The dynamics of city formation: finance and governance*

 

 

2/Aug/05

The dynamics of city formation:
finance and governance*

J. Vernon Henderson
Brown University

Anthony J. Venables
LSE and CEPR

Abstract:
This paper examines city formation in a country whose urban population is growing
steadily over time, with new cities required to accommodate this growth. In contrast
to most of the literature there is immobility of housing and urban infrastructure, and
investment in these assets is taken on the basis of forward-looking behavior. In the
presence of these fixed assets cities form sequentially, without the population swings
in existing cities that arise in current models, but with swings in house rents.
Equilibrium city size, absent government, may be larger or smaller than is efficient,
depending on how urban externalities vary with population. Efficient formation of
cities with internalization of externalities involves local government intervention and
borrowing to finance development. The paper explores the institutions required for
successful local government intervention.

JEL classification: R1, R5, O18, H7
Keywords: Urbanization, city size, urban developers, city governance.

* Henderson thanks the Leverhulme foundation for support as a visiting professor at the LSE.
Venables’ work is supported by the UK ESRC funded Centre for Economic Performance at LSE.
The authors thank G. Duranton, E. Rossi-Hansberg, J. Rappaport, W. Strange, and seminar
participants at Wisconsin, the NBER, and a CEPR conference on “European Integration and
Technological Change” for helpful comments.

J. V. Henderson
Dept. of Economics
Brown University
Providence RI
USA 02912

A. J. Venables
Dept. of Economics
London School of Economics
Houghton Street, London
UK WC2A 2AE

J_Henderson@brown.edu
http://www.econ.brown.edu/faculty/henderson/

a.j.venables@lse.ac.uk
http://econ.lse.ac.uk/staff/ajv/

I: Introduction

Understanding city formation and the financing requirements of cities is critical to effective policy

formulation in developing countries that face rapid urbanization. The rapid growth of urban

populations in developing countries is well known, but what is less well known is the growth in

the number of cities. Between 1960 and 2000 the number of metro areas over 100,000 in the

developing world grew by 185%, i.e. almost tripled (Henderson and Wang, 2005). Moreover the

UN’s projected two billion person increase in the world urban population over the next 45 years

ensures this growth in city numbers will continue. How do we start to think about whether the

proliferation of cities in developing countries is following an efficient growth path, and how

policies may assist or constrain achievement of better outcomes?

We start with two fundamental premises which define the research agenda. The first is

that city formation requires investment in non-malleable, immobile capital, in the form of public

infrastructure, housing, and business capital. Owner-occupied housing capital is immobile and

long lived, depreciating at a gross rate well under 1% a year and a net rate after maintenance of

almost zero. Urbanization also involves heavy investment in roads, water mains, sewers and the

like that are immobile and depreciate slowly. The second premise is that, in developing countries,

a key local public finance need is for cities to tax and to borrow, and/or to use central government

funds to finance infrastructure investments and subsidize the development of industrial parks so as

to attract businesses (World Bank, 2000).

Why does immobility of capital matter to the analysis of city formation? Consider a

context in which the urban population of a country is growing steadily with ongoing rural-urban

migration, as resources shift out of agriculture into urban industrial production. In most of the

literature city formation is modelled assuming perfect mobility of all resources (Henderson, 1974

and Anas 1992). Under this assumption, the analysis of the city formation process has three

unsatisfactory aspects, at odds with the data. The first is that initial urbanization is characterized

by huge swings in population of cities. In a country with just one type of city, urbanization

proceeds by the first city growing until at some point a second city forms, with the timing

depending on the details of the city formation mechanism and institutions. Regardless of that

timing, when a second city forms the first city loses half its population who migrate instantly to

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that second city. Then the first city resumes growth and the second city grows in parallel until a

third city forms, at which point both existing cities lose 1/3 of their population who migrate to

this new third city. And so on. Of course, when the number of cities gets very large, the

population swings of established cities may diminish. However, that still leaves the second

unsatisfactory aspect to this process: when new cities form in this context, they jump instantly to

some large size, rather than grow over time to a steady state size.

The third unsatisfactory aspect concerns outcomes under likely institutional environments.

With perfect mobility of resources, an efficient city formation process requires the intervention of

“large agents” such as city governments or large scale land developers who, through subsidies

and zoning, co-ordinate the en masse movement of population from old cities into a new city.

Many developing countries lack the local institutions required to co-ordinate en mass movements

of population. With no large agents and with perfect mobility of resources, there is massive

coordination failure in these models. New cities only form when existing cities become so

enormously oversized that living conditions deteriorate to the point where any individual would

be better off defecting from existing cities to form a “city” of size one. In these models national

urban population growth brings about this dismal Malthusian outcome of enormously oversized

cities, a not particularly plausible result.

This paper will start at the opposite extreme to the existing literature and assume that there

are substantial sunk capital costs of housing and associated urban infrastructure. Forward-looking

agents, in our case housing builders, anticipate income streams that will be earned in new and in

old cities and make investment decisions accordingly. These assumptions ensure that there are

neither swings nor jumps in urban population, and forward-looking behavior also resolves the

coordination problem.

Our setting is an economy with a single final good and with a continuous flow of migrants

moving out of agriculture into cities. Each new city starts off small and grows through rural to

urban migration until the next new city becomes the target of migrants. In the base case we have

pure sequential growth, where first one city grows to its final stationary size, then a second city

starts from zero population and grows continuously to its stationary size, and then a third and so

on. Extending the base case to include heterogeneity across cities and technological progress, we

show that, while cities still grow in sequence from scratch to some level, they also experience

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later repeated episodes of further growth, in parallel with other cities. We present evidence in

section 5.2 that urban growth in countries follows this broadly sequential pattern.

While having immobility of capital in our model gives patterns of city formation that are

more consistent with “reality”, the imposition of this key assumption has further interesting

economic implications. Immobility affects how housing prices within cities vary over time, as

growth proceeds. When a new city forms the prices of fixed assets in old cities may adjust in

order to maintain occupancy in both new and old cities. This analysis of how asset prices vary

within existing cities as a new city grows is a fundamental insight of the paper, as is the key role

of competitive housing builders in the city formation process.

Outcomes depend on the institutional regime under which urbanization occurs. We start

the paper by analyzing the benchmark case of socially efficient city, and show that efficient city

size is larger than in models in which resources are perfectly mobile. We then look at equilibrium

city formation under a regime in which there are no ‘large agents’ or city governments – all

individuals are price-takers in all markets. The simple coordination failures that arise in perfect

mobility models of city formation without city governments do not occur here. Agents are

forward looking, correctly anticipating population flows and housing market conditions in new

and old cities. In this context investment in durable capital provides a commitment device for

competitive builders to solve coordination failure problems. However, these agents can’t

internalize city externalities, so equilibrium city size may be larger or smaller than socially

optimal depending, in an intuitive manner, on the way in which externalities vary with city size.

We then turn to city formation with large agents: private developers or public city governments.

Such cities borrow in order to offer subsidies to attract migrants during the period in which urban

scale economies are not fully developed. As we will see, timely formation requires development

of institutions governing land markets, leases, and taxation.

The equilibria under the two market regimes, the first without and the second with city

governments, generate different income streams and housing price paths. As such, they yield

predictions about how housing price paths would differ across economies operating under the two

different regimes. For the regime without city governments, rental prices are constant in growing

cities. However rental and asset prices vary in stationary cities, starting high when another new

city first starts to grow, then declining, and then rising again towards the end of the growth

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interval of the new city. That in principle is a testable hypothesis, where one could compare price

paths in more stationary cities in developing countries with those in the current fastest growing

city. In the second case, with fully functional city governments, housing prices are invariant

across cities, whether growing or stationary. As such, one role of city governments with full tax

powers is to use their tax, tax exemption and subsidy powers to smooth out realized incomes, and

as a result also smooth housing price cycles. In summary, for developing countries with rapid new

city formation, the paper will predict housing price cycles in stagnant cities as new cities grow

under one set of institutional arrangements, but not under the other.

This idea of looking at housing price cycles is supported empirically by Glaeser and

Gyourko (2005), who examine the USA in the modern era in which the country is fully urbanized.

They look at the impact of city specific economic shocks. They find positive shocks are

associated with strong population growth of receiving cities, but have fairly modest effects on

housing prices in those cities. Cities that experience negative shocks have very sharp price

declines but rather modest or even zero population effects. While our situation is different – the

driving force is urbanization in developing countries not shocks in an urbanized world – the

implication is the same. In the face of growth of another city, stationary cities may retain

population through price changes, while growing cities have stable prices.

In terms of other relevant theoretical literature, there are growth models with city

formation (e.g. Black and Henderson 1999 and Rossi-Hansberg and Wright 2004), but they

assume that cities form with perfect mobility of all resources and without a local public sector that

borrows to finance development. Incorporating immobility and financing considerations requires

a different approach. The effect of having durable, immobile capital on single city growth has

been tackled in Brueckner (2000). However, the only papers that examine new city formation as

the population grows with durable capital are a thesis chapter of Fujita published in 1978 and

Cuberes (2004). Fujita examines planning, but not market solutions. Cuberes in a paper written

simultaneously and independently of ours has an empirical focus, with a motivating model that

has only two cities in total that ever form in the economy. Cuberes doesn’t analyze the role of

institutions in driving different types of equilibria, housing price cycles, and the general topics in

this paper. This paper develops a model of city formation under immobility of capital, building on

Venables (2005) who illustrates that population immobility will affect the city formation process.

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There is a complementary paper on city formation with durable capital by Helsley and Strange

(1994) in which large land developers form cities simultaneously in a static context, using durable

capital as a strategic commitment device. We have a dynamic context and for much of the paper

there are no large land developers; but the Helsley and Strange paper reinforces the idea that

durable capital functions as a commitment device to overcome coordination failure.

2. The model

In order to isolate the key elements in the urbanization process, we make four simplifying

assumptions. First we assume a small open economy where agents can borrow and lend at a fixed

interest rate * in world capital markets. We do not embed the process in a closed economy model

with capital accumulation and an endogenous interest rate. Second, we assume that the urban

sector grows in population by a constant amount, <, each instant, as if there were a steady stream of population out of agriculture and into the urban sector. Constancy of this rate is not critical to the concepts developed in the paper. For example, having the migration rate to cities respond to rural-urban income differences would affect the rate of population flow into cities and affect our precise calculations of the rate of cities’ population growth. But it would not affect the process of how new cities form or the analysis of policies and institutions. Third, in the base case, we abstract from ongoing technological change which would tend to increase equilibrium and efficient city sizes over time. Finally to derive the key results, we assume that all cities form under identical circumstances – technology, amenities, and industrial composition. While these four assumptions define our base case, in section 5.1 we demonstrate the implications of relaxing them, and show how are main insights are robust to these changes. We start with a description of a city in the economy, setting out both the urban agglomeration benefits and the urban diseconomies associated with city population growth. Cities form on a “flat featureless plain” with an unexhausted supply of identical city sites, and land is available for urban development at zero opportunity cost. There are n(t) workers in a particular city at date t and we define a worker’s income, y(t), as 5 (1) As we will see, the first term is the worker’s output, the second is land rent plus commuting costs, and the final term is any subsidy (or, if negative, a tax) that the worker receives. This real income expression contains all the components of earnings, subsidies, and expenditures that vary directly with city size. This sum is then available to be spent on final consumption and on housing. We discuss each of the components of (1) in turn, as well as housing. Production: Firms in a city produce a single homogenous good with internal constant returns to scale but subject to citywide scale externalities. With constant returns we simply assume that each worker is also a firm. The city work force is n and per worker output is x(n), with x!(n) > 0 and

bounded away from infinity at n = 0, and x”(n) < 0. This represents urban scale economies where per worker output rises at a decreasing rate with city population, as workers benefit from interaction with each other.1 Output per worker may continue to rise indefinitely with n, or may pass a turning point as congestion sets in. Commuting and land rent: The second term on the right-hand side of equation (1) is land rent plus commuting costs in a city of size n(t). It generalizes the standard approach in the urban systems literature (Duranton and Puga, 2004). All production in a city takes place in the city’s central business district (CBD), to which all workers commute from residential lots of fixed size. Free mobility of workers requires all workers in the city to have the same disposable income after rent and commuting costs are paid. Thus, there is a land rent gradient such that, at all points within the city, land rent plus commuting costs per person equal the commuting costs of the edge worker whose rent is zero. Edge commuting costs take the form (the term in (1)) which is derived, along with expressions for rent and commuting costs, in Appendix 1. The parameter c measures the level of commuting costs and ( combines relevant information on the shape and commuting technology of the city. If commuting costs per unit distance are constant then, in a linear city ( = 2, and in a circular or pie shaped city ( = 3/2. Our generalization encompasses these cases, and also allows commuting costs to be an iso-elastic function of distance, as shown in Appendix 1. We require merely that ( > 1, so average commuting costs, as well as average land rent, rise with city

population. Integrating over the commuting costs paid by people at each distance from the centre

and over their rents gives the functions TC(n) and TR(n) reported in Table 1.

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Housing: A plot of city land can be occupied by a worker only after a capital expenditure of H has

been incurred. This represents the construction of a house, although it could also include other

aspects of infrastructure such as roads and water supply. The housing construction, sale, and rental

markets are all assumed to be perfectly competitive, and the spot market rent of a house at time t is

denoted h(t), this paid in addition to the rent on land. Throughout the paper we assume that the two

rent components are separable; housing rent, h(t), is paid separately from land rent which is

determined by the city land rent gradient. We also assume that the two sources of income can be

taxed separately. Thus, house builders may rent land from land owners with an infinite lease and

pay land rents according to the perfectly foreseen city land rent gradient. Alternatively builders

could initially buy the land from the land owners, capitalizing the land rents. And a model with

owner-occupancy where residents buy land and housing would yield equivalent results. Land

owners are people outside the urban sector, although the same results on city formation hold if they

are nationwide Arrow-Debreu share holders in the land of all cities.2

Subsidies and taxes: The final term in equation (1) denotes a per worker subsidy at rate s(t) (tax if

negative) to workers in the city at date t. We will investigate use of this under different city

governance structures. We note that since workers are also firms, the subsidy could be viewed as

going to firms, a common element of city finance.

Table 1 summarizes key relationships in a city with population n. The left-hand block of the

table reports the basic relationships between commuting costs and land rent, derived in Appendix 1.

The right-hand block defines relationships which we will use repeatedly through the paper. Total

surplus, TS(n), is the output minus commuting costs of a city of size n; notice that this is defined

without including housing costs. Average surplus AS(n) and marginal surplus MS(n) follow from

this in the obvious way. LS(n) is the surplus per worker net of average land rent paid to

landowners, LS(n) / AS(n) – AR(n); it follows that

(from equation 1).

Finally, EX(n) = MS(n) – LS(n) is the production externality associated with adding a worker-firm to

the city: it is the increase in output of all other workers in the city when a further worker is

employed.

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