyard housing could thus decrease if the former effect dominates the latter. Brueckner et al. (2018), however, show that a standard utility function, such as the Cobb-Douglass, rules out this possibility altogether.

> 21Note that the minimum dwelling size _qmin_ is different from the basic housing need _q_ 0 that we introduced earlier in the utility function (with _qmin ≥ q_ 0).

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Because we have a minimum dwelling-size condition, we solve the system as follows: Let us denote _Q_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i_ ) and _Z_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i_ ) the optimal quantity of formal housing and composite good that households would want to consume in the absence of a minimum dwelling size requirement. Rearranging terms from the first two conditions in system (9) and plugging them into formula (4), we can express utility as: This implicitly defines _Q_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i | u_ ) as a function of _u_ . Note that, because α _<_ 1, _u_ increases with _Q_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i_ ), which implies that _Q_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i | u_ ) is an increasing function of _u_ . Because the SOC is verified (given that α and β _<_ 1), it is then easy to see that the constrained housing demand (i.e., the housing demand that is potentially constrained by the minimum dwelling-size condition) is _QFP_ ( **x** _, A_ ( **x** ) _, i, u_ ) = _max_ ( _qmin, Q_<sup>_∗_</sup> ( **x** _, A_ ( **x** ) _, i | u_ )).

Plugging back _QFP_ ( **x** _, A_ ( **x** ) _, i, u_ ) into the first condition in system (9) and inverting the resulting equation in the rent gives the bid rent: which expresses the maximum rent a type _i_ household would be ready to pay to reside in private formal housing in location **x** in order to attain utility _u_ .

From equation (10), we can see that _QFP_ ( **x** _, A_ ( **x** ) _, i, u_ ) is an increasing function of _u_ , a decreasing function of _A_ ( **x** ), and a decreasing function of _y_ ̃ _i_ . Therefore, the bid-rent ψ<sup>_i_</sup> _FP_<sup>(</sup><sup>**x**</sup><sup>_,u_) is an</sup> increasing function of _A_ ( **x** ), and an increasing function of _y_ ̃ _i_ . This implies that the bid-rent will be greater in locations with high amenities, and good accessibility to jobs. From equation (6), the quantity of housing produced per unit of land is an increasing function of rents, therefore it will also be greater in those locations.

_Formal subsidized housing._ Formal subsidized housing is offered in overall quantity _NFS_ = ∑ **x** _NFS_ ( **x** ) to a fraction of income group 1 households. The “demand” for subsidized housing will thus involve rationing as long as _NFS < N_ 1.<sup>22</sup> Note that the utility of a subsidized-housing recipient residing in **x** is:

> 22Note that, in the model, beneficiaries of subsidized housing have the option to reject the offer (see the equilibrium definition in Section 3.3). In practice and in our simulations, however, subsidized housing is sufficiently advantageous for all beneficiaries to always accept the offer.

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_Informal housing in backyards._ Backyard structures have a fixed size, _qI_ . Because individuals in backyard structures will spend all their income net of commuting and housing costs on the composite good, a household residing at location **x** obtains utility: Inverting Eq. (12) in the land rent gives the following bid rent: As in the case of formal private housing, the above formula measures the maximum rent an income group _i_ household would be willing to pay for backyard housing in **x** , while commuting to _c,_ in order to attain utility _u_ . Because the income net of commuting _y_ ̃ _i_ ( **x** ) decreases when moving away from jobs, it is easy to see from (13) that a household will be willing to pay more to reside in a backyard structure located closer to jobs. The supply of backyard structures will in turn positively respond to these higher bids as can be seen in equation (7).<sup>23</sup> _Informal settlements._ Finally, the same reasoning applies to the demand for informal settlement housing, leading to the following bid rent: which measures the maximum payment a household of income group _i_ would accept to pay to obtain utility _u_ while residing in an informal settlement in **x** and commuting to _c_ .<sup>24</sup>

# _3.3. The static equilibrium_

We can now define an equilibrium as follows:

**Definition.** An equilibrium is the set _ui_ ; _Ni_<sup>_h_(</sup><sup>**x**);</sup><sup>_Rh_(</sup><sup>**x**);</sup><sup>_Sh_(</sup><sup>**x**);</sup><sup>_Wic_,forall</sup><sup>_i_,</sup><sup>_h_and</sup><sup>**x**(wherethese</sup> functions are defined), where:

- _ui_ is the utility of income group _i_ ;

- _Ni_<sup>_h_(</sup><sup>**x**)isthedistributionofhouseholdsofincomegroup</sup><sup>_i_,housedinhousingtype</sup><sup>_h_,and</sup> residing in cell **x** ;

> 23In theory, because subsidized-housing beneficiaries will also obtain a higher wage income net of commuting costs from a closer location to jobs, the supply response to higher rents closer to jobs can be ambiguous because of the additional income effect discussed in footnote 18 (mathematically, see the _y_  ̃1( **x** ) term in (7)). Brueckner et al. (2018) show that if subsidized-housing beneficiaries are less attached to the labor market (i.e., if they commute less) than backyard structure renters, then _μ_ ( **x** ) will be greater in locations with greater job accessibility. The condition is verified in our case as subsidized-housing beneficiaries belong to group 1 which has the lowest employment rate. 24Observe that the bid rents for backyarding and informal settlement dwellings are identical, except for the housing externality term.

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- _Rh_ ( **x** ) is the market rent of housing type _h_ in cell **x** where these housing types are present, i.e. for **x** _∈ Xh_ = � **x** _| Nh_ = ∑ _i Ni_<sup>_h_(</sup><sup>**x**)</sup><sup>_>_0</sup> � and _h ∈{FP, IB, IS}_ ;

- _Sh_ ( **x** ) is the quantity of each housing type _h_ in cell **x** ;

- _Wic_ is the number of workers from group _i_ choosing to work in _c_ .

and satisfying the following constraints:

- (i) _Ni_ = ∑ _h_ ∑ **x** _Ni_<sup>_h_(</sup><sup>**x**);</sup>

- (ii) _P_ ( **x** ) _≥ PA_ for **x** _∈ XFP_ ;

Note that ( _i_ ) is a set of population constraints (which ensure that the city hosts all individuals in equilibrium). ( _ii_ ) is a city-edge constraint (which reflects the indifference of absentee landlords at the city fringe between selling their land to a developer or engaging in agricultural activities). ( _iii_ ) is a set of utility equalization constraints (which reflects indifference among individuals of each income group between locations and housing types). This utility equalization constraint does not involve beneficiaries of formal subsidized housing, as they benefit from a windfall transfer from the State and will have a higher equilibrium utility that non-beneficiaries in their income group.<sup>25</sup> ( _iv_ ) ensures that land is allocated to the highest bidder for each housing type in each cell (with the exception of subsidized housing beneficiaries who do not compete for land with anyone), and that housing demand and housing supply are equated in each location. 26 Note that ( _iv_ ) reflects competition for land within submarkets but not directly across market segments. Finally ( _v_ ) ensures labor-market clearing.

Observe that in equilibrium, formal and informal housing markets are connected in several ways. Firstly, there is a direct connection due to the fact that, with the exception of subsidizedhousing beneficiaries who receive a transfer from the State, other poor households from income group 1 and from income group 2 optimize across formal and informal residential options until their utilities are equalized (constraints (iii) and (iv) in the equilibrium definition). Secondly, the fact that informal settlements and backyarding locations are exogenously determined does not

> 25Note that, in equilibrium, we allow households to decline the subsidized housing they are offered and decide to live in an informal settlement, in a backyard structure, or in the private housing sector instead. If the utility from residing in a subsidized housing location is lower than for other housing types, then the household would be better-off declining the offer and the housing unit will remain vacant. In practice, however, this is very unlikely to happen given the advantageous conditions (free rent, relatively large dwelling, and possibility of renting out the backyard) under subsidized housing.

> 26Because we assume that only the poor may reside informally, note that _NiIB_ = _Ni_<sup>_IS_= 0 for</sup><sup>_i_= 3</sup><sup>_,_4 (income groups</sup> 3 and 4).

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imply that formal and informal housing developments occur in isolation of one another. In fact, they are linked through the choices of poor households across formal and informal housing options, and because formal developers’ building decisions respond to private formal housing prices (see equation (6) and constraint (iv) in the equilibrium definition), with private formal housing prices partially reflecting the sorting of low-income households across formal and informal housing market segments. Finally, there is an externality associated with the use of land for informal settlements and for publicly subsidized housing as these areas are somehow taken away from developable land that would otherwise be available for private formal development (see the land-use accounting equality (5)). This affects the supply and demand for formal housing by restricting the set of potential locations available for private formal development, while accommodating a potentially large number of urban residents in the informal sector.<sup>27</sup> _Existence and uniqueness of equilibrium._ Although the model is simple and has clearly laid-out supply and demand mechanisms, it is not possible to solve it analytically and we will resort to numerical simulations. As regards existence and uniqueness, it can be shown that the equilibrium exists and would be unique in the open-city case, as bid-rents, dwelling sizes, housing supply, and therefore population densities, are uniquely defined for given levels of utility. In our closed-city case, however, the unicity of the equilibrium is more complex to derive. Because of potentially non-monotonic residential sorting under a Stone-Geary specification function, one could suspect the possibility of multiple equilibria. Pfeiffer et al. (2019), however, theoretically show that with two income groups and one housing type, the equilibrium with Stone-Geary utilities is always unique. In our context with four income groups and four housing types, although we do not have a formal proof of equilibrium unicity, it is noticeable that running 250 simulations of our benchmark case (starting from a wide range of starting points), the algorithm always converged to the same equilibrium solution. Although we cannot prove it formally, this strongly suggests that the model has a unique equilibrium.

# _3.4. Dynamics_

Before describing how the model can be solved numerically (see Section 3.5), we first extend it to a dynamic version. In this dynamic version, the system is affected by exogenous variations in inputs over time (for example under a scenario of exogenous demographic changes) and the system responds with adjustments to these exogenous shocks that do not occur instantaneously. More specifically, we assume that the formal housing stock depreciates with time and that formal developers respond to price incentives with delay as in Viguié and Hallegatte (2012).<sup>28</sup> Mathematically, this implies that the stock of housing at time _t_ , _SFP_ ( **x** _| t_ ) may not equate the theoretical equilibrium quantity, _SFP_<sup>_eq_(</sup><sup>**x**</sup><sup>_| t_).Denoting τ the time lag for construction (</sup><sup>_i.e._, the time</sup>

> 27The net effect on formal housing prices is ambiguous as the restricted supply of formal land should raise formal housing prices in the center, while pushing away population to peripheral areas where prices will be lower. Housing in the informal sector reduces the demand for formal housing, which exerts a downward pressure on formal housing prices.

> 28We do not assume such delay for the informal sector, which, in practice, can respond very quickly to changing conditions.

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needed to complete a housing project) and θ the time lag for depreciation ( _i.e._ , the time needed for total depreciation of a building), the change in the housing stock between times _t_ and _t_ + 1 is: This law of motion reflects developers’ investments when the current stock of housing is below the equilibrium and the absence of investment if the reverse is true.<sup>29</sup>

# _3.5. Numerical solution and simulation algorithm_

In this subsection, we first present how the static equilibrium is solved in each period. We then describe how the dynamics is implemented.

_Static equilibrium._ We apply an iterative algorithm to converge towards a solution. Because we have a close city, the total population for each income group is fixed. Our algorithm then solves for all other variables. We start with an arbitrary set of initial utilities, from which we determine:

- (i) Housing demand for each housing type, using equation (10) for formal housing, and the fact that informal settlement dwellings have a fixed size _qI_ ;

- (ii) Rents, using equation (11) for formal private housing, equation 13 for informal backyarding and (14) for informal settlements;

- (iii) Housing supply, using equation (6) for formal housing and equation (7) for informal backyarding.

- (iv) Population in all locations for all housing types, using equilibrium condition (iv).

By summing populations across locations and housing types, we obtain the total population for each income group. Utilities are then incrementally adjusted and steps 1-4 iterated until the target population allocation is simulated. Graph E.11 in Appendix E summarizes the procedure.

_Dynamics._ We consider the state of the city in year _t_ . One year later, at _t_ + 1, we solve the equilibrium for a new set of input parameters (which may have exogenously changed, for instance if the population has increased). This determines housing supply without private construction inertia. We then apply equation (15) to determine the actual formal private housing supply at year _t_ + 1, accounting for inertia. Dwelling size and prices are then determined by deriving the new equilibrium given the period’s housing supply. This determines the new state of the city. We then reiterate the process for subsequent periods.

> 29Observe that the theoretical values _SFP_<sup>_eq_(</sup><sup>_x | t_)and</sup><sup>_S_</sup> _FP_<sup>_eq_(</sup><sup>_x | t_+1)willbeequalintheabsenceofanyexogenous</sup> variation between _t_ and _t_ + 1. The housing stock, however, may adjust so that _SFP_ ( _x | t_ ) and _SFP_ ( _x | t_ + 1) are not necessarily equal.

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# **4. Data sources and parameter estimation**

We apply the theoretical model to a grid of 100 x 80 km that largely encompasses the existing urban footprint of Cape Town. The grid is subdivided into 500m x 500m cells. Each cell represents a location **x** in our theoretical framework. Because the different data sets that we use are available at different spatial resolutions, we either spatially aggregated or disaggregated the information using cell areas as weights.

We make use of a variety of data sets as direct inputs into the model and in order to calibrate parameter values. More specifically, the inputs that are fed into the model consist of the total population in the city at an initial date and its decomposition into income groups ( _Ni_ ), the average income and employment rate (χ _i_ ) by income group, land use constraints ( _Lh_ ( **x** )), and the monetary and time cost of commuting (τ _m_ ) and (δ _m_ ) between cells and job centers, for 5 different modes (walking, minibus/taxi, train, bus, private car).<sup>30</sup> Exogenously chosen parameters include the size of subsidized housing plot ( _qFS_ ), the time lag of housing investment (τ), the housing stock depreciation parameter (θ), the financial depreciation rate of built capital (ρ), and the agricultural land rent ( _PA_ ). Estimated parameters include the gravity equation parameter (λ), wages at workplace ( _wic_ ), housing consumption elasticity in the utility function (β), the minimum housing consumption ( _q_ 0), the land elasticity of housing production ( _a_ ), the scale parameter of housing production (κ), and the index value of amenities _A_ ( **x** ). The disutility parameter of informal housing ( _B_ ) is calibrated to reproduce the distribution between housing types generated by the model.

Below, we describe in more detail data sources and the calibration process.

# _4.1. Data sources_

The spatial distribution of population is taken from National Censuses for the years 2001 and 2011. We define the four income groups by choosing income-group thresholds such that only the lowest income group is eligible for subsidized housing programs, and so that the two highest income groups are not observed to reside informally (see Appendix A for details).<sup>31</sup> We use the transport model used by the City of Cape Town to retrieve transport times between pairs of transport zones for each transport mode and job locations.<sup>32</sup> We also use aggregate statistics on modal shares and residence-workplace distances in Cape Town, that are derived from Cape Town’s 2013 Transport Survey.

Land availability is defined for each housing type. Areas of subsidized housing are identified from the cadastre of the City of Cape Town.<sup>33</sup> The area available for backyard housing is estimated as the yard size of these units. Informal settlement areas are obtained from the Enumerator Area

> 30For simplicity, in what follows, we do not index these variables with time.

> 31 The Census captures annual income, which we assume reflects the employment rate χ _i_ . Households eligible for subsidized housing are the ones with an annual income (χ _iwi,c_ in the model) below the threshold of R38,200.

> 32The origin-destination matrix is produced by the City of Cape Town’s four-step travel demand model, last updated in 2013 using the EMME/2 software. The model was designed by INRO Consultants at the University of Montreal and adopted by the City of Cape Town in 1991. The model implements an equilibrium route assignment based on the distribution of trip origins and destinations in relation to the transport network. The model is calibrated by means of the General Household Transport Survey, on-board surveys and cordon counts.

> 33This corresponds to the category “Single Residential 2 - Incremental Housing”.

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definition of the 2011 Census. Land available for formal private development corresponds to all land that is not constrained for construction.<sup>34</sup> The amenities that we consider include natural amenities (such as slope and proximity to the ocean) as well as historical amenities (such as the proximity to the historical center). The data sets used are listed in Appendix C.4.

For the estimation of the model’s parameters (see below), we also use property price data extracted from the City of Cape Town’s geocoded data set on property transactions for 2011, as well as data on dwelling sizes made available to us by the City of Cape Town.

# _4.2. Estimation of the parameters of the model_

The estimation of the model is done in three steps. First, we choose a first set of parameters using available information, without solving the model. These include the minimum dwelling size _qmin_ , the size of subsidized plots _qFS_ and of backyards _Y_ , the construction lag τ, the physical and financial depreciation of housing θ and ρ, the interest rate δ, and the agricultural rent _RA_ . The minimum dwelling-size for formal housing is set at _qmin_ = 31 _._ 6 _m_<sup>2</sup> , which is the minimum dwelling size observed in formal neighborhoods. Backyards of subsidized houses have a size _Y_ = 70 _m_<sup>2</sup> . We choose the time lag of housing investment τ to be 3 years and the physical depreciation time of building stock θ to be 100 years, as in Viguié et al. (2014). The financial depreciation rate of the built capital is ρ=.05 (i.e., 5%). We allow the interest rate δ and the agricultural land rent _PA_ to vary with time. For δ, we use the annual values for South Africa in the World Development Indicator database (World Bank, 2016). We set the agricultural price at the city border _PA_ at 807 _Rands/m_<sup>2</sup> (annual) in 2011, which corresponds to the ninth decile in the sales data sets, when selecting only agricultural properties in rural areas. In the dynamic simulations, we assume that the agricultural land is constant in real terms and have its nominal value increase at the same rate as the average household nominal income.

Second, we calibrate wages, housing production function parameters, utility function parameters and amenities using partial relations from our model. Following Ahlfeldt et al. (2015) and Tsivanidis (2018), we recover the vector of wages ( _wic_ ) using data on job locations and residential locations. The scale parameter in the commuting formula, λ, is estimated using the distribution of residence-workplace distances in Cape Town. We identify the land elasticity of housing production ( _a_ ) and the scale parameter of housing production (κ) by regressing the log of equation (6) (see Appendix B for details) . We then consider equations (9) and (11), which relate utility levels to dwelling sizes, rents and amenities. The amenity term, _A_ ( **x** ) is expressed as a score for all locations, and specified as _A_ ( **x** ) = ∏ _i ai_ ( **x** )<sup>ν</sup><sup>_i_</sup> , where the _ai_ ( **x** ) are measures for each amenity type (see Appendix C.4), and ν _i_ are their marginal valuation, to be estimated. We simultaneously estimate the system of equations by maximum likelihood and recover parameters _{_ β _, q_ 0 _,_ ν _i}_ . Appendix C.4 presents the procedure in more detail.

Third, we calibrate the disutility parameters of informal housing ( _BIB_ and _BIS_ ) by running the entire model to replicate the share of households in informal settlements and backyard housing in the 2011 Census data.

> 34Restrictions include areas used for formal subsidized housing, informal settlements, protected natural areas, large economic or industrial infrastructures (such as the Cape Town airport). A detailed list of sources is presented in Appendix B.3.

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# _4.3. Benchmark simulation and retrospective fit_

We run a benchmark simulation to compare the outputs of the model with the data. Past evolution of average income, total population and income distribution are derived from Census data. Results are presented on Figure D.8 of Appendix D, which graphically shows that the overall fit is reasonable. We run a retrospective simulation (i.e., running the model “backwards”) starting in 2011, and compare the outputs of the model 2001 estimation with Census and property price data for the same year. We present the details and results for this retrospective simulation in Appendix D, which shows that the model appropriately replicates changes in housing prices over time. The good fit provides confidence that the model can reasonably be used to simulate the future evolution of Cape Town’s spatial structure.

# **5. Scenarios and prospective simulations**

We consider scenarios of income growth and population trends aligned to those which inform the City’s Land Use Scenarios underpinning its medium-term infrastructure master plans (City of Cape Town 2017, Medium Term Infrastructure Investment Framework). The anticipated twentyyear supply of State-subsidized housing is based on the City of Cape Town’s Housing Pipeline as contained in its Integrated Human Settlement Framework (2013) (see Appendix F). All the other parameters remain constant.<sup>35</sup> We prospectively run simulations for the period 2011-2040.

# _5.1. Urban growth boundary_

There has long been discussions in policy circles of Cape Town being a sprawling city. Against a backdrop where Cape Town’s urban footprint was estimated to have expanded by over 1,000 hectares a year during an unprecedented housing boom during the late 1990s and early 2000s, the City introduced an Urban Growth Boundary (or ’Urban Edge’) as a policy guideline and then, in 2012, as a statutory instrument. It was delineated to include sufficient developable land to accommodate future growth for at least 10 years and was thus not immediately binding. We simulate two scenarios: the ’No Urban Edge’ scenario where the Urban Edge constraint is absent (and the city’s urban footprint is permitted to expand unhindered into its rural hinterland) and the ’Urban Edge’ scenario where it continues to be present (see Figure F.12 representing the Urban Edge).<sup>36</sup> 35In our dynamic simulations, we assume that, for each income group, the wage ratios across job sub-centers remain constant over time ( _wic/wi′c_ is the same for any _i_ , _i_<sup>_′_</sup> and _c_ , and _wic/wic′_ is the same for any _i_ , _c_ and _c_<sup>_′_</sup> ). In levels, the mean wage for each income group grows at a constant rate.

> 36In South Africa, the Urban Edge enjoyed support from policymakers, academics and environmentalists alike as a means to protect valuable agricultural land, natural amenities and the functioning of ecological services, while supporting a more compact urban environment. It has been opposed by local politicians and property developers. Developers claimed that the urban growth boundary would generate regressive distributive effects since the restriction of land supply to the housing market raises the cost of housing. Politicians claimed that, by encumbering greenfield development, the growth boundary invariably stifles economic growth and job creation. The Urban Edge is not mentioned anymore in the City of Cape Town’s current spatial development framework.

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We run the simulation until 2040, when the population of the City reaches 1,770,000 households (compared to 1,068,000 in 2011).<sup>37</sup> Maps (a) and (b) of Figure 2 show the population density and urban footprint under the two scenarios. Without the Urban Edge, the urbanized area would expand to 1,208 square kilometers, an urban footprint that is 40% greater than if spatial growth had been contained by the Urban Edge. Densities would be significantly greater, especially in the city center. Maps (c) and (d) of Figure 2 show that the lower footprint and greater density under the Urban Edge scenario would occur with significantly higher formal prices. Within 6 km of the CBD, we find that formal prices would be 21% higher under the Urban Edge Scenario than without the Urban Edge. Because formal housing would be less affordable to the poor, there would be an increase in the demand for informal housing, as shown on the histogram (e) of Figure 2. Because we assumed no spatial expansion of informal settlement footprints (only a densification in informal settlements), these would have become saturated by 2040 and backyard housing would absorb the informality differential between the two scenarios. In the Urban Edge scenario, we find that the number of households living in informal housing would be 27% higher than in the ’No Urban Edge’ scenario.

# _5.2. Public housing provision scenarios_

Subsidized housing has been an important part of post-Apartheid policies trying to address the housing backlog. The vast majority of the approximately 336,000 households who live in public housing as of 2016 (Statistics South Africa) live in dwellings transferred as part of the RDP and later the BNG program, delivered at a rate of approximately 10,000 per year in the 1990s and early 2000s, declining to about 5,000 per year by the late-2000s due to budget constraints and diversification to in situ upgrading of informal settlements.<sup>38</sup> We consider three scenarios for future public housing provision. In the first scenario (entitled ’business as usual’ or BAU), we assume that the provision of public subsidized housing follows the current pace of 5,000 dwellings per year. In the ’low’ scenario, we assume that construction of public housing is slowed down to a pace of 2,500 per year from 2019 onward. In the ’high’ scenario, we assume that construction of public housing is accelerated to a pace of 10,000 dwellings per year starting in 2019. The sites for future public housing replicate the pipeline of projects known to the City of Cape Town starting with ’short-term projects’, before considering ’long-term projects’ (see map of Figure F.13).<sup>39</sup>

> 37The demographic growth projections used in the model correspond to the base projection used by the Cape Town metropolitan authority as of 2019. The 1.77 million households in 2040 will correspond to a total population of 5.3 million inhabitants.

> 38Although government housing estates have featured in Cape Town since at least the 1920s (Wilkinson 2000), today, apartheid-era "Council housing" only includes 43,500 rental units, 21,000 homeownership dwellings, 11,000 hostel beds and 11 old-age home complexes (City of Cape Town Integrated Human Settlements 5-year Strategic Plan 2013/2014 Review).

> 39The information on projects is extracted from a 2015 data set provided by the City of Cape Town, which gives the location and number of dwellings of future RDP/BNG projects, corresponding to a total of 255 _,_ 000 dwellings to be built. We assume that projects indicated as ’short term’ will be built before 2025, while ’long term’ projects will be built from 2025 onwards.

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# **6. Conclusion**

The paper lays out the foundation for a simple urban simulation tool that can easily be implemented in developing country contexts and used by urban planners to generate broad urban development trends. An important contribution is the explicit modeling of both formal and informal housing markets and their interaction, which is a key feature of many cities in the developing world. Having a realistic model of that interaction makes it possible to more accurately simulate city structure in cities where informal housing accommodates a significant portion of the population (and as formal and informal housing have different land use implications). Such a model also makes it possible to simulate the evolution of informal housing over time, an important policy issue that is of course impossible to assess in a model with only formal housing. As a proof of concept, our simulations of zoning policies (with and without an urban growth boundary) and subsidized housing policies (different scenarios of the subsidized housing program) illustrate that point as demand for informal housing responds to land supply restrictions and the higher formal land prices that ensue. The increase in housing informality following an urban growth boundary is an unintended systemic effect that was not previously envisioned in the literature, which only focused on developed country contexts. As for the subsidized housing simulations, they show that income inequality and population dynamics are such that housing informality is likely to persist over time despite policy efforts to reduce it, confirming a theoretical result first derived by Cai et al. (2018). Interestingly, in the Cape Town case, the substitution of backyarding to traditional informal settlements–a trend present in the past data and confirmed in our simulations for the future–stresses the changing nature of informal housing in South African cities. This is a noticeable trend as ongoing discussions in South Africa revolve around the facilitation of such dwelling arrangements to increase access to affordable housing and to stimulate densification (see Brueckner et al., 2018, for a more in-depth discussion).

The model will be available to the wider public on an open source basis and is expected to be further refined and applied to other city contexts in the future. Specific features may be added or removed from the model depending on the context and the policy focus. Two important modifications, in particular, that we intend to prioritize for future versions of the model are the integration of endogenous transportation costs (that may change as congestion will be modified by changes in land use and transportation patterns)<sup>40</sup> and a specific modeling of public infrastructure expansion costs and their funding, an important policy challenge for expanding cities in developing countries.

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> 40See Larson, Liu and Yezer (2012) and Larson and Yezer (2015) for endogenous transportation costs in a radial version of the standard monocentric model. Introducing endogenous congestion in a polycentric simulation model such as ours would add significant complexity and is thus left for future development.

23
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26
# **Appendix A. Housing types and income groups**

|**Income**<br>**group**|**Annual income range**<br>**in 2011 (ZAR)**|**Average 2011 income**<br>**(ZAR, estimated**<br>**using Census)**|**Percentage of the**<br>**total population in**<br>**2011 (estimated using**<br>**Census)**|
|---|---|---|---|
|1|1 - 38,200|19,580|38.6%|
|2|38,200 - 76,400|57,300|16.7%|
|3|76,400 - 307,600|170,140|28.9%|
|4|> 307,600|780,723|15.8%|

Table A.1: Income groups used in the simulation

|**Housing**<br>**types**|**Income group(s)**|**Location**|**Dwelling size**<br>**(plot size for**<br>**formal**<br>**subsidized)**|**Price**|
|---|---|---|---|---|
|Formal private<br>(FP)<br>Formal<br>subsidized<br>(FS)|1,2,3,4<br>1|Endogenous<br>Exogenous|Endogenous,<br>with minimum<br>dwelling size<br>Fixed (40_m_<sup>2</sup>)|Endogenous<br>Free|
|Informal in|1,2|Endogenous|Fixed (20_m_<sup>2</sup>)|Endogenous|
|backyard (IB)||within the<br>backyards of FS<br>plots|||
|Informal in<br>informal|1,2|Exogenous<br>settlement|Fixed (20_m_<sup>2</sup>)|Endogenous|
|settlement(IS)||locations|||

Table A.2: Modeling assumptions regarding housing 27
# **Appendix B. Model inputs**

_Appendix B.1. Employment rates_ The exogenous employment rates are the same for households within a given income group. We calibrate the parameters χ _i_ as the fraction of employed workers in each income group using cross tabulations for the City of Cape Town as a whole in the Census 2011 data. For each income group, we calculate the distribution of educational attainment. We then use the distribution of employment status for each educational attainment to derive the average employment status of each income group. Table B.3 summarizes the values of χ _i_ .

|Income group<br>Parameterχ_i_|1<br>0.57|2<br>0.97|3<br>0.96|4<br>0.97|
|---|---|---|---|---|

Table B.3: Values of the employment rate parameter _Appendix B.2. Employment centers and transport costs_ We extract employment center locations, their composition, and transport times to these centers from the City of Cape Town transport model, as detailed below: _Employment centers._ We use the number of jobs per income group at the Transport Zone level, estimated by the City of Cape Town in 2015 to recover local wages ( _wic_ ). For simplicity and speed of computation, we restrict the employment locations to Transport Zones with more than 2 _,_ 500 jobs (185 job centers). Figure B.4 shows these employment centers and relative employment size by income group.

28
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|Amenity_an_(**x**)|Data source|
|---|---|
|Distance to the ocean|Shoreline shapefle from City of Cape Town<br>Open Data portal|
|Distance to an Urban Heritage Site|Sites shapefle from City of Cape Town OD<br>portal|
|Distance to a district park|District parks shapefle from City of Cape<br>Town OD portal|
|Distance to a protected natural area|Shapefle of protected area layer (SAPAD Q4)<br>from the Environment Department of South<br>Africa.|
|Distance to a train station|Train stations from OpenStreetMap open data|
|Average slope|USGS Digital Elevation Model|
|Presence in the Airport Noise Cone|Shapefle from City of Cape Town OD portal|

Table B.4: List of amenities used in the model 31
# **Appendix C. Calibration**

# _Appendix C.1. Other data sets used for the calibration_

We use the housing sales registry of the City of Cape Town. This data set includes records of housing transactions, including sales year, area, type, price and location of properties. We only consider transactions that took place in 2011 and aggregate the sales information at the Sub-Place level. We calculate a median price (denoted _Ps_ ) per square meter of land for Sub-Places with more than 20 transactions recorded. We also use a data set of average formal dwelling sizes provided by the Municipality of Cape Town, at the Transport Zone level. We aggregate the values at the Sub-Place level (denoted _qs_ ).

# _Appendix C.2. Estimation of construction function parameters_

We estimate the relation between the estimated stock of formal private housing and housing prices in cross-section to identify the parameters of the construction function _{_ κ _, a}_ . Combining (6) and the equilibrium condition (iv), we have: 4 where ∑ _Ni_<sup>_FP_</sup> ( **x** ) = _N_<sup>_FP_</sup> ( **x** ) is the number of household living in formal dwellings. We regress _i_ =1 the log of the previous equation at the sub-place level, denoted _s_ : where _Ns_<sup>_FP_</sup> is the number of households in formal housing at the Sub-Place level, _Ps_ is the median price per unit of land, _Ls_<sup>_FP_</sup> is the amount of available land for formal housing and _qS_ is the dwelling size. From equation (C.1), we expect coefficients γ3 to be close to 1 and γ4 to be close to _−_ 1. Note that equation (C.1) theoretically only applies to formal private housing. However, because housing in low-income neighborhoods is a mix of formal private, formal subsidized housing and informal housing, we restrict our sample for the estimation, by excluding the Sub-Places in the bottom quintile of property prices _Ps_ and for which more than 5% of dwellings are reported to live in informal housing<sup>43</sup> We also exclude rural sub-places (i.e., those that are large, with a small share than can be urbanized). We find: log � _Ns_<sup>_FP_</sup> � = _−_ 3 _._ 51 (0 _._ 97) + 0 _._ 25 (0 _._ 07) log ( _Ps_ ) _−_ 0 _._ 98 (0 _._ 07) log ( _qs_ ) + 0 _._ 92 (0 _._ 08) log � _Ls_<sup>_FP_</sup> �, with standard errors in parenthesis. We find that coefficient γ3 is close to 1 and coefficient γ4 is close to _−_ 1. Coefficient κ and _a_ in the construction function are retrieved from γ1 and γ2, with _a_ = 1 _−_ γ2 and

> 43Our data set for dwelling sizes only provides the average dwelling size at the Sub-Place level, aggregating formal and informal housing.

32
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Combining equations 4 and 9 at the Sub-place level, we obtain the following relationship, for all _s_ : where _ui_ ( _s_ ) is the (constant) utility of income group _i_ ( _s_ ), and _As_ is the amenity index in _s_ .

The relationship between rents and dwelling size is given by (9): We define a range of possible values for β and _q_ 0, knowing that 0<β<1 and the minimum consumption of housing _q_ 0 must be below the size of informal dwellings _qIS_ = _qIB_ = 20 _m_<sup>2</sup> .

For each possible value of β, _q_ 0, and _ui_ we calculate the amenity index as: In each Sub-Place, the amenity index is an aggregate of several amenities, as given by _As_ = ∏ _n_ ( _an,s_ )<sup>θ</sup><sup>_i_�</sup> ε _A,s_ . We estimate the equation _log_ ( _As_ ) = ν0 + ∑θ _nlog_ ( _an,s_ )+ _log_ (ε _A,s_ ) to identify � the set of (θ _n_ ) _q_ 0 _,_ β conditional on _q_ 0 and β. As for the simulated dwelling sizes, they can be written as � _qs_ = β _R_<sup>_<u>y</u>_ ̃</sup><sup>_<u>s</u>_</sup> _s_<sup>+(1</sup><sup>_−_β)</sup><sup>_q_0</sup><sup>_Rs._We further denote ε</sup><sup>_q,s_the ratio of the dwelling sizes in the data (</sup><sup>_qs_) and</sup> of the simulated dwelling sizes ( _q_ � _s_ ), with _qs_ = � _qs_ ε _q,s_ . Finally, for a set of _{_ β _, q_ 0 _, ui}_ , we estimate the log-likelihood that the model predicts the correct income sorting. To do that, we calculate the likelihood that the model reproduces income sorting as the likelihood of a discrete-choice logit model of land allocation to the highest bidder (consistently with competition for land within the formal private sector). Identifying the group with the highest bids as the dominant group in the data, we can write the log-likelihood for income sorting as _l_ = ∑ _s_ � ψ _i_ λ( _sinc_ <u>)(</u> _s_ )� _−_ ∑ _s_ log ∑ _j e_ ψλ _incj_ ( _s_ ) , � � where λ _inc_ is the scale parameter of a Gumbel maximum distribution.

We identify the coefficients _{_ β _, q_ 0 _,_ θ _i}_ by maximizing the sum of log-likelihoods of the distributions ε _A,s_ and ε _q,s_ (assuming that ε _A,s_ and ε _q,s_ follow a log-normal law of mean 1) plus the log-likelihood _l_ . We first scan a discrete set of values for the parameters. From the best solution, we then run Matlab’s ’interior-point’ algorithm to find the maximum. The obtained values for the parameters θ _i_ are presented in table C.5. We use these coefficients to generate a map of the amenity index for every location of the grid (see Figure C.7).

34
Table C.5: Result of the regression on amenities

||Residual utility (log)|
|---|---|
|Proximity to a district park (< 1 km)|-0.015 (0.02)|
|Proximity to the ocean (< 2 km)|0.11<sup>_∗∗∗_</sup>(0.02)|
|Proximity to the ocean (2< . < 4 km)|0.08<sup>_∗∗∗_</sup>(0.02)|
|Proximity to Urban Heritage Site (< 2 km)|-0.01 (0.02)|
|Airport Noise Cone (within)|-0.04 (0.03)|
|Slope (between 1 and 5%)|0.09<sup>_∗∗∗_</sup>(0.02)|
|Slope (_>_5%)|0.15<sup>_∗∗∗_</sup>(0.02)|
|Proximity to a biosphere reserve (< 2 km)|0.004(0.02)|
|Proximity to a train station (< 2 km)|-0.013 (0.02)|
|Constant|-1.96<sup>_∗∗∗_</sup>(0.02)|
|Observations|307|
|R<sup>2</sup>|0.30|
|Adjusted R<sup>2</sup>|0.28|
|F Statistic|14.2<sup>_∗∗∗_</sup>(df = 297)|

_Note:_<sup>_<u>∗</u>_</sup> p _<_ 0.1;<sup>_<u>∗∗</u>_</sup> p _<_ 0.05;<sup>_<u>∗∗∗</u>_</sup> p _<_ 0.01 35
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|Maximum fraction of ground surface<br>devoted to housing|0.7|
|---|---|
|Transport times and costs|cf. Appendix B.2|
|Built capital depreciation rate|2_._5%|
|Cost associated with travel time|Equal to income per minute|
|Dimension of an RDP/BNG house|Interior space 40_m_<sup>2</sup><br>Backyard space 70_m_<sup>2</sup>|
|Minimum dwelling size for formal<br>private housing<br>Table C.6: Chosen|_qmin_=31_._6_m_<sup>2</sup><br>parameters|
|Households utility function parameter|β=0_._25|
|Basic need in housing in the utility<br>function|_q_0=4_._1|
|Coeffcients of development function for<br>formal private housing|_a_=0_._75, andκ=0_._04|
|Disamenity for living in a backyard<br>shack|_BBY_ =0_._74|
|Disamenity for living in a settlement<br>shack|_BIS_=0_._70|
|Agricultural land price (in 2011)|807R/_m_<sup>2</sup>|

Table C.7: Estimated parameters

> 44Recall that the amenity for living in formal housing is normalized to 1.

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<!-- Start of picture text --> 12000<br>Data - interquartile range<br>') Simulation Simulation - interquartile interquartile range<br>——~ Simulation average average<br>40000. +sss«Data average average<br>_ | ee<br>E 8000-4:44<br>€ sooo =|<br>8 4000 3<br>a<br>2000 eS ITT oe entttteeast 3 eee,<br>0 a ‘ Pa iz _ .<br>0 10 20 30 40 50<br>Distance to the the center (km) (km)<br><!-- End of picture text -->

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<!-- Start of picture text --> o* 10* 10*<br>8. J Simulation: 57.6 % 57.6 % % of total total poputation<br>2 4 ‘/ \<br>vw yi td<br>% 5 10 1s 20 2 30 eR s if 40<br>Distance to the center (km)<br><!-- End of picture text -->
<!-- Start of picture text --> W ——2001___ , 10 ——2006___ , 10 1 .<br>z<br>-_—<br>S a { 8 O . ‘ a , ot 4 a )<br>£ ¥ JOR OS Sete O<br>a37° Q aeist 4 ¥ 7 a; cern Oe oy Sos2 2. eer,xa 7} < OF Micsfr PSKere ¢~><br>ef a & “Op Yc gra eee ws<br>3 6) ye | ¢<br>Vv]<br>><br>=<br>4“ 5 6 7 8 9 10 “4 5 6 7 8 9 10 “ $ 6 7 8 9 10<br><!-- End of picture text -->

Log housing prices from data (R/m2)
<!-- Start of picture text --> Initial set of<br>utilities (u;)<br>Set of (u;)<br>Formal private Informal backyard _ Informal settlement<br>Housing demand Qrp(ui, i) wlBey y. WS (x u;<br>Housing supply Spp(xve ) Sip1B(x) N/S(x)<br>Nj (%) Nir (x)<br>Sum across housing Population at<br>types location x<br>nj(x) Adjust utilities<br>Sum across locations For all 1 NO<br>Compare with target |S ni(x) - Ni| <e?<br>YES, the model<br>converged<br><!-- End of picture text -->
<!-- Start of picture text --> b * a x 71 ee ‘<br>DL, ht ace ae Spe ay<br>pe AS ar pues yt<br>F : > . a<br>= % rr ire a - rn<br><!-- End of picture text -->
<!-- Start of picture text --> units<br>© 1000 |<br>es ugh, ene<br><!-- End of picture text -->

<!-- Start of picture text --> \ ‘ 3 units<br>5 © 1000 |<br>ago ‘ Tir *Reatye<br><!-- End of picture text -->