A TRACTABLE CITY MODEL FOR AGGREGATIVE ANALYSIS

• Date: 01 February 2017
• Published date: 01 February 2017
• DOI: http://doi.org/10.1111/iere.12211
• Author: Burcu Eyigungor,Satyajit Chatterjee

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 1, February 2017

A TRACTABLE CITY MODEL FOR AGGREGATIVE ANALYSIS∗

BYSATYAJIT CHATTERJEE AND BURCU EYIGUNGOR1

Federal Reserve Bank of Philadelphia,U.S.A.; Federal Reserve Bank of Philadelphia, U.S.A.

An analytically tractable city model with external increasing returns is presented. The equilibrium city structure is

either monocentric or decentralized. Regardless of which structure prevails, intracity variation in endogenous variables

displays exponential decay from the city center, where the decay rates depend only on parameters. Given population,

the equilibrium of the model is generically unique. Tractability permits explicit expressions for when a central business

district (CBD) will emerge in equilibrium, how external increasing returns affect the steepness of downtown rent

gradients, and how wages and welfare vary with population. An application to urban growth boundary is presented.

It seems to me that the force we need to postulate for the central role of cities in economic life is of exactly

the same character as the external human capital I have postulated as a force to account for certain features

of aggregative development. If so,land rents should provide an indirect measure of this force. ...What can

people be paying Manhattan or downtown Chicago rents for,if not for being near other people?

Robert E. Lucas Jr. (1988), “On the Mechanics of Economic Development,” p. 39.

1. INTRODUCTION

Increasing returns are a key element in several explanations of economic growth (Arrow,

1962; Romer, 1986; Lucas, 1988). Often times, external increasing returns accompany and

sustain the spatial concentration of industries, as famously noted by Alfred Marshall (1890)

long ago.2Indeed, when placing external human capital at the center of the process of economic

development, Lucas (1988) linked cities and growth. His intent (expressed in the quote above)

was to bring facts regarding the internal structure of cities—in particular, the concentration

of businesses and the generally high value of land in the center of cities—to both validate the

existence of increasing returns and learn about their empirical importance.

Our article is motivated by this link between cities and growth. It presents a tractable model of

a city in which industry scale improves the efficiency of firms depending on how close physically

firms are to each other. Our model has the same setup as the Lucas and Rossi-Hansberg (2002)

circular city model but alters the way physical proximity between firms is defined. Specifically,

the proximity between two firms located at different points in the plane is measured as the

∗Manuscript received March 2014; revised July 2015.

1This article combines material from two earlier papers circulated as “A Tractable Circular City Model with En-

dogenous Internal Structure” and “Agglomeration Economies, Geography and the Value of Urban Land.” The authors

thank Roc Armenter, Jeffrey Brinkman, Daniele Coen-Pirani, Russel Cooper, Gilles Duranton, Jeffrey Lin, Esteban

Rossi-Hansberg, Maisy Wong, and seminar participants at the Federal Reserve Bank of Philadelphia, the Wharton

School of the University of Pennsylvania, Iowa State University, Pennsylvania State University, and participants at the

SED 2012, RSAI 2014, HULM Fall 2014, and CURE 2014 conferences for thoughtful comments. Detailed comments

from the editor, Guido Menzio, and two anonymous referees are also gratefully acknowledged. The views expressed in

this article are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Philadel-

phia or the Federal Reserve System. Please address correspondence to: Satyajit Chatterjee, Research Department,

Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 19106. Phone: 215-574-3861. E-mail:

satyajit.chatterjee@phil.frb.org.

2Principles of Economics, Book IV, Chapter 10: “The Concentration of Specialized Industries in Particular

Localities.”

127

C

(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

128 CHATTERJEE AND EYIGUNGOR

sum of the lengths of the rays connecting each point to the city center.3With this change,

we can show that the city either has a “monocentric” structure with businesses concentrating

in the city center, or it has a “decentralized” structure in which firms locate next to their

workers. Regardless of which form prevails, the intracity variation in all endogenous variables—

residential and commercial rents, employment and residential densities, and wages—displays

(over their relevant domains) exponential decay from the city center, in which the rates of decay

depend only on preference and technology parameters. Furthermore, for a given population

size, the equilibrium outcome is generically unique.

Analytic tractability has several useful consequences. First, we are able to give explicit condi-

tions on preference and technology parameters under which a business core, or central business

district (CBD), will emerge in equilibrium. Second, the explicit expressions for the CBD rent

gradients reveal how the strength of increasing returns, in conjunction with other parame-

ters, affects the steepness of downtown rent gradients. Third, we are able to characterize the

equilibrium relationship between downtown rents and wages and population.

Analytic tractability also allows us to characterize the equilibrium relationship between wel-

fare and population. The shape of this relationship is important for explaining why modern

economies are organized around a finite number of large cities. If city welfare is monotonically

increasing or decreasing in population, we should expect all economic activity to concentrate in

one giant city or be dispersed across an infinity of miniscule locations. The fact that it is neither

suggests that welfare is initially increasing but eventually declining in population. Visually, the

relationship must resemble an inverted-U. Our characterization of the relationship gives ex-

plicit conditions under which the inverted-U shape will emerge for both the mononcentric and

decentralized cities.4

Finally, tractability permits informative comparative static analyses. We study the impact

of an urban growth boundary on equilibrium outcomes in the context of rising demand for

urban land.5Intuition might suggest that urban growth controls reduce housing affordability

by increasing rents. We show that this intuition ignores the local increasing returns that are

central to urban agglomerations. When productivity is increasing in population, cities that can

expand will experience larger increases in rents relative to cities whose expansion is constrained

by a growth boundary. On the other hand, if factors other than land, such as structures, are

also fixed—as they might be in the short run—then productivity is declining in population, and

rents are predicted to rise more in the constrained city. Thus, the model predicts that the effects

of urban growth boundaries on housing affordability will vary with the time horizon under

consideration. Empirical evidence on this point is discussed later in the article (Section 7).

There are several precursors to the theory presented in this article. The paper by Fujita and

Ogawa (1982) is an early precursor that examined for a linear city (and mostly numerically)

the conditions under which one or more business districts could emerge in equilibrium.6Lucas

(2001) studied the connection between the magnitude of increasing returns and the steepness

of business rent gradients in a model in which the residential location choice of workers was

suppressed. Thus, the conditions under which a business core would emerge were not addressed.

3In most studies, including Lucas and Rossi-Hansberg, proximity is taken to mean Euclidean distance between two

points. It is unclear, however, what measure of proximity is relevant for urban agglomerations. For instance, if distance

between firms matters because of commuting costs, one might need to consider that most cities have a radial highway

network. Our definition is consistent with this and has the benefit of tractability.

4Studies on the system of cities derive this relationship in a reduced form fashion: It is assumed that firms locate

at the city center to benefit from local increasing returns, and it is (typically) assumed that the city area grows with

population (see the survey by Abdel-Rahman and Anas, 2004).

5Constraints on urban development are growing trends in the United States and a well-established policy in most of

Western Europe (Pendall et al., 2002). In emerging economies, where the rate of urbanization is expected to increase

dramatically in the coming years, urban growth containment policies are of prime interest.

6There is extensive literature in urban economics that theoretically examines residential land use in a city—see,

for instance, the surveys by Brueckner (1987) and Duranton and Puga (2014). This literature assumes that businesses

locate in the city center and, therefore, makes no predictions regarding the emergence, size, or rent gradients of business

districts.