An Urban Cost-based Theory of Central Places
K. Behrens* , A.V. Sidorov†, J.-F. Thisse‡
Consider an economy with one sector and one production factor, labor. The
economy is populated with Lconsumers who are to be distributed across cities.
Each consumer supplies inelastically one unit of labor. Each region can be
urbanized, i.e., it can develop a city that accommodates firms and consumers. For
simplicity, we assume that, whenever a city exists it is monocentric – it has an
exogenously given dimensionless central business district (CBD). Cities emerge
endogenously, the only exception being the capital city, which is exogenously
given but its size is endogenous.
Each city is characterized by its rank r  0 in the urban hierarchy, and
several cities may have the same rank. The capital city has the highest rank 0. In
what follows, we refer to all cities of rank ras layer-r cities. Let h denote the
depth of the urban hierarchy, or the number of layers of ranks lower than the
capital city, and let M r > 0 be the number of cities in layer r, for each 0  r  h .
For simplicity, we follow the literature in club theory and urban economics by
allowing for non-integer numbers of cities. By definition, the urban system
displays a hierarchy if each layer r involves M r > M r-1 cities. If all cities
belonging to the same layer have the same population size lr , given M r layer-r
cities, the full population condition implies that
h
l0   M r lr  L.
r 1
There are four types of consumption goods. First, there are n (public and
private) goods that must be consumed where they are produced. The second
good, q  {qi , i [0, N ]} , is produced as a continuum of varieties of a horizontally
differentiated good under monopolistic competition and increasing returns. The
third good is land, which is used as a proxy for housing. Each consumer uses
inelastically a residential plot of fixed size, normalized to unity. The fourth good,
Q, is unproduced, homogeneous and chosen as the numéraire. We assume that
all goods are costlessly tradable. This assumption captures the well-known fact
that the cost of shipping goods has dramatically decreased ever since the
beginning of Industrial Revolution.
Following Christaller (1933) and successors, we assume that
nontradables can be gathered in a way such that the type-r nontradables are
supplied in cities having a rank higher than or equal to r. Let Gr be the utility
derived from consuming all the nontradables provided by layer-s cities with
*Canada Research Chair, Département des Sciences Économiques, Université du Québec à Montréal (uqam),
Canada; CIRPÉE; and CERP. E-mail:behrens.kristian@uqam.ca, kristian.behrens@gmail.com
† National Research University, Higher School of Economics, Center for Market Studies and Spatial
Economics. E-mail: alex.v.sidorov@gmail.com
‡National Research University, Higher School of Economics, Center for Market Studies and Spatial
Economics; CORE, UniversitéCatholique de Louvain; and CERP. E-mail: jacques.thisse@uclouvain.be
s = 0,1,...,r only. Therefore, we have Gr > Gr+1 for r = 0,1,...,n -1, that is, higher-
rank cities provide a wider range of nontradables than lower-rank cities.
All consumers have identical quasi-linear preferences defined on the
tradable goods (q,Q) given by U(q,Q) = u(q) + Q where u(q) is a well-behaved
subutility function. Consumers commute from their residence, x, to the CBD. In
doing so, they incur a unit commuting cost t>0. Commuting costs are then given
by t|x|. Let Rr x denote the land rent prevailing at location x in a layer-r city. Let
()
also ALR r  
xX
Rr ( x)dx denote the aggregated land rent in a layer-r city. In
what follows, we follow standard practice and assume that the aggregate land
rent is equally redistributed among all residents of the city.The budget
constraint of a consumer living at location xof layer-r cityis then given by
N
ALR
0 pi qi di  Q  Rr ( x)  t | x | wr  lr r  Q
where wr denotes the wage and Q the consumer's initial endowment of the
numéraire. Without loss of generality, we normalize the opportunity cost of land
to zero.
Firms are located in the CBD of a city, where they use no space. Producing
q units of variety i requires a fixed amount f of labor. It then follows from the
labor market clearing condition that the mass of firms/varieties in the economy
is equal to N = L / f .Each firm being negligible to the market, it accurately treats
the price aggregator(s) p parametrically. Because preferences are symmetric and
trade is costless, profit-maximizing prices and outputs are identical across firms
and cities; they are denoted p * and q* = D( p* ,p* ). Let S  pq hs0 M sls be the
revenue (sales) of a firm. Thus, the profit of a layer-r firm can be written as
follows:
Pr (wr ) = S - f wr .
Turning to labor markets, free entry implies that firms' operating profits are
completely absorbed by the wage bill. Hence, the equilibrium wage wr in a layerr city must satisfy the condition  r ( wr )  0 . Equilibrium wages are then given by
S
wr  w 
for all r  1, 2, h .

Since prices are equal across cities, the consumer surplus CS*r = CS* is the same
for all workers irrespective of the city in which they live. Thus, the indirect utility
of a worker residing in a layer-r city is equal to
ALR r
l
l
Vr = CS* + w* + Gr +
- t r = CS* + w* + Gr - t r
lr
2
4
For any given number of layers, h  n , and any given number of cities, in each
layer r, a spatial equilibrium of depth his a distribution lr  0, r  0,1,, h , of
consumers across layers such that the indirect utility of consumers in each city of
each layer is equalized, Vr = V0 for all r , and such that the full population
constraint is satisfied.
Exogenous spatial equilibrium
The next result states a necessary and sufficient condition for a spatial
equilibrium of depth hto exist.
Proposition 1.For any given hierarchy Gr , M r , r  0, , h there exists a unique
spatial equilibrium of depth h if and only if
4 h

L   M r (Gr  Gh )  .
t  r 0

Exogenous spatial equilibrium
Now we assume that there is no given hierarchy and the building/maintenance
of the central place in the layer r (or, corresponding Public Good) require to
spend the given total cost Pr , which is financed by taxes. Let’s denote per capita
tax at layer r as cr . We also assume that there is Social Planner, who collects
these taxes and then redistributes them across layers on the base of the
following equity principles:
Self-Financing: For each layer r the local Public Good is locally subsidized, i.e.,
Pr = cr lr , where lr is equilibrium population for each r  0, , h .
Egalitarianism: Public Good expenditures Pr M r are equalized across all layers
r  0, , h .
Then the indirect utility of a worker residing in a layer-rcity is as follows
P
l
l 
Vr (lr )  CS  w  Gr  cr  t r  CS  w  Gr   r  t r  .
4
4
 lr
We also assume that Social Planner is benevolent and maximizes the total Social
Welfare Function
h
SW   M r lrVr .
r 0
Proposition 2. Let inequalities
Pr 1t  Pr t
Gr 1  Gr 
, G0  Gr  P0t  Pr t
2 P0t  G0


holds for all r  1, 2,, h . Then there exists a unique stable endogenous spatial
equilibrium lr , M r , r  0, , h , which satisfies
l0  l1    lh  0 , M 1    M h .
Moreover, there exists well-defined endogenous depth of hierarchy
h*  max h | M 1  M 0  1 .