metadata: authors: 0: author: Cai,Yongyang
1: author: Selod,Harris
2: author: Steinbuks,Jevgenijs
count: World
docna: 0: docna: Urbanization and property rights
docty: Policy Research Working Paper
subtopic: Economic Theory & Research,Social Policy,Legal Products,Judicial System Reform,Industrial Economics,Agricultural Economics,Common Property Resource Development,Economic Growth,Legal Reform,Legislation,Regulatory Regimes,Energy Demand,Energy and Environment,Energy and Mining,Armed Conflict
teratopic: Communities and Human Settlements,Poverty Reduction
abstracts: cdata!: Since the industrial revolution, the
            economic development of Western Europe and North America was
            characterized by continuous urbanization accompanied by a
            gradual phasing-in of urban land property rights over time.
            Today, however, the evidence in many fast urbanizing
            low-income countries points towards a different trend of
            “urbanization without formalization”, with potentially
            adverse effects on long-term economic growth. This paper
            aims to understand the causes and the consequences of this
            phenomenon, and whether informal city growth could be a
            transitory or a persistent feature of developing economies.
            A dynamic stochastic equilibrium model of a representative
            city is developed, which explicitly accounts for the joint
            dynamics of land property rights and urbanization. The
            calibrated baseline model describes a city that first grows
            informally, with the growth of individual incomes leading to
            a phased-in purchase of property rights in subsequent
            periods. The model demonstrates that land tenure informality
            does not necessarily vanish in the long term, and the social
            optimum does not necessarily imply a fully formal city,
            neither in the transition, nor in the long run. The welfare
            effects of policies, such as reducing the cost of land
            tenure formalization, or protecting informal dwellers
            against evictions are subsequently investigated, throughout
            the short-term transition and in the long-term stationary state.
repnme: repnme: Urbanization and property rights
display_title: Urbanization and property rights
listing_relative_url: /research/2015/11/25260752/urbanization-property-rights
topicv3: Economic Growth m1326241 471,Personal Property Rights m1326571 1907,Energy and Extractives m1326595 507,Land Tenure m1326631 888,Revolutionary Violence m1327774 2465
publishtoextweb_dt: 2015-11-10T00:00:00Z
historic_topic: Communities and Human Settlements,Poverty Reduction
no_of_pages: 42
author_keys: 0: author_key: 1911374
1: author_key: 750638
2: author_key: 666646
model_extractions: WPS7486 Policy Research Working Paper 7486 Urbanization and Property Rights _Yongyang Cai Harris Selod Jevgenijs Steinbuks_ Development Research Group Environment and Energy Team November 2015
Policy Research Working Paper 7486

# **Abstract**

Since the industrial revolution, the economic development of Western Europe and North America was characterized by continuous urbanization accompanied by a gradual phasing-in of urban land property rights over time. Today, however, the evidence in many fast urbanizing low-income countries points towards a different trend of “urbanization without formalization”, with potentially adverse effects on long-term economic growth. This paper aims to understand the causes and the consequences of this phenomenon, and whether informal city growth could be a transitory or a persistent feature of developing economies. A dynamic stochastic equilibrium model of a representative city is developed, which explicitly accounts for the joint dynamics of land property rights and urbanization. The calibrated baseline model describes a city that first grows informally, with the growth of individual incomes leading to a phasedin purchase of property rights in subsequent periods. The model demonstrates that land tenure informality does not necessarily vanish in the long term, and the social optimum does not necessarily imply a fully formal city, neither in the transition, nor in the long run. The welfare effects of policies, such as reducing the cost of land tenure formalization, or protecting informal dwellers against evictions are subsequently investigated, throughout the short-term transition and in the long-term stationary state.

This paper is a product of the Environment and Energy Team, Development Research Group. 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://econ.worldbank.org. The authors may be contacted at hselod@worldbank.org.

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

Yongyang Cai<sup>_†_</sup> , Harris Selod<sup>_‡_</sup> , and Jevgenijs Steinbuks<sup>_§_</sup>

**JEL classification** : O43, P14, R14

**Keywords** : agglomeration, economic development, informality, land property rights

> _∗_ we are grateful to Gilles Duranton, Remi Jedwab, Kenneth Judd, David Palmer, Gulnaz Sharafutdinova, Anthony Yezer, Yves Zenou and participants to the World Bank/George Washington University 2nd Urbanization and Poverty Reduction Research Conference for useful comments. The views expressed in this paper are those of the authors and do not necessarily reflect those of the world bank, its board of directors or the countries they represent.

> _†_ Yongyang Cai: Hoover Institution, Stanford University and Becker-Friedman Institute, University of Chicago. Email: yycai@stanford.edu.

> _‡_ Harris Selod: Energy and Environment Team, Development Research Group, The World Bank. Email: hselod@worldbank.org.

> _§_ Jevgenijs Steinbuks: Energy and Environment Team, Development Research Group, The World Bank. Email: jsteinbuks@worldbank.org.
# **1 Introduction**

Over the past two centuries, the world has witnessed unprecedented changes in the form of massive population growth, continuous urbanization, and economic development. With the advent of the Industrial Revolution (1760-1840), countries in Western Europe and North America began to shift from agrarian towards industrialized societies. This involved migration from rural areas to cities, which, combined with a decline in urban mortality, caused urban populations to grow at an unprecedented pace. As a result, urbanization rates began to pick up steadily. The proportion of people residing in cities increased dramatically over that period, starting from a low rate of approximately one-tenth throughout the Middle Ages and the early modern period (Bairoch 1988) and reaching today 81.5 percent in North America and 73.4 percent in Europe (United Nations, 2014). Simultaneously, North America and Western Europe experienced unprecedented economic development as urbanization accompanied a profound structural transformation of their economies whereby employment in the agricultural sector decreased and a large manufacturing sector emerged in cities (Bairoch and Goertz 1986, Michaels et al. 2012). At the same time, declining transport costs facilitated trade between cities, which further stimulated industrial growth. To the extent that cities also facilitated innovation, in particular through dynamic knowledge and human capital externalities (Lucas 1988, Romer 1990, Glaeser et al. 1992, Black and Henderson 1999, Bertinelli and Black 2004), it is fair to say that they acted as engines of growth (Henderson 2005 and 2010, Quigley 2009). The dynamics of urbanization both stimulated and were accelerated by economic development. Urban population growth contributed to increase labor productivity (and thus income) thanks to pervasive and significant agglomeration effects whereby workers become more productive when cities become larger (Duranton, 2014a). This positive relation is represented on the cross-sectional graph plotting per capita GDP against the urbanization rate for 189 countries (see Figure 1, Panel A below). In accordance with this graph, it is roughly estimated that a one percentage point increase in the urbanization rate is associated with a five point increase in per capita gross domestic product (see Duranton 2014b). A similar relation emerges when plotting U.N. estimates of per capita production for a cross section of large cities against the corresponding country’s urbanization rate (see Figure 1, Panel B below). The latter graph suggests that cities tend to be more productive in countries which are more urbanized.

Several authors contend that these evolutions in the cities of Western Europe and North America were made possible by the emergence of institutions conducive to economic growth (North 1990 and 1991, Chong and Calder ́on 2000), such as financial development (Bondenhorm and Cuberes 2010), the degree of democratization (Henderson and Wang 2007), and property rights (Acemoglu et al. 2005). Improvement in land property rights, in particular, played a crucial role as it favored the development of manufacturing activities by lowering transaction costs for the transfer of land to the emerging industries that were locating in cities and by making investments not only more secure but also more productive (Knack and Keefer 1995). By stimulating the development of 2
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impersonal markets (Arru ̃nada 2012) and enabling the reallocation of land to more productive uses, institutions defining and facilitating the enforcement of land property rights very likely improved economic efficiency (as argued by Coase 1960 and Demsetz 1967). Even though only a few empirical studies investigated the role played by property rights on urban development, these studies all consistently point towards a significant and positive effect. DeLong and Shleifer (1993) estimate that during the 800 years prior to the Industrial Revolution, West European cities in which property rights were more secure—because they were ruled by an elite made of merchants rather than princes—grew faster than other cities. Acemoglu et al. (2005) show that both urbanization and income growth occurred faster in West European countries than in East European countries because of the emergence of property rights institutions in the former group of countries that served the interests of Atlantic trade between 1500 and the Industrial Revolution. Looking at population density in Britain in the decades surrounding the industrial revolution (between 1660 and 1830), Bogart and Richardson (2009) show that the counties where the transfer of property rights were facilitated by Acts of Parliament experienced faster urbanization.<sup>1</sup> It is important to note that the emergence or reform of property rights institutions is in many instances endogenous and responds to the changing economic context. In theory, property rights institutions will emerge or be strengthened as the benefits of property rights increase (Demsetz 1967, Anderson and Hill 1975, Bubb 2013) and as the political groups that benefit from property rights become sufficiently powerful to weigh in on the legal and institutional framework governing property rights (Libecap 1986, Feder and Feeny 1991). In the historic examples that we already mentioned, this is illustrated by the pressure exerted by Atlantic traders in Western Europe post 1500 to improve institutions that facilitated their activities (Acemoglu et al. 2005) or, in the specific context of Great Britain around the Industrial Revolution, by the public demand for Parliament acts to simplify rules governing land inheritance (Richardson and Bogart 2008). Both in Western Europe and North America, however, the emergence of efficient urban land property rights institutions took considerable time and property rights were only progressively phased in (Konig 1974), possibly due to various political, technical and economic constraints. The political constraints that hindered the development of property rights systems mainly involved the opposition of landowners to registration because of their fear of increased taxation (Offer 1981, Bourillon et al. 2008). It has been theorized, more generally, that landowners may have opposed any form of public protection of property rights (Sonin 2003) as they favored private and selective enforcement of property rights—for which they had a comparative advantage—over public enforcement mechanisms (Kerekes and Williamson, 2010). In addition to such political factors, technical challenges also played an important role in delaying the completion of functioning land property rights systems as illustrated by the repeated attempts over different centuries to establish a complete cadaster in France where manually mapping properties remained extremely costly and time-consuming until the 19th century (Bourillon et al. 2008). Last but not least, a large

> 1The authors focus on “estate acts” which relaxed “strict settlements” whereby land had to descend intact from one generation to the other and was encumbered by a complex system of assignment of stipends to relatives.

4
part of the demand for property rights may have been constrained by the high costs of access to property rights (e.g. registration fees) until a decrease in the costs or until economic development and the ensuing increases in wealth made property rights affordable to a large fraction, if not all the population.<sup>2</sup> It is clear that the dynamic process of urbanization and the development of urban land property rights systems have been intertwined, with emerging property rights institutions facilitating urbanization and, in turn, with urbanization creating the conditions for further improvements of property rights institutions and for the dissemination of property rights. These dynamics are illustrated by the spatial expansion of European cities which often occurred outside the jurisdiction of the existing property rights system until coverage of the system was later extended to these areas or until the demand for property rights in these areas was expressed and met. In this context, residential informality (defined as the lack of formal property rights) initially rose with the spatial expansion of the city before being progressively resorbed over time. Today, nearly all urban dwellers hold a property right in the cities of the developed world.

There are important differences, however, when comparing the process of urbanization and evolution of land property rights systems in developing countries to that of North America and Western Europe. First, the timing and pace of urbanization turned out to be different in the developing world. Contrary to Western Europe and North America, it is only in the twentieth century that developing countries began to urbanize (see Figure 2), with a pace of urbanization that proved much higher than that of earlier industrializing countries at similar stages of urbanization. Strikingly, Latin America rapidly caught up with developed countries and is now more urbanized than Europe (with 79.5 percent of the Latin America’s population residing in cities). As for the Middle East, it has been more than half urban for more than three decades. Today, only Asia and Africa remain predominantly rural (with respective urbanization rates of 47.5 and 40.0 percent). It is in these still predominantly rural regions, however, that the urban population is growing at the fastest rate, with many large cities growing by 4 or 5 percent annually.<sup>3</sup> In Africa, in particular, as a result of these high population growth rates, the urban population is expected to increase almost threefold over the next 35 years, reaching 1.3 billion inhabitants in total. This will raise tremendous challenges for African cities as their net population gain is expected to outweigh in just 35 years the current urban population of Europe and North America combined.

Second, in many parts of the developing world, dysfunctional urban land property rights systems continue to prevail (Durand-Lasserve and Selod, 2009). In African cities in particular, the inconsistent and inappropriate legal frameworks governing land, the poor capacity of land administrations, the lack of political will to reform land laws and practices, and the lucrative

> 2In different country contexts, it is often the case that the cost of accessing property rights is inflated by the rent capture behaviors of land administrations or by “palliative” service industries (e.g. lawyers, notaries) who favor dysfunctional systems that increase the demand for their own service (Offer 1981, Arru ̃nada 2012).

> 3Most of this growth originates from rural urban migration as two thirds of the growth can be attributed to migration to cities and only one third to natural growth (see Jedwab et al. 2015).

5
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rent-seeking behavior of numerous stakeholders in the land sector combine to prevent large fractions of the population from accessing land under secure property rights (Durand-Lasserve et al. 2015). In this context, informal settlements have become a key feature of urbanization in developing countries, reflecting both the low income levels and the institutional weaknesses of these countries. In Sub-Saharan Africa in particular, it is believed that between 60 and 80 percent of the urban population live in informal settlements (UN-Habitat 2010). As informal settlements are very often slums<sup>4</sup> , they may hinder the economic development of cities and exacerbate poverty through a variety of channels including lack of investment in education (Benabou 1993), adverse labor market outcomes (Rospab ́e and Selod 2006), high criminality (Glaeser and Sims 2015), and a negative impact on health (Sclar et al. 2006). The lack of formal land property rights in informal settlements further immobilizes capital that is “locked” and cannot be used for productive activities (de Soto 2000, Safavian et al. 2006).<sup>5</sup> More generally, distorted land markets in cities are a major source of inefficiency that hinder economic activities (Duranton et al. 2015). As a result, some countries (mostly African ones) are urbanizing with insecure land rights and in the absence of economic growth.<sup>6</sup> In view of all the above, the key question raised in our paper is the extent to which the dynamics of urbanization in developing countries will replicate or will diverge from the experience of North America and Western Europe. It is indeed not clear a priori whether the combination of rapid urbanization with distorted property rights systems will affect the timing of urbanization and growth along a path that is comparable to that of industrialized countries, or, on the contrary, if it will set developing economies on a different path of “messy urbanization” with persisting slums and limited long-run economic growth. Projecting the experience of North America and Western Europe onto urbanization in the developing world, there seems to be an implicit consensus among development experts that the first scenario will be happening. In this respect, the “modernization” view of slums sees them as transitory (Marx et al. 2013) and bound to disappear with increases in income as in the case of developed countries: slums, which were quite common in 19th century cities in Europe (Offer 1981), have become almost inexistent in the cities of developed countries. Figure 3 (Panel A) plots the percentage of urban dwellers living in slums on the urbanization rate for a selection of developing countries. In support of this assumption, it shows a negative cross-sectional relation, and thus suggests that the pursuit of urbanization will indeed reduce slums. Similarly, longitudinal data over a short time period (15 years) also indicate a decrease in the prevalence of slums in almost all the countries in the

> 4There is a high correlation between informal settlements and slums. The UN definition of slums includes a high prevalence of households residing under insecure tenure due to weak property rights as one of the possible criteria that determine slums.

> 5De Soto’s intuition that real property rights should enable their holders to use their property as collateral to obtain credit rests on the assumptions that low-income households in slums are credit constrained and that financial institutions would agree to provide mortgage credit to these households and would accept their properties as collateral (Woodruff 2001).

> 6See Fay and Opal (2000) for the seminal paper identifying the puzzle of “urbanization without growth”, and Jedwab (2015) for an explanation based on the resource curse leading to the development of “consumption cities” that do not trigger a virtuous circle of industrialization.

7
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# sample (Figure 3, Panel B).

The likelihood of the first scenario to occur, however, hinges upon the assumption that property rights will continue to become increasingly accessible (as evidenced by Figure 3, Panel B), either under existing institutions or because improved institutions will emerge and facilitate access to property rights. Whereas much progress has been achieved to improve land governance in the developing world (see Deininger et al. 2012), some countries seem to be marred in a status quo of inadequate property rights systems and are unable to push forward institutional reforms in the land sector. There are a number of reasons for this inertia. As noted by North (1990), bad institutional systems can persist over time with the creation of new organizations that have no incentive to create more efficient rules. Moving away from weak property rights systems may be difficult as those who benefit from the weaknesses (e.g., palliative service industries or land grabbers) exercise political clout to prevent reforms. Weak property rights systems may also persist given path dependency in the building of property rights institutions (Libecap et al. 2011, Arru ̃nada 2012). The specific context of high inequality and corruption in developing countries may further make the collective design and enforcement of adequate property rights systems particularly problematic. Even when appropriate legal frameworks are in place, the degree of enforcement of property rights may be weak (Coase 1960, Firmin-Sellers 1995) or the pricing of property rights can be excessive in comparison with private benefits (Feder and Feeny 1991), limiting the “take up” of property rights.

These observations imply that different paths are possible. In the case of countries which, for some reason, have reached efficient land property rights institutions, can it be anticipated that urbanization will lead to a virtuous circle where a functioning land market with secure property rights will enable growth-enhancing agglomeration? For countries with inefficient land property rights institutions, will the same occur with a delay? Or will these economies follow a different urbanization trajectory with limited growth and persisting slums over the long run?

To our knowledge, the joint dynamics of urbanization and of the evolution of land property rights has not been studied. The small empirical literature on urban property rights has mainly been concerned with assessing the capitalization of characteristics associated with formal vs. informal land tenure such as tenure security (Jimenez 1982 and 1984, Friedman et al. 1998, Kim 2004) or transferability (Lanjouw and Levy 2002), or on measuring the effect of formalization (i.e. transforming informal into formal tenure) on a variety of social and economic outcomes including increased labor market participation, reduced child labor, housing improvement, health, and the facilitation of market transactions (see Field 2005 and 2007, Field and Torero 2006, Galiani and Schargrodsky 2010, Macours et al. 2010). As for the emerging theoretical literature, it has mainly been concerned with explaining the demand for informal land tenure. In the wake of Jimenez (1985), a series of papers have explained how squatting can be sustained in an equilibrium where eviction may occur with some probability (Jimenez 1985, Hoy and Jimenez 1991, Turnbull 2008) or where eviction is absent as squatters take action to guard against eviction (Brueckner and 9
Selod 2009, Brueckner 2013, Shah 2014).<sup>7</sup> Departing from the restricted focus on squatters, Selod and Tobin (2015) model the formalization choices of households in a context where land plots may be appropriated by others due to multiple sales. In their model, higher endogenous land prices in the core city provide greater incentives to formalize than at the periphery. Because all these models are static—with the only exception of Turnbull (2008), who explores the strategic choices of landowners regarding the timing of squatter eviction—none of them accounts for the transformation of property rights over time under evolving land prices.

Our paper fills this important gap by proposing a theory that accounts for the joint dynamics of urbanization and the evolution of property rights. We model urbanization through the migration of heterogeneous agents from a representative rural area to a representative city. The prevalence of land tenure informality is determined by urban dwellers’ decisions whether and when to purchase a property right. As land prices increase with agglomeration, formalization becomes more desirable. However, formalizing may be too costly for less productive workers. These workers who may not be able to afford both a land plot and a land title, will postpone the decision to formalize, making themselves potential prey to land grabbers. The model can flexibly account for the variety of situations regarding urbanization dynamics as its behavior critically depends on key parameters regarding institutional quality (the cost and enforcement strength of land property rights), the potential for agglomeration effects, and macroeconomic uncertainty. Contrary to the modernization view of slums, and in line with the observation that “informality may be here to stay”, it allows us to exhibit plausible contexts in which land tenure informality remains in the long run. Deconstructing common-sense intuitions, the model also sheds lights on situations where some level of informality is actually socially desirable. This is because, under the parameters of the economy, informality provides access to the city to low income workers who would not be able to contribute to the urban economy under compulsory formal property rights. In this context, enforcement of strict eviction policies would be counter-productive. The model also provides insights into leaving urbanization to the market, which can be suboptimal under adverse structural parameters, in particular when the potential for agglomeration is low and when land administration charge high fees.

The rest of the paper is organized as follows. Section 2 below presents the model which takes the form of a discrete-time dynamic stochastic game under infinite time but a finite number of states and actions. Section 3 then presents our concepts of social planner and decentralized solutions. Section 4 explains our calibration before Section 5 presents the main results from simulations.

# **2 The model**

We consider an economy with a set of _N_ individuals. Individuals live forever and may either reside in the rural area or in the city. In the city, individuals are exposed to a risk of conflict

> 7For a detailed survey, see Brueckner and Lall (2014).

10
over land that may result in their eviction from the plot they occupy (a “land grab”). Urban dwellers can nevertheless pay to establish a property right on their plot instead of holding it informally. Whereas informally holding a plot does not provide tenure security, formally held plots are protected against land grabs and their owners cannot be evicted. Formally, let us denote _x_<sup>_i_</sup> _t_<sup>=(</sup><sup>_xU,i_</sup> _t , x_<sup>_F,i_</sup> _t_<sup>)the</sup><sup>_vector_ofstatevariablesforarepresentativeindividual</sup><sup>_i_attimet.</sup> _x_<sup>_U,i_</sup> _t_ is the location variable for individual _i_ . It takes value 1 if the household resides in the urban area and 0 if the individual resides in the rural area at time _t_ . _x_<sup>_F,i_</sup> _t_ is the property right variable. _x_<sup>_F,i_</sup> _t_ takes value 1 if the household holds a property right at time _t_ and 0 otherwise. We assume that property rights can only be obtained in the urban area so that living in the rural area with a property right is never a possibility in the model.<sup>8</sup> When living in the city over the period ( _t, t_ + 1), the individual _i_ earns a wage _wt_<sup>_U,i_</sup> for his work during the period but faces a congestion costs _ct_ , which we assume to be the same for all individuals. If residing in the rural area, the individual earns rural income _wt_<sup>_R_whichweassume</sup> is the same for all rural dwellers.<sup>9</sup> There is no congestion cost in the rural area. Wages and the urban congestion cost will be endogenous in the model. In addition, if settling in the city at time _t_ , an individual must purchase a land plot at a price _Rt_ on the land market. Although the price of land is endogenous in the city, because land is abundant in the rural area, we assume for simplicity that the price of land in the rural area is equal to zero without any loss of generality.<sup>10</sup> In our dynamic setting, the land price in the city is determined at the beginning of each period by a macro-economic relation that accounts for city composition and economic conditions (see below). The prevailing land price then drives location choices and grabbing decisions within the period, which in turn affects the fundamentals of the land price for the next period. We assume that there are absentee landlords who, in each period, sell land to urban in-migrants upon arrival to the city and purchase it from voluntary out-migrants upon departure from the city, as well as from land grabbers who resell grabbed land. In our modeling, absentee landlords are price takers over a single period of time. They play the role of “market makers” allowing for a simplified representation of the land market.<sup>11</sup>

> 8The prevalence of property rights in rural areas is very low in many countries. This is the case for instance when tenure in rural areas is overwhelmingly customary. Some countries may not have a land registration system or a cadastre for rural areas. Relaxing this assumption would not change the main outcomes of the model while adding unnecessary complexity.

> 9Productivity in the rural area is low and does not depend on individual intrinsic characteristics.

> 10 _Rt_ could also be interpreted as the time-varying difference between the urban and rural land prices. Under this alternative definition—which would assume an initial land endowment in the rural area for each individual— _Rt_ would be much smaller than under the previous definition as it would only represent the fraction of income needed to complement the proceed of the rural land sale to purchase an urban plot. The two definitions for _Rt_ are mathematically equivalent in our setting under the assumption of a zero price for rural land.

> 11The existence of absentee landlords is also a standard assumption in static urban economics model where they sell land to households competing for land (see Fujita 1989).

11
# **2.1 Decisions**

At the beginning of each period ( _t, t_ + 1), an individual makes two choices: whether to migrate and whether to purchase a property right.

Let us denote **d**<sup>**i**</sup> **t**<sup>= (</sup><sup>_dU,i_</sup> _t , d_<sup>_F,i_</sup> _t_<sup>)thevectorofchoicevariablesmadebyindividual</sup><sup>_i_,where</sup><sup>_dU,i_</sup> _t_ is the migration decision and _d_<sup>_F,i_</sup> _t_ is the land tenure decision.

_d_<sup>_U,i_</sup> _t_ takes the value 1 when the individual decides to migrate to (or remain in) the urban area between time _t_ and _t_ + 1, and the value 0 if the individual decides to migrate to (or remain in) the urban area between time _t_ and _t_ + 1.

Because property rights provide protection against land grabs, formality is preferable over informality. It is, however, costly to establish: an individual residing on an informal plot in the city at time _t_ needs to decide whether to pay a formalization fee _ft_ to obtain a property right between time _t_ and _t_ + 1 or not to pay anything and continue to hold the plot informally.<sup>12</sup> The fee is proportional to the value of the land: with 0 _< μ <_ 1. It is a one-time payment (e.g. registration in a cadastre) and does not need to be paid again in subsequent periods as long as the individual uses the plot, i.e. as long as the individual remains in the city. _d_<sup>_F,i_</sup> _t_<sup>,thelandtenurechoicevariable,takesthevalue1whenthe</sup> individual decides to establish a property right between time _t_ and _t_ + 1 and takes the value 0 otherwise.

# **2.2 Shocks**

Individuals move in and out of the urban area and in and out of residential formality throughout their infinite lifetime. Movements between locations and between land tenure situations are governed by migration and land tenure decisions (as individuals may seek property rights when residing in the city) in the face of productivity shocks and land grab shocks.

Let us start with shocks and denote _ε_<sup>_i_</sup> _t_<sup>= (</sup><sup>_εP_</sup> _t_<sup>_, εG,i_</sup> _t_ ) the _vector_ of shocks faced by individual _i_ in period ( _t, t_ + 1).

Firstly, _ε_<sup>_P_</sup> _t_<sup>isanegativeproductivityshock(suchasaweathershock)thataffectsthewagein</sup> the rural area in period ( _t, t_ + 1). _ε_<sup>_P_</sup> _t_<sup>takesthevalue1whentheshockoccursand0otherwise.</sup> This occurs with exogenous probability _π_<sup>_P_</sup> .

> 12In practice, fees can be paid for the formalization of an informally held plot, or following transactions of formal plots to ensure that they do not revert to informality (for instance by paying to update the registry with the new owners’ name).

12
Secondly, _ε_<sup>_G,i_</sup> _t_ is a “land grab opportunity” that affects informal urban dwellers to the potential benefits of land grabbers. _ε_<sup>_G,i_</sup> _t_ takes the value 1 if an opportunity to grab the plot of individual _i_ arises between time _t_ and _t_ + 1 and 0 otherwise. The shocks _ε_<sup>_G,i_</sup> _t_ for _i_ = 1 _, .., I_ are assumed to be _i.i.d._ and occur with exogenous probability _π_<sup>_G_</sup> . _π_<sup>_G_</sup> reflects tenure insecurity on informal plots. It may reflect the probability that an agent of the land administration or a judge will be willing to facilitate an eviction. It also inversely reflects the degree of protection that informal dwellers may have under current legislation and practices (e.g. if evictions are banned by local authorities and if this ban is implemented, then _π_<sup>_G_</sup> = 0). Only plots faced with a land grab opportunity ( _ε_<sup>_G,i_</sup> _t_ = 1) and which are not held formaly ( _x_<sup>_F,i_</sup> _t_ = 0) will be grabbed or evicted. We will present this in more detail below, after a brief presentation of the timing of decisions and shocks..

# **2.3 Timing of decisions and shocks**

In the model, state variables ( _x_<sup>_U,i_</sup> _t_ and _x_<sup>_F,i_</sup> _t_<sup>)aredefinedatdiscretepointsintimegenerically</sup> denoted _t_ . Decisions ( _d_<sup>_U,i_</sup> _t_ and _d_<sup>_F,i_</sup> _t_<sup>) and shocks (</sup><sup>_εP_</sup> _t_<sup>and</sup><sup>_εG,i_</sup> _t_ ) occur _within_ time periods generically denoted [ _t, t_ +1).<sup>13</sup> We specify here the exact timing of events within a period [ _t, t_ +1). We assume the following sequence: at time _t_ , individuals are located either in the city or the rural area and may or may not have a property right. Shortly after time _t_ , individuals simultaneously decide where they want to locate (possibly relocate) and whether they want to purchase a property right (if they have not done so earlier for the plot they occupy). Immediately after this decision is made, they may then migrate (possibly involving the sale or purchase of land from absentee landlords) and pay for the property right for those settling in the city and wishing to formalize. In the first half period ( _t, t_ + 0 _._ 5), the city faces land grab opportunity shocks (idiosyncratic probabilities of land grab opportunities are drawn). At time _t_ + 0 _._ 5, land plots exposed to grabbing are grabbed if unprotected (see below). . Land grab activities can affect urban wages for the second half of the period (through a change in the city composition as individuals that lose their plot are forced to move to the rural area shortly after time _t_ +0 _._ 5).<sup>14</sup> The rural productivity shock occurs at _t_ + 0 _._ 5 and affects location decisions in the next period. It is important to note here that, in order to be consistent with the nature of our shocks and our modeling of the land market, we assume a period length of several years (exactly 10 years in our simulations).<sup>15</sup>

> 13We index these middle-of-the-period shocks, along with intermediary variables introduced later, with subscript

> _t_ .

> 14The relocations that occur shortly after time _t_ +0 _._ 5 only reflect land grabs and do not result from the location decision which is made at time _t_ and implies relocation shortly after time _t_ .

> 15We assume that land can be fully purchased within one period, which requires a mid-term time horizon. Choosing relatively long periods does not qualitatively alter the results of the model. Calibrated parameters are chosen accordingly.

13
# **2.4 Migration**

Individuals may choose to migrate between the rural area and the city in response to the difference in the urban and rural incomes (net of housing and congestion costs) and given tenure insecurity in the city and the possibility to formalize and the cost of formalization. In our model, in each period, the rural-urban difference in income is affected by the rural productivity shock at time t+0.5, implying: where _w_<sup>_R_</sup> is agricultural productivity in the “good” state of the nature, and _w_<sup>_R_</sup> _._ (1 _− ηε_<sup>_P_</sup> _t_<sup>)is</sup> agricultural productivity in the “bad” state of the nature when the rural productivity shock _ε_<sup>_P_</sup> _t_ (e.g. low rainfall) kicks in. _η_ is the rural income elasticity to the shock with 0 _< η <_ 1.

# **2.5 Land grabs / evictions**

Formal and informal residents in the city may grab land from other informal residents after a land grab opportunity arises ( _ε_<sup>_G,i_</sup> _t_ =1) for an urban plot that is informally held. Let us denote _It_ the set of informal dwellers in the city during the first half period ( _t, t_ + 0 _._ 5) before the land grab opportunity shocks happen. These are households who choose to live in the city at the beginning of the period ( _d_<sup>_U,i_</sup> _t_ = 1), who did not hold a formal property right at the beginning of the period ( _x_<sup>_F,i_</sup> _t_ = 0), and who did not become formal during the period ( _d_<sup>_F,i_</sup> _t_ = 0). By definition, _It_ = � _i|d_<sup>_U,i_</sup> _t ._ (1 _− x_<sup>_F,i_</sup> _t_<sup>)</sup><sup>_._(1</sup><sup>_−dF,i_</sup> _t_<sup>) = 1</sup> � or _It_ = � _i|d_<sup>_U,i_</sup> _t ._ (1 _− x_<sup>_F,i_</sup> _t_ +1<sup>) = 1</sup> �. The land grab opportunity shocks are drawn at a time in ( _t, t_ + 0 _._ 5), defining a subset _It_<sup>_R_</sup> ⫅ _It_ of informal residents at risk of being grabbed, with _It_<sup>_R_=</sup> � _i|ε_<sup>_G,i_</sup> _t .d_<sup>_U,i_</sup> _t ._ (1 _− x_<sup>_F,i_</sup> _t_ +1<sup>) = 1</sup> �. At time _t_ + 0 _._ 5, if a plot _i∈I_<sup>_R_</sup> _t_<sup>, is grabbed, grabbers can immediately sell off the plot on the land market (to absentee</sup> landlords) at the prevailing market land price _Rt_ . The set of informal dwellers who get grabbed _It_<sup>_G_⫅</sup><sup>_I_</sup> _t_<sup>_R_isdefinedby</sup>

where _Ht_ = _i|d_<sup>_U,i_</sup> _t_ = 1 is the set of city dwellers over ( _t, t_ + 0 _._ 5].<sup>16</sup> � � The aggregate profit from land grabbing in period _t_ is

> 16 _ItR_<sup>=</sup><sup>_Ht_isthepathologicalcasewhereallindividualsshouldbegrabbedbutnoonewouldbeleftinthecity</sup> to be a grabber, which cannot be possible. In this (unlikely case), we simply assume that no one gets grabbed in the period.

14
We assume that the land grabbing profit is redistributed to all remaining city-dwellers _i∈J t_ .<sup>17</sup> Because richer (and more powerful) households are more likely to exert grabbing, we assume that the share of the grabbing profit redistributed to each individual is proportional to that individual’s wage _wt_<sup>_i_.18Eachindividualremaininginthecityafter</sup><sup>_t_+ 0</sup><sup>_._5(</sup><sup>_i ∈Jt_)thusobtains</sup> an additional income from grabbing activities:

# **2.6 Agglomeration and congestion**

Considering that an individual resides in the city during time sub-interval ( _t, t_ + 0 _._ 5] if and only if _d_<sup>_U,i_</sup> _t_ = 1, and during time sub-interval ( _t_ + 0 _._ 5 _, t_ + 1] if and only if _x_<sup>_U,i_</sup> _t_ +1<sup>=1,thepopulation</sup> size in the city for those two time intervals can be expressed by the formula There are benefits and costs associated with an increasing urban population, which we present sequentially. Benefits are captured through agglomeration effects, where the productivity (and thus the wage) of each individual in the city depends on the size and composition of the urban labor force (see Duranton, 2013) as well as on land tenure. The latter assumption reflects the fact that informal plot owners have a reduced labor supply as shown for example by Field (2007) in the case of Lima, Peru. Assuming that each individual is characterized by an exogenous ability _ai >_ 0 and contributes _ai_ units of labor to production when residing formally in the city and (1 _− κ_ ) _.ai_ with 0 _< κ <_ 1 when residing informally in the city, we can further define the total efficient labor in the city as If residing in the urban area, the wage of individual _i_ can then be specified as

> 17The set of non-grabbed dwellers is _Jt_ = _Ht_ ⧹ _ItG_<sup>.Itincludesallurbanhouseholdswhowerenotgrabbedin</sup> period _t_ . It is easy to see that _Jt_ = � _i|d_<sup>_U,i_</sup> _t ._ �1 _− ε_<sup>_G,i_</sup> _t ._ (1 _− x_<sup>_F,i_</sup> _t_ +1<sup>)</sup> � = 1� or _Jt_ = � _i|x_<sup>_U,i_</sup> _t_ +1<sup>= 1</sup> <u>�.</u>

> 18Alternatively, we could have considered that redistribution occurs with equal shares _|J_ <u>1</u> _t|_<sup>orthatitreflects</sup> an idiosyncratic ability or willingness to exert corruption.

15
where _σ >_ 0, and _γ >_ 0.<sup>19</sup> Let us now turn to congestion. For simplicity, we assume that congestion translates into a monetary cost borne by city dwellers and is expressed in monetary units. It is measured by a congestion function which depends on the size of the city.<sup>20</sup> We have where _Ns_ is defined by (6), for _s ∈_ ( _t, t_ + 1]. Here, _ws_<sup>_i_,</sup><sup>_cs_,</sup><sup>_W i_</sup> _t,_ 1<sup>,</sup><sup>_W i_</sup> _t,_ 2<sup>,</sup><sup>_Ct,_1,</sup><sup>_Ct,_2aremeasured</sup> per period. Because of the existence of possible shocks, both _Wt,_<sup>_i_</sup> 2<sup>and</sup><sup>_Ct,_2arerandom.</sup> Equation (9) captures the negative externalities associated with agglomeration.<sup>21</sup>

# **2.7 Land prices**

As for land costs, they are only incurred upon arrival in the city (when the individuals need to purchase a land plot). We assume that each individual consumes one unit of land and that there is a unique price for land in the city at any given point in time.<sup>22</sup> We assume that the ratio of the land price to the average income is constant. This long-term relation has been established at the city level both theoretically and empirically in the literature (see Ried and Uhlig, 2009, Malpezzi, 1999, and Leung, 2014). Since transactions only occur during the first half of the period (households purchase or sell land at the beginning of the period, and grabbers sell off grabbed land at time _t_ + 0 _._ 5), the price is a function of the population and wage that prevail during ( _t, t_ + 0 _._ 5], so we have

> 19The formula can be derived by assuming a production function

with _wt_<sup>_i_=</sup><sup>_ai dY_</sup> _dL_<sup>.</sup>

> 20Congestion here reflects the various externalities associated with city size, including travel distance and time (Izraeli and McCarthy, 1985), pollution (Lamsal _et al._ , 2013) or criminality (Glaeser and Sacerdote, 1999).

> 21The formula can be derived as the average transport cost in a mono-centric city with convex transport costs (see Fujita, 1989).

> 22This can be viewed as the average price of land in the city. We could adopt a more sophisticated representation of land with prices that vary depending on within-city location as in Selod and Tobin (2015) but this would make the model more complex without affecting the main results.

16
where 0 _< λ <_ 1 .<sup>23</sup>

# **2.8 Land transactions**

Land is transferred whenever there are population movements between the city and the rural area or when land is grabbed (and immediately sold off).

Migrants to the city only purchase land upon arrival. We can thus define a variable that measures land purchases as _Pt_<sup>_i_=</sup><sup>_Rt_ifindividual</sup><sup>_i_arrivesinthecityshortlyaftertime</sup><sup>_t_and</sup><sup>_P i_</sup> _t_<sup>=0</sup> otherwise. This is modeled using the formula Conversely, individuals who voluntarily decide to leave the city shortly after time _t_ sell their land back upon departure at a price _Rt_ . We can define a variable that measures land sales as _St_<sup>_i_=</sup><sup>_Rt_</sup> if individual _i voluntarily_ leaves the city during period ( _t, t_ + 1] and _St_<sup>_i_=0otherwise.24Thisis</sup> obtained for<sup>25</sup>

# **2.9 Location and land tenure transitions**

We now turn to the general definition of transitions, which are defined by the correspondence Γ : **x**<sup>_i_</sup> _t_<sup>_�−→_</sup><sup>**x**</sup><sup>_i_</sup> _t_ +1<sup>asafunctionof</sup><sup>**x**</sup><sup>_i_</sup> _t_<sup>,</sup><sup>**d**</sup><sup>_i_</sup> _t_<sup>and</sup><sup>_εi_</sup> _t_<sup>where</sup><sup>**x**</sup><sup>_i_</sup> _t_<sup>_∈{_0</sup><sup>_,_1</sup><sup>_}_2,</sup><sup>**d**</sup><sup>_i_</sup> _t_<sup>_∈{_0</sup><sup>_,_1</sup><sup>_}_2and</sup><sup>_εi_</sup> _t_<sup>_∈{_0</sup><sup>_,_1</sup><sup>_}_2.</sup> This correspondence completely defines what happens between time _t_ and time _t_ + 1 to the location and tenure (state variables) of an individual in all possible states and facing all possible combinations of shocks.

We can now infer the laws of motion that fully characterize the transitions between _t_ and _t_ + 1 as functions of states at time _t_ and decisions and shocks between time _t_ and time _t_ + 1. We have

> 23

Although the land price fluctuates within each period because of involuntary population movements between the first and the second half of each period, transactions only occur at the beginning of the first half of the period following voluntary location decisions. We also assume that sales of grabbed land occur at the current land market price, which can be justified by stickiness.

> 24Individuals who are victim of a land grab must leave the city and cannot recoup their land investment.

> 25Note that in each period the total net payment received by absentee landlords is Π _t_ =<sup>�</sup><sup>_N_</sup> _i_ =1 � _Pt_<sup>_i−S_</sup> _t_<sup>_i_</sup> � _−_ � _i∈It_<sup>_GRt_,whichcanbepositiveornegative.Inthelongrun,whenthepopulationofthecityandtheshareof</sup> informal dwellers stabilize, this is expected to be zero as there will be as many sales as purchases.

17
Transition formula (13) states that in order to reside in the city at time _t_ + 1, individual _i_ must have decided to remain in or to move to the city between time _t_ and _t_ +1 and either already held a property right for his urban plot at time _t_ , decided to formalize between time _t_ and time _t_ + 1, or did not face a land grab shock between time _t_ and _t_ + 1. Transition formula (14) recognizes that to have a property right at time _t_ + 1, the individual must have decided to remain in or to move to the city between time _t_ and _t_ + 1 and either already had a property right at time _t_ or acquired one between time _t_ and _t_ + 1.

# **3 Stochastic game and social planner solution**

We will now determine how the system evolves under two different optimization problems and associated solution concepts (stochastic game and social planner optimum). This involves determining how states at the beginning of one period are followed by policy functions during the period (decisions in our framework) that determine states at the beginning of the following period. To do this, let us define the _strategy_ for individual _i_ as the function from the vector of states **x** _t_ = ( _x_<sup>_U,i_</sup> _t , x_<sup>_F,i_</sup> _t_<sup>)</sup><sup>_i_=1</sup><sup>_,...,N_</sup> into individual decisions **d**<sup>_i_</sup> _t_<sup>=(</sup><sup>_dU,i_</sup> _t , d_<sup>_F,i_</sup> _t_<sup>),whichwesimply</sup> � � denote _Dt_<sup>_i_(</sup><sup>**x**</sup><sup>_t_).26</sup> To understand how strategies are determined, we now need to understand how individuals are affected by decisions. To do that, observe that at time _t_ , when individual _i_ decides to live in the rural area during the period ( _t, t_ + 1] (i.e. _d_<sup>_U,i_</sup> _t_ = 0), he will receive � _Wt,_<sup>_R_</sup> 1<sup>+</sup><sup>_W R_</sup> _t,_ 2� _/_ 2 + _St_<sup>_i_in</sup> the period. When individual _i_ decides to live in the urban area during the period, he will pay _d_<sup>_F,i_</sup> _t_<sup>_.ft_+</sup><sup>_P i_</sup> _t_<sup>atthebeginningoftheperiod,receivewage</sup><sup>_W U,i_</sup> _t,_ 1<sup>_/_2andincurthecongestioncost</sup> _Ct,_ 1 _/_ 2 during the first half period. During the second half period, if the individual is forced to leave the city because he faced a land grab opportunity at _t_ + 0 _._ 5 (i.e. _ε_<sup>_G,i_</sup> _t_ = 1) and did not have a property right (i.e., _x_<sup>_F,i_</sup> _t_ +1<sup>=0),thenhewillreceive</sup><sup>_W R_</sup> _t,_ 2<sup>_/_2;otherwise,hewillreceivea</sup> wage _Wt,_<sup>_U,i_</sup> 2<sup>_/_2</sup><sup>_,_incuracongestioncost</sup><sup>_Ct,_2</sup><sup>_/_2,andreceivehisfractionofthelandgrabproceeds</sup> _Gt_<sup>_i_.Thenetincomeoftheindividualwhodecidestoresideinthecityin(</sup><sup>_t, t_+ 1)isthus</sup> Thus, we have the utility of individual _i_ over the period ( _t, t_ + 1]

> 26Note that because decisions are made before shocks occur in each period, they are only functions of the current states but not of the realization of the shocks.

18
The relevant part of the expression when the individual decides to reside in the rural area does not include the term _d_<sup>_F,i_</sup> _t_<sup>_.ft_because,bydefinition,ahouseholdwillnotsimultaneouslydecide</sup> to formalize and migrate to the rural area. The specification reflects risk-aversion through the parameter _α_ .

With the above framework, we can now solve the two different optimization problems that respectively correspond to the social planner solution or the stochastic game solution that results from individual choices.

# **3.1 Social planner solution**

At the beginning of each period _t_ , the planner assigns to individuals a sequence of location and formalization decisions so as to maximize the expected value of the sum of discounted stream of aggregated utilities and profits, considering the exogenous shock probabilities (and likelihood of transitions between states). The social planner’s objective function at time _t_ is where _U_ ( **x** _t,_ **d** _t_ ) =<sup>�</sup><sup>_N_</sup> _i_ =1<sup>_u_</sup> _t_<sup>_i_(</sup><sup>**x**</sup><sup>_t,_</sup><sup>**d**</sup><sup>_t_)aggregatesindividualpreferencesandEistheexpectation</sup> operator<sup>27</sup> and _β ∈_ (0 _,_ 1) is the discount factor and the expectation is taken over the random productivity and land grab opportunity shocks. Note that the shocks enter in (16) through **x** _τ_ which, as we have seen, is a function of **x** _τ −_ 1, **d** _τ −_ 1 and _ετ −_ 1. The transition laws of **x** _t_ are given by (13) and (14) for each individual. For simplicity, we denote the transition laws as **x** _t_ = _g_ ( **x** _t−_ 1 _,_ **d** _t−_ 1 _, εt−_ 1).

Because it is an infinite-horizon problem, the decision depends only on the current state and we can write the corresponding Bellman equation (Bellman, 1957) as where **x** is the current-period state, and **x** + is next-period state dependent on **x** , decision **d** , and the shocks _ε_ , i.e., **x** + = _g_ ( **x** _,_ **d** _, ε_ ). The optimal policy function is _D_<sup>_i_</sup> ( **x** ) for individual _i_ .

_Definition of social planner solution_ : We define the social planner solution as the value function _V_<sup>_s_</sup> ( **x** ) and the set of individual policy functions **D** ( **x** ) = � _D_<sup>_i_</sup> ( **x** ) : _i_ = 1 _, ..., N_ � that solve the Bellman equation (17) given the Social Welfare Function _U_ .

> 27We consider a utilitarian social welfare function but any other social welfare function can be used.

19
The right-hand side of the Bellman equation (17) can be expressed as _Γ_ ( _V_<sup>_s_</sup> )( **x** ) where _Γ_ is the Bellman operator. We see that the value function _V_<sup>_s_</sup> ( **x** ) is a fixed point of the Bellman operator, so we can use an iterative method to find _V_<sup>_s_</sup> ( **x** ). We apply the value function iteration, that is, we first choose an initial guess of _V_<sup>_s_</sup> ( **x** ), denoted as _V_ 0<sup>_s_(</sup><sup>**x**),andthencompute</sup><sup>_V_</sup> _t_<sup>_s_(</sup><sup>**x**) =</sup><sup>_Γ_(</sup><sup>_V_</sup> _t_<sup>_s_</sup> _−_ 1<sup>)(</sup><sup>**x**)</sup> for _t ≥_ 1 until the iteration converges. A detailed discussion of value function iteration is available in Judd (1998).

# **3.2 Stochastic game**

Here we are trying to solve the stochastic game problem where every individual wants to maximize the present value of his own expected stream of utilities. Mathematically, instead of (16), we now have the multi-objective optimization problem: Note that **x** _t_ and **d** _t_ are still states and decision vectors for all individuals in the economy.

Since every individual has his own value function _V_<sup>_i_</sup> ( **x** ) and policy function _D_<sup>_i_</sup> ( **x** ) which depend on the states of all individuals, we also need to replace (17) with the following set of Bellman equations (one per individual): where **x** + = _g_ ( **x** _,_ **d** _, ε_ ).

In this context, our concept of equilibrium is the stationary Markov-perfect equilibrium as in Maskin and Tirole (1987, 1989a, and 1989b).<sup>28</sup> It is a natural equilibrium concept to adopt in a dynamic setting such as ours where individuals optimize their own discounted utility taking into account other individuals’ decisions.

_Definition of equilibrium_ : We define a stationary Markov-perfect equilibrium as the set of value functions **V**<sup>_all_</sup> ( **x** ) = � _V_<sup>_i_</sup> ( **x** ) : _i_ = 1 _, ..., N_ � and the set of individual policy functions **D** ( **x** ) = � _D_<sup>_i_</sup> ( **x** ) : _i_ = 1 _, ..., N_ � such that for any individual _i_ = 1 _, ..., N_ and for any vector of states **x** and other players’ actions **d**<sup>_−i_</sup> = � **d**<sup>1</sup> _, ...,_ **d**<sup>_i−_1</sup> _,_ **d**<sup>_i_</sup> _, ...,_ **d**<sup>_N_�</sup> with **d**<sup>_j_</sup> = _D_<sup>_j_</sup> ( **x** ) for _j_ = _i_ , the value function _V_<sup>_i_</sup> ( **x** ) solves the Bellman equation in (19), and the individual strategy _D_<sup>_i_</sup> ( **x** ) solves the right-hand side of the equation (19).

We denote the right-hand side of the system of Bellman equations _F_ ( **V**<sup>_all_</sup> )( **x** ), where _F_ is the Bellman operator vector. **V**<sup>_all_</sup> ( **x** ) is the fixed point of _F_ . Thus, as for the the social

> 28The same equilibrium concept has been widely applied in the literature on industry dynamics, see e.g., Ericson and Pakes (1995), Maskin and Tirole (2001), and Doraszelski and Satterthwaite (2010).

20
planner problem, we can also solve for **V**<sup>_all_</sup> ( **x** ) by applying an iterative method. However, it is time consuming to compute _F_ for problems with large _N_ , because of three levels of “curseof-dimensionality”, whereby computational cost increases exponentially in the number of state variables, the number of decision variables, and the number of shock variables for computing expectation. To address this issue Pakes and Mcguire (1994) introduced a computational method for computing Markov-perfect Nash equilibria for stochastic games with continuous choices. In a related work, Doraszelski and Judd (2012) analyze continuous-time stochastic games to avoid the curse of dimensionality in computing expectations. In this paper, we use a variant of the method of Pakes and Mcguire (1994) to find a stationary Markov-perfect equilibrium for our discrete-time, infinite-horizon stochastic game with discrete states and discrete choices. That is, we choose the optimal policy functions of the social planner problem as our initial guess of **D** ( **x** ) for the stochastic game, denoted as **D** 0( **x** ) = � _D_ 0<sup>_i_(</sup><sup>**x**) :</sup><sup>_i_= 1</sup><sup>_, ..., N_</sup> �. We then let the initial guess of **V**<sup>_all_</sup> ( **x** ) be **V** 0<sup>_all_(</sup><sup>**x**)=E</sup> � _u_<sup>_i_</sup> ( **x** _,_ **D** 0( **x** ))� _/_ (1 _− β_ ). Denoting ( **D**<sup>_−_</sup> _t_<sup>_i_(</sup><sup>**x**)</sup><sup>_,_</sup><sup>**d**</sup><sup>_i_)=</sup> ( _Dt_<sup>1(</sup><sup>**x**)</sup><sup>_, ..., D_</sup> _t_<sup>_i−_1</sup> ( **x** ) _,_ **d**<sup>_i_</sup> _, Dt_<sup>_i_+1</sup> ( **x** ) _, ..., Dt_<sup>_N_(</sup><sup>**x**)),wecompute</sup> for each _i_ = 1 _, ..., N_ , and let _Dt_<sup>_i_(</sup><sup>**x**) be the optimizer for individual</sup><sup>_i_at the</sup><sup>_t_-th iteration for</sup><sup>_t ≥_1.</sup> At iteration _t_ , we know the value function and policy functions at the previous iteration _t −_ 1 for every individual. Each individual determines his own optimal decision under the assumption that the other individuals make their decisions at _t_ that replicate their policy functions at _t −_ 1. Note that although the _F_ operator has to be solved jointly for all individuals, equation (20) can be solved separately for each individual _i_ . This makes equation (20) much easier and faster to solve. We iterate over _t_ until both **V** _t_<sup>_all_</sup> and **D** _t_ ( **x** ) converges. The converged value and policy functions are the solution to the stochastic game.

From Maskin and Tirole (2001), we know that there exists a Markov-perfect equilibrium for our stochastic games with finite action spaces. However, our stochastic games may still have multiple equilibria, although Maskin and Tirole (2001) point out that the Markov-perfect equilibrium concept, a refinement of the Nash equilibrium, can be applied to eliminate or reduce the multiplicity of equilibria in dynamic games. Recently, Borkovsky et al. (2010) applied the homotopy approach to detect equilibrium multiplicity for games with continuous strategies, and Judd et al. (2012) developed all-solution homotopy methods to find all equilibria of static and dynamic games with continuous strategies. Yeltekin et al. (2015) provide a numerical method for computing all subgame-perfect equilibria of dynamic games with both discrete and continuous strategies, but their examples consist of only two or three-player dynamic games even with massive parallelization. As our examples are stochastic games and have a large number of players (in our example _N_ = 5 players and 2 _N_ = 10 dimensions of states), it is still challenging to find all equilibria for such a large-dimensional problem, so we leave this for future research.

21
# **4 Model calibration**

We choose values for the model parameters based on econometric estimates in the economic literature on urbanization and development. When econometric estimates are not available, we calibrate the remaining model parameters to match observed patterns of urbanization in developing countries. For most significant model parameters, we conduct a sensitivity analysis over the range of their plausible values. Table 1 lists the model’s parameters and shows the chosen values.

We set the number of economic agents to 5 in order to balance the model’s computational feasibility with tractability. We interpret the number of agents as population quantiles. We consider a uniform distribution of abilities over (0 _,_ 1] with _ai_ = _i/N_ for _i_ = 1 _, ...N_ . Agent _N_ is thus the individual of highest ability _aN_ = 1. We normalize the before-productivity-shock rural wage _w_<sup>_R_</sup> to 1. We set the informal residency penalty on productivity _κ_ equal to .2, consistently with Field (2007). We choose a value of 3 for _σ_ . In developed economies for which state-ofthe-art studies control for sorting according to ability, an increase in city size of 10 percent causes productivity to increase by between .3 and .8 percent (Duranton, 2009). We thus choose a benchmark value of _γ_ of 11, which yields productivity increases of .4 to .7 percent depending on city size.<sup>29</sup> We choose a value of .2 for the rural wage elasticity to adverse shocks, _η_ , based on Jayachandran (2006) who estimated an rural wage elasticity of 17 percent to a weather shock. Wet set the probability of adverse income shock in rural area, _π_<sup>_P_</sup> , to .3 based on Damberg and AghaKouchak (2013), who report 12 episodes between 1979 and 2013 where more than 20 percent of the land in the Southern Hemisphere was under moderate to severe drought. This corresponds to a drought every three years or a yearly probability of a drought of approximately 1/3.<sup>30</sup> As regards the congestion cost parameters, we choose a value of 1 for _b_ and .3 for _δ_ . These numbers are based on the average of Kahn (2010) estimates, who reports city-size elasticities of .14, .25 and .49 for commute times, carbon monoxide pollution and for violent crime respectively. Our choice of _λ_ is based on Malpezzi (1999), who reports a median housing price-to-income ratio in 133 US MSAs of 4.5. Assuming that land represent two-thirds of total housing costs and considering that our period is 10 years, this leaves us with a value of .3.

For the formalization cost parameter, _μ_ , we choose a value of .3. There is a wide variety of situations in terms of formalization contexts, with costs ranging from a few percentages of land value to up to 100 percent or more of the value of the land (see Durand-Lasserve et al _._ , 2015). For _π_<sup>_G_</sup> we assume a value of .1 which is consistent with the high prevalence of conflicts in the cities of developing countries.

> 29In our model, wage elasticity with respect to urban labor force size is _γ._ (1+ _<u>L</u> L_ )<sup>.Since</sup><sup>_L_iscomprisedbetween</sup> 1 and 5, a value of _γ_ comprised between 10.4 and 16.6 ensures that wage elasticities are comprised between 0.03 and 0.08 in any state of the economy.

> 30.3 is a conservative estimate as other shocks affecting incomes such as conflicts, natural disasters or health crises can also occur (see Zceleczky and Yosef, 2014).

22
Table 1: Parameter Values of Calibrated Model

|Parameter|Description|Value|
|---|---|---|
|_I_|Number of economic agents|5|
|_ai_|Agents’ ability|_{_0.2,0.4,0.6,0.8,1_}_|
|_κ_|Informal residency productivity penalty|0.2|
|_σ_|Agglomeration scaling factor|3|
|_γ_|Income efect of agglomeration|5, 11, 100|
|_η_|Rural wage elasticity of adverse shocks|0.2|
|_b_|Congestion scaling factor|1|
|_δ_|Congestion elasticity of agglomeration|0.3|
|_λ_|Land price to rent ratio|0.3|
|_μ_<br>|Cost of land tenure formalization|0.3, 0.9|
|_π_<sup>_P_</sup><br>|Probability of adverse shock in rural area|0.3|
|_π_<sup>_G_</sup>|Probability of land grab attempt / eviction enforcement|0.05, 0.1, 0.5|
|_α_|Risk aversion parameter<br>|2|
|_β_|Discount rate (10 year period)|0_._86|

We set the value of the risk aversion parameter, _α_ , to 2, which is consistent with reported estimates in the meta-analysis conducted by Havranek et al. (2015). We use an annual discount rate of .985, which is consistent with the number adopted by long run dynamic computational forward-looking economic models, e.g., Nordhaus’ DICE model (Nordhaus, 2008). Given that a period spans over 10 years in our model, this translates to ( _._ 985)<sup>10</sup> = _._ 86.

Given a significant range of heterogeneity for some model parameters, we conduct a sensitivity analysis to demonstrate the model’s robustness to chosen parameter values. Specifically, we analyze our results’ sensitivity to variations in following model parameters: (a) income effect of agglomeration, _γ_ , of 5 (high agglomeration potential) and 100 (low agglomeration potential); (b) cost of land tenure formalization, _μ_ , of .9 (high formalization cost); and (c) probability of land “grab”, _π_<sup>_G_</sup> of .05 (low chance of eviction) and .5 (high chance of eviction).

# **5 Simulation Results**

We solve equations (17) and (19) as explained in section 3 for a set of parameter values described in section 4 to obtain the value functions that can be used to infer stationary states and corresponding optimal decision rules. We then simulate dynamic urbanization paths corresponding to the social optimum and the Markov perfect equilibrium based on the realization of randomly drawn shocks _ε_<sup>_P_</sup> _t_<sup>and</sup><sup>_εG,i_</sup> _t_ .

Figure 4 illustrates one of such simulated paths that corresponds to solving for a Markov-perfect equilibrium under a low formalization cost, low agglomeration potential and low risk of eviction. For all simulations we start with an empty set of the city with no residents.<sup>31</sup> We see that

> 31We have also run simulations for an alternative assumption of one, the most productive, informal urban

23
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and calculate the expected urbanization and land formality rates.

Figures 5 and 6 show expected urbanization and land formality rates that correspond to the simulated Markov-perfect stationary equilibria (panel a) and social optima (panel b).<sup>32</sup> Figure 5 illustrates the variations in urbanization and land formality outcomes given different sizes of agglomeration potential and probabilities of land eviction when the land formalization cost is low ( _μ_ = 0 _._ 3). Figure 6 shows the same simulation results when the land formalization cost is high ( _μ_ = 0 _._ 9).

Beginning with Figure 5 (panel a), we see that when agglomeration potential is high and eviction probabilities are low to moderate, the Markov-perfect stationary equilibria is consistent with an expected urbanization rate of 60 percent. When the agglomeration potential is low or moderate and eviction probabilities are high, the expected urbanization rate declines to 40 percent. Figure 6 (panel a), shows a similar pattern. These results lead to the following proposition: _Proposition 1a: The urbanization rate in the Markov-perfect stationary equilibrium increases when (a) eviction enforcement is weak, and (b) agglomeration potential is high._ The underlying intuition behind Proposition 1a is as follows. The urbanization dynamics in our model are driven by the realized benefits of moving to the city. Agents with lower abilities can only find it individually optimal to move to the city when agglomeration potential is high enough to compensate for the urban rent and congestion costs. Even in these situations the decision to move to the city is not warranted. As their income is not sufficient to pay for formalization costs, agents choose to stay informal for some time before they can afford paying for land titles. When the probability of land eviction is high, low productive and risk-averse agents will find it suboptimal to move to the city.

Now, let us turn to Figure 5 (panel b), which represents simulated results for the social optimum, and let us compare these results to those from panel (a). We see that irrespective of variations in model parameters, the expected urbanization rate is 60 percent. The expected urbanization rates are the same for the social optimum and the Markov-perfect stationary equilibrium when agglomeration potential is high and eviction probabilities are low to moderate. The expected urbanization rates are, however, higher for the social optimum that for the Markov-perfect stationary equilibrium for low or moderate agglomeration potential and high eviction probabilities. A similar pattern is observed when we compare panels (a) and (b) of Figure 6, although the expected urbanization rate of 60 percent is not always sustained when land tenure formalization costs are higher. These results lead to the following proposition: _Proposition 1b: The urbanization rate under the Markov-perfect stationary equilibrium is lower than in the social optimum when (a) agglomeration potential is low, and (b) eviction enforcement is strong._

> 32The numerical values of expected urbanization and land formalization rates that correspond to the model simulations are shown in appendix Table A.1.

25
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To understand the logic behind Proposition 1b, it is important to recall the important distinction between the allocation of urban residents under the social optimum and the Markov-perfect equilibrium. In the Markov-perfect equilibrium, all agents move sequentially and make their land formalization decisions simultaneously. In the social optimum all decisions are made by the joint welfare-maximizing social planner, who has discretion over the timing of individuals moving into the city. The social planner may then choose to move less productive agents into the city first and allow them to obtain land titles before they can be evicted by more productive agents entering the city. Proposition 1b makes an important claim that the path of urbanization itself matters for final urbanization outcomes - an issue completely overlooked in static models of urbanization.

Combining propositions 1a and 1b yields the following corollary: _Corollary 1: In cities that have low agglomeration potential and strong eviction enforcement, the urbanization rate is low and lower than the social optimum._ This finding resonates well with the observed over-enforcement of evictions in the developing countries. The model suggests that, in some contexts, local governments may have done more harm than good by checking migration to cities through eviction of informal settlements (see Lall et al., 2006, on the causes of over-restrictive internal migration policies and Durand-Lasserve and Selod, 2009, on eviction policies). Over-enforcement of evictions may have caused some cities to be undersized (i.e., overlooking the potential benefits of further agglomeration) in a way that is similar to the enforcement of migration restrictions (see Au and Henderson, 2006, on the case of Chinese cities).

The next important issue to investigate is that of land tenure formalization. We see from Figure 5 (panel a) that when land tenure formalization costs are low, all agents that choose to move to the city also choose to formalize. This does not always happen when formalization costs are high. Figure 6 shows that informal residency can be sustained in the Markov-perfect stationary equilibrium when the eviction probability is low. If the city exhibits medium to high agglomeration potential, the observed informality rate is two-thirds. When the city’s agglomeration potential is low, all residents are informal. These observations constitute the basis for the next proposition: _Proposition 2a: Informality can be sustained in the long run in the Markov-perfect stationary equilibrium. Informality is all the more likely to prevail as formalization costs are higher, eviction enforcement is weak, and agglomeration potential is low. For low land formalization costs, informality is resorbed._ The underlying explanation for Proposition 2a is fairly straightforward. Agents’ decisions to formalize are determined by the difference between the expected marginal benefits of avoided eviction and potential gains from evicting other informal residents, and the realized marginal cost of formalizing land tenure. As the probability of land eviction declines, the latter effect 28
tends to dominate and no one formalizes. When the city’s agglomeration potential is low, the city is smaller (Proposition 1a), and the expected gain from evicting other informal residents diminishes. This, in turn, leads to an even higher share of informal residents in the city.

Figure 5 (panel b) shows that even when formalization costs are low, informality is desirable in the social optimum when both the city’s agglomeration potential and the probability of eviction are low. In this case, the resulting urbanization rate (59 percent) is higher compared to that observed in the Markov-perfect equilibrium (40 percent). We observe a similar phenomenon when the formalization cost is high. Figure 6 (panel b) shows that for low to medium values of the city’s agglomeration potential and the eviction probability, informality is more prevalent under the social optimum, and the expected urbanization rate is higher (56 percent) compared to that observed in the Markov-perfect equilibrium (40 percent). This finding leads to the following proposition:

# _Proposition 2b: The social optimum does not necessarily require a fully formal city._

To understand this proposition it is important to point out that our model considers an utilitarian social optimum defined as the discounted sum of of expected utilities of all economic agents. Under this utilitarian social optimum, the social planner may choose to relocate an individual to the city even if it is individually optimal for that agent not to migrate into the city. In this case, the social planner’s choice is driven by the positive agglomeration externalities this agent imposes on other city residents. If the expected utility gain from agglomeration externalities for other city residents exceeds the utility loss for that individual, the social planner will find it optimal to relocate the agent to the city. However, in view of the high formalization costs, the social planner may not find it optimal to require that individual to formalize, leading to greater informality.

29
Table 2: Simulated Value Functions

|_γ_|_μ_|_π_<sup>_G_</sup>|_V_ <sup>1</sup>|_V_ <sup>2</sup>|_V_ <sup>3</sup>|_V_ <sup>4</sup>|_V_ <sup>5</sup>|�_V i_|_V _<sup>_s_</sup>|
|---|---|---|---|---|---|---|---|---|---|
|5|0.3|0.05|-7.38|-7.38|-2.93|-1.92|-1.43|-21.04|-21.04|
|5|0.3|0.1|-7.38|-7.38|-2.93|-1.92|-1.43|-21.04|-21.04|
|5|0.3|0.5|-7.38|-7.38|-2.93|-1.92|-1.43|-21.04|-21.04|
|5|0.9|0.05|-7.38|-7.38|-3.41|-2.09|-1.44|-21.70|-21.50|
|5|0.9|0.1|-7.38|-7.38|-2.93|-1.92|-1.43|-21.04|-21.04|
|5|0.9|0.5|-7.38|-7.38|-7.38|-1.94|-1.45|-25.54|-21.04|
|11|0.3|0.05|-7.38|-7.38|-3.64|-2.32|-1.70|-22.42|-22.42|
|11|0.3|0.1|-7.38|-7.38|-3.64|-2.32|-1.70|-22.42|-22.42|
|11|0.3|0.5|-7.38|-7.38|-7.38|-2.26|-1.68|-26.08|-22.42|
|11|0.9|0.05|-7.38|-7.38|-4.31|-2.49|-1.69|-23.25|-23.16|
|11|0.9|0.1|-7.38|-7.38|-3.64|-2.32|-1.70|-22.42|-22.42|
|11|0.9|0.5|-7.38|-7.38|-7.38|-2.26|-1.68|-26.08|-24.25|
|100|0.3|0.05|-7.38|-7.38|-7.38|-2.54|-1.87|-26.55|-23.77|
|100|0.3|0.1|-7.38|-7.38|-7.38|-2.54|-1.87|-26.55|-23.73|
|100|0.3|0.5|-7.38|-7.38|-7.38|-2.54|-1.87|-26.55|-23.73|
|100|0.9|0.05|-7.38|-7.38|-7.38|-2.70|-1.96|-26.81|-24.35|
|100|0.9|0.1|-7.38|-7.38|-7.38|-2.54|-1.87|-26.55|-24.08|
|100|0.9|0.5|-7.38|-7.38|-7.38|-2.54|-1.87|-26.55|-25.85|

Once we have established how urbanization and informality rates respond to the model parameters under different Markov-perfect equilibria and social optima, we can turn our attention to the welfare analysis. Table 2 shows the values of the simulated value functions of the five economic agents under Markov-perfect equilibria (columns 4-8), the sum of their value functions (column 9), and the value function corresponding to the social optimum (column 10). We see from Table 2 that the sum of agents’ utilities is highest in cities with the highest agglomeration potential ( _γ_ = 5) and the lowest formalization cost ( _μ_ = 0 _._ 3) irrespective of the values of the eviction probability. In this case the sum of agents’ expected utilities is the same as the expected sum of utilities, which implies that the Markov-perfect equilibria are also the social optima. This result leads to the following proposition: _Proposition 3: Simulated Markov-perfect stationary equilibria Pareto dominate other equilibria and are socially optimal under (a) high agglomeration potential and (b) low formalization costs._ Proposition 3 implies that the equilibria observed in developed countries, where the agglomeration potential is high and formalization costs are low, are likely to be socially optimal. Urbanization can thus be left to the market when the parameters of the economy are favorable. The proposition also highlights that in the absence of effective policies to improve those parameters in developing countries (e.g., making formalization more affordable), urbanization will likely be 30
Careful examination of Table 2 shows that other equilibria may not be ranked unanimously as economic agents rank them differently. The expected agents’ utilities are non-monotonous functions of (a) agglomeration potential, (b) eviction enforcement and (c) formalization costs. More productive agents that choose to reside in the city may face a trade-off between maximizing their wages (which depend on the city’s agglomeration potential and agent’s abilities) and increasing their gains from potential land grabs of informal residents’ land plots as these two effects may or may not go in the same direction.

To further elaborate on this point, consider a city with medium agglomeration potential ( _γ_ = 11). We can see from Table 2 that, regardless of formalization costs, the social optimum is reached for low to medium probabilities of eviction. Strict eviction enforcement deters in-migration of less productive agents who cannot afford to pay for land tenure formalization before they can establish themselves in the city. However, by looking at individual payoffs under different land eviction regimes, we see that more productive agents (4 and 5) achieve higher levels of expected utility when the eviction probability is high. For these agents, the marginal costs exceed the marginal gains of having an additional agent migrate to the city. As these agents are likely to be economic and political elites that establish themselves earlier in the city, their preferred eviction regime may not necessarily reflect the socially optimal outcome. These observations combined yield the following proposition: _Proposition 4: Strict eviction policies can be welfare reducing. It can be in the interest of the society but not necessarily in the interest of city elites to accept some level of informality._ In line with Corollary 1, our model provides a theoretical insight into the political economy of over-eviction policies. Although evictions remain widespread in many developing countries, it is interesting to note that some countries have moved away from strictly enforced evictions and modified their legal frameworks to provide protection against eviction (Durand-Lasserve 2007). Proposition 4 suggests that more democratic contexts are favorable to such changes.

# **6 Conclusion**

We analyze the interplay between urbanization and evolution of land property rights, two features of economic development that interact in a very important way. Understanding this interaction is crucial for developing countries as dysfunctional urban land property rights systems continue to prevail in many parts of the developing world, and their implications for the path of urbanization in the long run are not well understood. The model we develop provides a novel analytical framework to study this issue. It is a dynamic stochastic model of a representative city that jointly determines urbanization and evolution of land property rights. Each period, heterogeneous households decide whether to live in the city or in a rural area, and whether to pay for land property rights that will protect them from future eviction attempts. Holders of informal plots are randomly exposed to land seizure by other city dwellers, and seized plots can be resold 31
on the land market. Population movements in and out of the city are thus affected both by land grabbing and productivity shocks. Agglomeration and congestion effects are endogenous to migration, affecting wages and land prices.

Our main result is that land property rights institutions, which are very difficult to reform, have a significant effect on the dynamics of urbanization in the long run. Contrary to the developed world experience, leaving urbanization to the market may be suboptimal because of the constraints imposed by dysfunctional property rights systems in developing countries. We also demonstrate that, under certain conditions, informality can be socially optimal, as it makes it possible for low-income residents to contribute to the urban economy. In these situations, strict urban eviction policies can be welfare reducing.

It is important to note that our major insights are conditional on structural fundamentals of prevailing land property rights systems. While our approach is justified given the persistence of land property rights institutions and apparent difficulties in reforming them, our comparative dynamics provides further insights into the political economy of changing land property rights systems. In future research it would be interesting to understand how the parameters of land property rights systems themselves evolve endogenously with urbanization and possibly shed light on the “institutional trap” that seems to be plaguing much of the developing world.

# **References**

- [1] Acemoglu, D., S. Johnson and J. Robinson (2005) The Rise of Europe: Atlantic Trade, Institutional Change and Economic Growth. _American Economic Review_ , 95(3), 546-79.

- [2] Anderson, T. and P. Hill (1975) The Evolution of Property Rights: A Study of the American West, _Journal of Law and Economics_ , 18, 163-79.

- [3] Arru ̃nada, B. (2012) _Institutional Foundations of Impersonal Exchanges. Theory and Policy of Contractual Registries_ . University of Chicago Press: Chicago and London, 299 pages.

- [4] Au, C.-C. and J. V. Henderson (2006) Are Chinese Cities too Small, _Review of Economic Studies_ , 73, 549-576.

- [5] Bairoch (1988) _Cities and Economic Development. From the Dawn of History to the Present_ . The University of Chicago Press.

- [6] Bairoch, P. and G. Goertz (1986) Factors of Urbanization in the Nineteenth Century Developed Countries: A Descriptive and Econometric Analysis, _Urban Studies_ , 23, 285-305.

- [7] Bellman, R. (1957). Dynamic Programming. Princeton University Press. Benabou, R. (1993) Workings of a City: Location, Education and Production, _Quarterly Journal of Economics_ , 108, 619-52.

32
- [8] Benabou, R. (1993) Workings of a City: Location, Education and Production, _Quarterly Journal of Economics_ , 108, 619-52.

- [9] Bertinelli, L. and D. Black (2004) Urbanization and Growth, _Journal of Urban Economics_ , 56(1), 80-96.

- [10] Black, D, and J.V. Henderson (1999) A Theory of Urban Growth, _Journal of Political Economy_ , 107, 2, 252-84.

- [11] Bogart, D. and G. Richardson (2009) Making Property Productive: Reorganizing Rights to Real and Equitable Estates in Britain, 1660 to 1830, _European Review of Economic History_ , 13, 3-30.

- [12] Bondehorm, H. and D. Cuberes (2010) Financial Development and City Growth: Evidence from Northeastern American Cities, 1790-1870, _NBER Working Paper 15997._

- [13] Borkovsky, R.N., U. Doraszelski, and Y. Kryukov (2008) A User’s Guide to Solving Dynamic Stochastic Games using the Homotopy Method. _Operation Research_ , 58(4), 11161132.

- [14] Bourillon, F., P. Clergeot and N. Vivier (eds.) (2008) _De l’estime au cadastre en Europe. Les syst`emes cadastraux aux XIXe et XXe si`ecles. Colloques des 20 et 21 janvier 2005._ Paris: Comit ́e pour l’Histoire Economique et Financi`ere de la France, 423 p.

- [15] Brueckner, J. (2013) Urban Squatting with Rent-Seeking Organizers, _Regional Science and Urban Economics_ , 43, 561-569.

- [16] Brueckner, J. and S. Lall (2015) Cities in Developing Countries: Fueled by Rural-Urban Migration, Lacking in Tenure Security, and Short of Affordable Housing, in _Handbook of Regional and Urban Economics_ , vol 5., chapter 21, in G. Duranton, J.V. Henderson, and W. Strange eds., 1399-1455.

- [17] Brueckner, J. and H. Selod (2009) A Theory of Urban Squatting and Land-tenure Formalization in Developing Countries, _American Economic Journal: Economic Policy_ , 1, 28-51.

- [18] Bubb, R. (2013) The Evolution of Property Rights: State Law or Informal Norms?, _Journal of Law and Economics_ , 56, 555-594.

- [19] Chong, A. and C. Calder ́on (2000) Causality and Feedback between Institutional Measures and Economic Growth, _Economics and Politics_ , 12, 69-81.

- [20] Coase, R.H. (1960) Problem of the Social Cost, _Journal of Law and Economics_ , 3, 1-44.

- [21] Damberg, L. and A. AghaKouchak (2014) Global Trends and Patterns of Drought from Space, _Theoretical and Applied Climatology_ , 117, 441-448.

33
- [22] Deininger, K., H. Selod and A. Burns (2012) _The Land Governance Assessment Framework. Identifying and Monitoring Good Practice in the Land Sector_ , The World Bank: Washington DC, pages 147.

- [23] Demsetz, H. (1967) Toward a Theory of Property Rights, _American Economic Review_ , 57, 347-59.

- [24] DeLong, J. Bradford and A. Shleifer (1993) Princes and Merchants: European City Growth Before the Industrial Revolution, _Journal of Law and Economics_ , 36, 671-702.

- [25] De Soto, H. (2000) _The Mystery of Capital: Why Capitalism Triumphs in the West and Fails Everywhere Else_ . New York: Basic Books.

- [26] Doraszelski, U., & Judd, K. L. (2012). Avoiding the Curse of Dimensionality in Dynamic Stochastic Games. _Quantitative Economics_ , 3(1), 53-93.

- [27] Doraszelski, U. and M. Satterthwaite (2010) Computable Markov-perfect industry dynamics, _RAND Journal of Economics_ , 41, 215-243.

- [28] Durand-Lasserve, A. and H. Selod (2009) The Formalization of Urban Land Tenure in Developing Countries, in Lall S., Freire M., Yuen B., Helluin J.-J. (eds), _Urban Land Markets: Improving Land Management for Successful Urbanization_ , Springer, October 2009, chapter 5, 101-132.

- [29] Durand-Lasserve, A., Durand-Lasserve, M. and H. Selod (2015) _Land Delivery Systems in West-African Cities. The Example of Bamako, Mali_ , Washington, DC: World Bank and Agence Fran ̧caise de D ́eveloppement, 106 pages.

- [30] Duranton, G. (2009) _Are Cities Engines of Growth and Prosperity for Developing Countries_ , in Urbanization and Growth, Spence, Annez and Buckley eds., chapter 3, 67-113, The World Bank.

- [31] Duranton, G. (2014a) _Growing through Cities in Developing Countries_ , World Bank Policy Research Working Paper 6818.

- [32] Duranton, G. (2014b) _The Urbanization and Development Puzzle. The Buzz in Cities: New Economic Thinking._ 2014. Shahid Yusuf (ed) Growth Dialogue, Washington DC, 1-19.

- [33] Duranton, G., Ghani, E., Goswami, A.G., and W. Kerr (2015) The Misallocation of Land and Other Factors of Production in India, _World Bank Policy Research Working Paper 7221_ .

- [34] Ericson J. and A. Pakes (1995) Markov-perfect Industry Dynamics: A Framework for Empirical Work, _Review of Economic Studies_ , 62, 53-82.

34
- [35] Fay, M. and C. Opal (2000) Urbanization without Growth. A Not So Uncommon Phenomenon, _World Bank Policy Research Working Paper 2412_ .

- [36] Feder, G. and D. Feeny (1991) Land Tenure and Property Rights: Theory and Implications for Development Policy. _The World Bank Economic Review_ 5(1), 135-153.

- [37] Field, E. (2005) Property Rights and Investment in Urban Slums, _Journal of the European Economic Association, Papers and Proceedings_ , 3 (2/3), 279–290.

- [38] Field, E. and M. Torero (2006) Do Property Titles Increase Credit Access among the Urban Poor? Evidence From A Nationwide Titling Program, Unpublished manuscript, Harvard University, Department of Economics.

- [39] Field, E. (2007) Entitled to Work: Urban Tenure Security and Labor Supply in Peru, _Quarterly Journal of Economics_ , 122, 1561-1602.

- [40] Firmin-Sellers, K. (1995) The Politics of Property Rights, _American Political Science Review_ , 89, 4, 867-81.

- [41] Friedman, J., Jimenez, E., & Mayo, S. (1988) The Demand for Tenure Security in Developing Countries, _Journal of Development Economics_ , 29 (2), 185–198.

- [42] Fujita, M. (1989). _Urban Economic Theory: Land Use and City Size_ . Cambridge University Press.

- [43] Galiani, S., & Schargrodsky, E. (2010) Property rights for the Poor: Effects of Land Titling, _Journal of Public Economics_ , 94(9-10), 700-29.

- [44] Glaeser, E., H. Kallal, J. Sheinkman and A. Shleifer (1992) Growth in Cities, _Journal of Political Economy_ , 100(6), 1126-1152.

- [45] Glaeser, E. and H. Sims (2015) Contagion, Crime and Congestion: Overcoming the Downsides of Density, _International Growth Center Growth Brief_ , May, 8 pages.

- [46] Glaeser, E. and B. Sacerdote (1999) Why is There More Crime in Cities?, _Journal of Political Economy_ , 107(S6), S225-58.

- [47] Havranek, T., Horvath, R., Irsova, Z., and M. Rusnak (2015) Cross-country Heterogeneity in Intertemporal Substitution. _Journal of International Economics_ , 96(1), 100-118.

- [48] Henderson, J. V. (2005) _Urbanization and Growth_ , in Handbook of Economic Growth, vol. 1, P. Aghion and S. Durlauf eds., Elsevier, Chapter 24, 1543-1591.

- [49] Henderson, J. V. (2010) Cities and Development, _Journal of Regional Science_ , 50(1), 515–540.

35
- [50] Henderson, J.V. and H. Wang (2007) Urbanization and City Growth: The Role if Institutions, _Regional Science and Urban Economics_ , 37, 283-313.

- [51] Hoy, M. and E. Jimenez (1991) Squatters Rights and Urban Development: an Economic Perspective, _Economica_ , 58, 79-92.

- [52] Izraeli, O. and T. McCarthy (1985) Variations in Travel Distance, Travel Time and Modal Choice among SMAs, _Journal of Transportation Economics and Policy_ , 139-160.

- [53] Jayachandran, S. (2006) Selling Labor Low: Wage Responses to Productivity Shocks in Developing countries, _Journal of Political Economy_ , 114, 538-575.

- [54] Jedwab, R. (2015) Urbanization without Structural Transformation: Evidence from Consumption Cities in Africa, mimeo.

- [55] Jedwab, R., L. Christiaensen and M. Gindelsky (2015), Demography, Urbanization and Development: Rural Push, Urban Pull and. . . Urban Push?, _Journal of Urban Economics_ , forthcoming.

- [56] Jimenez, E. (1982) The Value of Squatter Dwellings in Developing Countries, _Economic Development and Cultural Change_ , 30 (4), 739–752.

- [57] Jimenez, E. (1984) Tenure Security and Urban Squatting, _Review of Economics and Statistics_ , 66 (4), 556–567.

- [58] Jimenez, E. (1985) Urban Squatting and Community Organization in Developing Countries, _Journal of Public Economics_ , 27 (1), 69–92.

- [59] Judd, K.L. (1998) _Numerical Methods in Economics_ . The MIT Press.

- [60] Judd, K.L., P. Renner, and K. Schmedders (2012) Finding All Pure-Strategy Equilibria in Games with Continuous Strategies. _Quantitative Economics_ 3, 289-331.

- [61] Kahn, M. (2010) _New Evidence on Trends in the Cost of Urban Agglomeration_ , in Agglomeration Economics, Glaeser ed., National Bureau of Economic Research, 339-354.

- [62] Kim, A. (2004) A Market without the “Right” Property Rights, _Economics of Transition_ , 12 (2), 275–305.

- [63] Knack, S. and P. Keefer (1995) Institutions and Economic Performance: Cross-Country Tests Using Alternative Institutional Measures, _Economics and Politics_ , 1, 207-28.

- [64] Kerekes, C. and C. Williamson (2010) Propertyless in Peru, Even with a Government Land Title, _American Journal of Economics and Sociology_ , 69, 3, 1011-33.

- [65] Konig, D. (1974) Community Custom and the Common Law: Social Change and the Development of Land Law in Seventeenth-Century Massachussetts, _American Journal of Legal History_ , 18, 137-77.

36
- [66] Lall, S., H. Selod and Z. Shalizi (2006) Rural-Urban Migration in Developing Countries: a Survey of Theoretical Predictions and Empirical Findings, _World Bank Policy Research Working Paper 3915_

- [67] Lamsal, L., R. Martin, D. Parrish and N. Krotkov (2013) Scaling Relationship for NO2 Pollution and Urban Population Size: A Satellite Perspective, _Environmental Science and Technology_ , 47, 7855-7861.

- [68] Lanjouw, J. and P. Levy (2002) Untitled: A Study of Formal and Informal Property Rights in Urban Ecuador, _Economic Journal_ , 112 (482), 986–1019.

- [69] Leung, C. (2014) Error Correction Dynamics of House Prices: An Equilibrium Benchmark, Federal Reserve Bank of Dallas, Globalization and Monetary Policy Institute, _Working Paper No. 177_ .

- [70] Libecap, G. (1986) Property Rights in Economic History: Implications for Research, _Explorations in Economic History_ , 23, 227-52.

- [71] Libecap, G., D. Lueck and T. O’Grady (2011) Large-Scale Institutional Changes: Land Demarcation in the British Empire, _Journal of Law and Economics_ , 54, S295-S327.

- [72] Lucas, R. (1988) On the Mechanics of Economic Development, _Journal of Monetary Economics_ 22, 3–42.

- [73] Macours, K., A. de Janvry and E. Sadoulet (2010) Insecurity of property rights and social matching in the tenancy market, _European Economic Review_ , 54(7), 880-99.

- [74] Malpezzi, S. (1999) A Simple Error Correction Model of House Prices, _Journal of Housing Economics_ , 8, 27-62.

- [75] Maskin, E. and J. Tirole (1987) A Theory of Dynamic Oligopoly, III: Cournot Competition, _European Economic Review_ , 31, 947-968.

- [76] Maskin, E. and J. Tirole (1988a) A Theory of Dynamic Oligopoly, I: Overview and Quantity Competition with Large Fixed Costs, _Econometrica_ , 56, 549-569.

- [77] Maskin, E. and J. Tirole (1988b) A Theory of Dynamic Oligopoly, II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles, _Econometrica_ , 56, 571-599.

- [78] Maskin, E. and J. Tirole (2001) Markov Perfect Equilibrium, I: Observale Actions, _Journal of Economic Theory_ , 100, 191-219.

- [79] Marx, B., T. Stoker and T. Suri (2013) The Economics of Slums in the Developing World, _Journal of Economic Perspectives_ , 27, 4, 187-210.

- [80] Michaels, G., F. Rauch and S. Redding (2012) Urbanization and Structural Transformation, _Quarterly Journal of Economics_ , 127(2), 535–586.

37
- [81] Nordhaus, W.D. (2008) A Question of Balance: Weighing the Options on Global Warming Policies. Yale University Press.

- [82] North, D. (1991) Institutions, _Journal of Economic Perspectives_ 5:97-112.

- [83] North, D.F. (1990) _Institutions, Institutional Change and Economic Performance_ , Cambridge: Cambridge University Press, 156 p.

- [84] Offer, A. (1981) _Property and Politics 1870-194. Landownership, Law, Ideology and Urban Development in England_ , Cambridge University Press: Cambridge, 445 pages.

- [85] Pakes, A., and P. McGuire (1994) Computing Markov-perfect Nash equilibria: Numerical Implications of a Dynamic Dfferentiated Product Model, _RAND Journal of Economics_ 25(4), 555-589.

- [86] Quigley. (2009) _Urbanizaton, Agglomeration, and Economic Development_ , in Urbanization and Growth, Spence, Annez and Buckley eds., chapter 4, 115-132, The World Bank.

- [87] Rauch, J.E. (1993) Economic Development, Urban Underemployment, and Income Inequality, _Canadian Journal of Economics_ , 26(1), 901-918.

- [88] Ried, S. and H. Uhlig (2009) The Macroeconomics of Real Estate. Unpublished manuscript, 30 pages.

- [89] Richardson, G. and D. Bogart (2008) Institutional Adaptability and Economic Development: The Property Rights Revolution in Britain, 1700 to 1830, _NBER Working Paper 13757._

- [90] Romer, P. (1990) Endogeneous Technological Change, _Journal of Political Economy,_ 98, 5, S71-S101.

- [91] Rospab ́e, S. and H. Selod (2006) _Urban Unemployment in the Cape Metropolitan Area, in Poverty and Policy in Post-Apartheid South Africa_ , H. Bhorat and R. Kanbur, eds., Human Science Research Council: Pretoria, chapter 7, 262-287.

- [92] Safavian, M., H. Fleisig and J. Steinbuks (2006) Unlocking Dead Capital. How Reforming Collateral Laws Improve Access to Finance, _Public Policy for the Private Sector_ , note 307, The World Bank.

- [93] Sclar, E., P. Garau and G. Carolini (2005) The 21st century health challenge of slums and cities, _The Lancet,_ Vol. 365, March 5, 901-903.

- [94] Selod, H. and L. Tobin (2015) The Informal City: Property Rights and Sorting, _World Bank Policy Research Working Paper_ , forthcoming.

- [95] Shah, N. (2014) Squatting on Government Land, _Journal of Regional Science_ , 54, 114-136.

38
- [96] Sonin, K. (2003) Why the Rich May Favor Poor Protection of Property Rights, _Journal of Comparative Economics_ , 31. 715-31.

- [97] Turnbull, G. (2008) Squatting, Eviction and Development, _Regional Science and Urban Economics_ , 38, 1-15.

- [98] United Nations (2014) _World Urbanization Prospects – Highlights_ , 27 pages.

- [99] UN-Habitat (2010) _The State of African Cities. Governance, Inequality and Urban Land Markets_ , UN-Habitat: Nairobi, 268 pages.

- [100] Woodruff, C. (2001). Review of de Soto’s The Mystery of Capital, _Journal of Economic Literature_ , 39(4), 1215–1223.

- [101] Yeltekin, S., Y. Cai, and K.L. Judd (2015) Computing Equilibria of Dynamic Games. Unpublished manuscript.

- [102] Zseleczky and Yosef (2014) Are Shocks Really Increasing? A Selective Review of the Global Frequency, Severity, Scope and Impact of Five Type of Shocks, _IFPRI 2020 Conference Paper No.5_ , 24 pages.

39