Deindustrialization: Vanishing Cities Revisited∗

∗
Deindustrialization: Vanishing Cities Revisited
Maxim Goryunov†and Sergey Kokovin‡
Abstract
Anas' (2004) impossibility theorem vanishing cities states that monopolistic competition
economies of scale alone are insucient to explain the growth of cities in response to growing
country population or decreasing trade costs; cities decline. Is this conclusion an artifact of
unrealistic assumptions?
Instead of Anas' normative approach, we consider a positive ap-
proach: considering migration and developers' equilibria. Still, vanishing remains robust to
our more realistic modications! Ultimately, we argue that vanishing is a realistic outcome:
industries, free of externalities, can be expected to locate in rural areas. Generalizing our com-
parative statics, we conclude that simultaneous growth of the number and size of industrial
cities is implausible.
Key words: city sizes, growth, agglomeration, trade
JEL codes: D62, F12, R12
1
Introduction
Studies of contemporary urban development show a rather typical tendency of many manufacturing facilities to locate in small cities or even in rural areas. The examples include industries
from food production to heavy industries such as auto manufacturing. Generally, nowadays manufacturing has become one of the least urbanized activities (see, for instance, Holmes and Stevens
∗
We gratefully acknowledge grants RFBR 12-06-00174a and 11.G34.31.0059 from the Russian Government. We
are indebted to Jacques-Francois Thisse and Kristian Behrens for valuable advice and to Takatoshi Tabuchi for
fruitful critique.
†
CERGE-EI and NRU Higher School of Economics. Email: [email protected]
‡
Sobolev Institute of Mathematics SBRAS, Novosibirsk State University, NRU Higher School of Economics.
Email: [email protected]
1
(2004) on the U.S. and Canada or Kolko (2010) on the comparison between manufacturing and
service industries).
As a result, big cities are becoming more and more deindustrialized, spe-
cialized in exporting services rather than goods.
The services include governance of territories
by governments, governance of multi-plant rms by headquarters, education, banking, etc. Small
cities, instead of exporting such services, rely on manufacturing and tend to decrease in size. Can
we say that the decreasing size of small cities is a
natural
outcome of market evolution, driven by
noticeably reduced trade costs?
Urban theory and economic geography (see, for example, Fujita and Thisse, 2002) has much
to say about the agglomeration forces driving rms and workers together into cities, and about
countervailing dispersion forces. Among the latter, commuting costs, land rent and other diseconomies of size understandably restrict city size. Alternatively, the well-known Krugman's CorePeriphery model uses agricultural population as a dispersion force instead of commuting costs. It
predicts that large trade costs can support many small cities, whereas
agglomeration into few large cities
decreasing trade costs force
. This view has become popular.
The opposite tendency an evolution towards smaller and smaller cities is also predicted
by market theory in several settings. In particular, Starrett's (1978) impossibility theorem states
that without economies of scale or externalities, equilibrium with trading cities is impossible: an
economy would generate backyard capitalism with every household producing everything for its
own consumption. However, even accounting for externalities, Henderson (1974) has achieved a
somewhat similar prediction. He assumes intra-industry externalities, but ignores inter-industry
ones, and considers commuting costs as a dispersion force. Henderson's model world would generate
mono-industry cities. Describing competition between two cities or regions, Helpman (1998) has
explored the tension between the agglomeration force stemming from a preference for variety and
a dispersion force stemming from a limited housing supply.
Somewhat similarly treating two
cities, Tabuchi (1998) introduces competition for land as a dispersion force. Both models reach
similar results on
arising dispersion
: equal distribution of population between two regions when
transportation costs become low enough.
Extending this line of reasoning to an emerging system of cities instead of two given cities,
Anas (2004) studies a normative setting. World (or country) population given, a social planner
maximizes the per-consumer welfare by choosing the number of symmetric cities, taking into
account consecutive equilibrium arising under monopolistic competition à la Dixit-Stiglitz with
2
iceberg trade costs. The agglomeration force amounts to economies of scale, whereas the dispersion
force stems from commuting costs (a square root of the city size). Anas' main theorem describes
optimal cities under growing world population as follows: their number increases but the size of
each
decreases
, and eventually
drops down
to a technologically admissible minimum (sucient for
the production of only one variety of manufacturing good). The explanation is that the benet
of living in a big city (close to many producers) decreases when more and more varieties are
imported from other cities of a growing world, while commuting costs remain the same.
interprets his surprising result as a failure of the new economic geography paradigm:
externalities, a simple monopolistic competition mechanism is
cities
Anas
without
insucient to drive the growth of
in response to growing population, and/or decreasing trade costs, thus, the new economic
geography is insucient for understanding of the evolution of cities.
Our goal is to check the
robustness of this idea and to expand it.
Our setting uses Anas' (2004) model as a baseline. We have suspected that his normative setting
and restrictive assumptions drive his unexpected result.
Does it remain valid in more realistic
settings? Instead of a model featuring a central planner, we explore two positive alternatives: (1)
migration equilibrium, where each citizen can voluntarily choose a city to live in, or can settle in
a new city (understanding how production and trade will respond to her choice); (2) developers'
equilibrium, where each city decides to invite additional citizens or not (also understanding the
production/trade consequences).
These two versions remind one of two cases in the theory of
clubs: the migration setting resembles the open clubs theory as everybody can join irrespective
of the will of city residents; the developers' equilibrium can be related to a closed clubs setting,
where the admission decision is made by the current members.
1
As in Anas (2004), we use the familiar Dixit-Stiglitz preferences and iceberg trade costs, and
assume a technological minimum of city size called village.
2
Commuting costs also remain the
same and we similarly maintain symmetry, and ignore the discrete nature of cities, rms and
citizens, a continuous model being more tractable.
We start presentation of our results with
1 We
Migration equilibria
. These equilibria are multiple,
cannot directly rely on club theory, because our clubs-cities are interacting : they inuence each other
through trade.
2 Although we ignore integer problem throughout the paper, we employ the notion of the minimal technological
size to have well dened equilibrium under any parameters. It can be dened as maximum between one person and
a number of people necessary for the production of one variety.
3
not uniquely determined by preferences and costs. One can study only the zone of equilibria.
Formally, under given world population
L
and trade freeness
φ,
every symmetric composition of
n
cities (of equal sizes) is a migration equilibrium, in the sense that a consumer's utility is the same
everywhere. However, we consider only
stable
equilibria. Stability means that small perturbation
of the population distribution should be reversed by migration forces.
Stability includes two
conditions: (i) no dispersion: nobody wants to settle in a new city; (ii) no agglomeration: nobody
wants to migrate to another similar city (such movement should make larger city less attractive).
Propositions 1 and 2 describe vanishing cities.
The zone of
(L, φ, n)
admissible for stable
cities turns out to be bounded, and cities must disappear in 3 cases: (i) when the current number
n
of cities becomes larger than certain uniform bound
becomes greater than certain uniform bound
than certain uniform bound
Ld∗ ;
n∗ ,
or (ii) when the world population
or (iii) when trade freeness
φ
L
becomes larger
φd∗ .
Therefore, other parameters given, whatever the historical city system is, growth of
L, φ
or
n
makes our city system abruptly switch to complete dispersion, due to individual migration. This
conclusion on migration is more interesting than Anas' global optimality of cities. It surprisingly
contrasts with Krugman's agglomeration outcome, because Krugman's agglomeration force and
dispersion force decrease together with trade freeness, the latter decreasing faster, which seems a
the agglomeration force is the
benets of combining rms and consumers, it decreases as trade freeness increases However, our
dispersion force is the commuting cost and it does not change with trade freeness, which explains
the outcome
doubtful assumption for modern cities. In our case, as in Krugman,
.
. Similarly, under constant city sizes, the dispersion force does not change with world
population but the agglomeration force weakens, due to an increasing share of imported varieties
in consumption, meaning that domestic production becomes less important to consumers.
Of course, both Anas' and our analysis of this general tendency are too stylized, in comparison
to richer urban models that describe a heterogenous city system (for instance, Tabuchi and Thisse,
2006, Behrens et al., 2010).
We rather provide a link between these developed models and the
two-regional models that economic geography has previously relied on.
Our goal here is more
modest; we just make a step in analysis of endogenous city systems and show that some industries
may become dispersed, due to the tendency revealed by Anas. We also interpret our results as a
realistic outcome for a certain type of cities, in contrast to his interpretation of theory failure.
However, instead of limiting results of Propositions 1 and 2, detailed comparative statics can be
4
more interesting: What precedes the abrupt disintegration of the city system into villages? How do
economic parameters matter for
gradual
changes in cities? Proposition 3 states that under growing
population the migration stability restrictions can cause city size to either gradually shrink or to
abruptly collapse, but not to increase.
To get additional predictions, we must limit equilibria multiplicity and dene a reasonable
selection
among equilibria.
Our developers' equilibrium imposes new restrictions on cities, in
addition to migration stability.
We assume that citizens are able to restrict entry to their city,
or to attract new people by granting small privileges, and thereby increase average welfare. Such
collectively rational behavior is represented by a benevolent city government called local developer
or city mayor (unlike Anas' global planner but like entities that maximize the price of land by
trying to conform to the citizens wishes). Such developers may launch a new city. When attracting
a citizen, the developer is myopic enough to ignore the consequences of this development to the
wider world: depopulating other cities and changing their economic variables. Caring only about
changes in her city, correctly anticipating production and trade, she optimizes average welfare with
respect to her city size
N.
We explore two versions of such equilibria: myopic and wise, both
displaying similar features.
It turns out that the zone of (symmetric) stable developers' equilibria is a curve
migration equilibria, it is bounded near the origin, for higher
L
N (L)
within
it disappears and the result is
disintegration of cities (Proposition 4). Additionally (whenever cities are multiple), the equilibrium
city size
gradually decreases
with world population or trade freeness, before dropping down to its
technological minimum. This version of the model diers from Anas' global optimization in that
its local government ignores the interests of other cities. However, the main conclusion remains:
the tendency of (manufacturing, industry-specic) towns to decrease and eventually collapse down
to villages.
cities comprised of
industries lacking intra- or inter-industries externalities have a tendency to decrease and eventually
reach their minimal technological size
Overall, our study generally supports this prediction and concludes that
.
The rest of the paper is organized as follows: Section 2 presents our baseline model, Section
3 studies migration equilibria.
Section 4 deals with developers' equilibria, and the nal section
concludes. The Appendix is devoted to proofs and possible extension of the model.
5
2
Model: System of Cities with Migration
We introduce a model of city system very close to that of Anas', except for: (1) possibly asymmetric
cities and (2) migration process instead of social planner setting. We start with the description
of internal city structure and then embed it into a system of cities.
We construct the general
equilibrium one sector model and relegate to Appendix B discussion of possible extension of the
model to two sectors.
City.
Traditionally, we consider monocentric and circular cities endowed with a Central Busi-
ness District (CBD) where production and trade take place. The only production input is labor
supplied by consumers citizens. Each consumer needs one unit of land and possesses a unit of
time which she spends commuting to her workplace (CBD) and laboring. The cost of commuting
is
s
sx
units of time per unit of distance, therefore, consumer living at distance
units of time for commuting and supplies
there are
N
h(x) = 1 − sx
x
from CBD spends
units of labor for production. Suppose
residents in a given city. Then, the radius of such a city becomes
r=
p
N/π .
Given
individual labor supply and uniform distribution of citizens within a city, overall labor supply for
production
H
is given by
ˆ
r
2πxh(x)dx = πr2 − 2πr3 /3 = N − kN 3/2 ,
H(N ) =
(1)
0
where
√
k ≡ 2s/3 π
is a constant, summarizing commuting cost.
We denote the average (per-
citizen) labor supply in a city as
θ(N ) ≡ H(N )/N.
In addition, we assume zero opportunity value of land and free reallocation within a city.
Now, we explain how redistribution of rent makes income proportional to the labor supply.
Suppose that the city size is
r.
We have normalized the land rent on the edge of the city to zero.
Since consumers face the same price vector, free reallocation of consumers should lead to disposable
income equalization among them. In addition, we assume that the local government collects the
land rent and distributes it equally among citizens in a form of lump-sum transfer. The disposable
income of every citizen in a city of size
N
is
√
I(N ) = θ(N )w ≡ (1 − k N )w,
which is decreasing in size.
3 To
3
see details of rent redistribution, see that at any location within a city the sum of rent cost and commuting
6
System of cities and goods.
Suppose there are
n cities with population masses (N1 , N2 , ..., Nn ).
There is only one dierentiated good in the economy. Each variety is produced by only one rm
residing in some city. All cities trade with each other through some common hub, i.e., transport
costs for each pair of cities is the same. The market for varieties is monopolistically competitive.
So, the mass
mi
of varieties (and hence, rms) produced in city
i
is endogenous.
Consumers have identical Constant Elasticity of Substitution (CES) preferences over the set
of varieties:
"
U=
n ˆ
X
mi
0
i=1
,
where
xki(j)
σ>1
denotes the elasticity of substitution. A consumer of type
is a single purchase of variety
j
#σ/(σ−1)
(σ−1)/σ
xki(j) dj
produced in city
i
(2)
and consumed in city
k
maximizes in
xk
k.
Parameter
her utility (2)
subject to the budget constraint
n ˆ
X
pki(j)
numeraire,
is the price of variety
I(Nk )
pki(j) xki(j) dj ≤ I(Nk ),
j
produced in city
i
and consumed in city
demand function X(·)
"
1−σ
Xki(j) (pki(j) , Ik , Pk ) = p−σ
ki(j) I(Nk )/Pk
Pk =
n ˆ
X
i=1
Pk
k.
Labor is the
denotes income. Taking the rst-order conditions and expressing the Lagrange
multiplier from the budget, we obtain the consumer
with
(3)
0
i=1
where
mi
0
in the form:
#1/(1−σ)
mi
1−σ
pki(j)
dj
,
(4)
being a price index. It is perfect, as a price of one unit of utility, in the sense that the
indirect utility of a consumer in the city
Extension.
k
is
Vk = I(Nk )/Pk .
In Appendix B we also explain the two-sector version of the model, where prefer-
ences over the two goods have a standard Cobb-Douglas form:
"
n ˆ
X

Ǔ =
i=1
0
mi
#σ/(σ−1) µ
(σ−1)/σ
xki(j)
dj
 M 1−µ .
Since the budget shares remain constant, the qualitative features for this extension remain the
same as the below analysis of a one-sector model.
The extension explains how manufacturing
cities can coexist with large cities that provide services, which is important for interpretation.
cost must be the same. Hence, the rent at any point x must be R(x) = s(r − x)w, where w is wage per unit of
time, and total rent in the city is T R =
´r
0
2πxs(r − x)wdx = πswr3 /3 = kwN 3/2 /2. Since the local government
distributes the rent equally, we can now calculate disposable income of every citizen in a city of size N as I(N ) =
√
√
w(1 − 3k N /2) + kwN 3/2 /2N = (1 − k N )w.
7
Producer.
Each producer is a price-maker for her variety. As is standard for monopolistic
competition literature, we assume that a producer in city
i.e.
F
the producer has xed labor cost
production. Wage is normalized to
i has a cost function C(y) = (cy + F )wi ,
to set up a plant and marginal labor requirement
wi = 1
c
of
when we consider symmetric cities, but it may dier
when we analyze deviations from symmetry. Trade within a city is costless, whereas trade with
other cities requires iceberg transportation costs. This means that supplying one unit of a good
from city
i
to city
k
requires
τki = τ > 1
these assumptions each producer
πi(j) =
n
X
j
in city
units of good when
i
i 6= k
but
τki = 1
if
i = k.
Under
has a prot function
[pki(j) − τki cwi ]Xki(j) (pki(j) , Ik , Pk )Nk − F wi
(5)
k=1
which she maximizes in prices subject to demand functions (4) taking the price indexes as given. As
is standard, under CES preferences, such a prot function is concave and has a unique maximum.
symmetric
Then, symmetry of producers leads to
us to drop index
j
pricing by all rms from a given city. This allows
from further discussion. Standard rst order condition for producer's problem
delivers
στki cwi
pki =
σ−1
n
X
τki cxki Nk
πi =
σ−1
k=1
!
−F
wi .
Free entry into the market drives rms' prot in every city to zero.
(6)
Combining zero-prot
conditions (6) with labor market clearing yields an equilibrium mass of varieties in every city:
mi =
Finally, equilibrium wages,
wi ,
Ni θ(Ni )
Fσ
∀i.
(7)
and corresponding prices,
pki ,
and incomes,
I(Nk ),
can be
obtained from market clearing for a representative variety produced in each city:
n
X
τki p−σ I(Nk )Nk
ki
k=1
Trade equilibrium
a bundle
{mi }
{xki }k=1,n
i=1,n
Pk1−σ
= (σ − 1)F/c ∀i
associated with a system of
n
cities of sizes
of consumption values, a bundle of prices
and vector of wages
{wi }
(8)
(N1 , N2 , ..., Nn )
{pki }k=1,n
,
i=1,n
is dened as
vector of varieties masses
such that: (i) consumption values solve consumers problems (2)
subject to budget constraint (3) under given prices, wages and available varieties; (ii) prices solve
producers problems (5) given demand function (4), price indexes and wages; (iii) rms earn zero
prot (free entry); (iv) labor market and market for every variety clear.
8
We do not discuss the existence of such general equilibria, pointing out later on the existence
of symmetric ones (whose behavior under small perturbations we study).
equilibrium delivers indirect utility
world population amounts to
As is standard, we call
Pn
i=1
Ni = L
cities:
Vk = Vi
L
n
Vk = θ(Nk )wk /Pk
Further, every trade
to any consumer in city
k.
Suppose the
consumers.
(N1 , N2 , ..., Nn )
cities of sizes
a
migration equilibrium
if (1)
and (2) related trade equilibrium yields the same level of indirect utility across
∀k, i.
Naturally, the symmetric distribution of population across an arbitrary number of cities is a
migration equilibrium. Indeed, the case of symmetric cities implies symmetric trade equilibrium.
In this case cities are interchangeable, and utility is equalized across cities. However, our goal is to
understand, when such symmetric migration equilibrium is
stable
, in the sense that small pertur-
bations are not amplied. Therefore, we shall consider mainly symmetric (or close to symmetric)
population distributions across cities. Let us reserve notation
with
n
3
Migration Stability
cities of size
N,
so that
(n, N ) for the symmetric equilibrium
L = nN .
In this section we discuss the stability of any symmetric equilibrium
(n, N ) against small perturba-
tions in population distribution. First, we dene two stability conditions based on dierent kinds
of perturbations: migration to villages and migration to other cities.
1. Consider a system of slightly asymmetric cities. Starting from
that a new
n
cities of size
N,
suppose
(n+1)-st city of size ε is created, with one of the old cities hence taking size Ñ = N −ε.
If in new trade equilibrium indirect utilities
we say that this migration equilibrium
Vi
(n, N )
evaluated at point
is (strictly)
ε≈0
satisfy
Vn+1 (ε) < Vn (Ñ ),
stable against dispersion, otherwise
it is not.
2. Consider a system:
of size
N.
1-st city of size N1 = N + ε, 2-nd city of size N2 = N − ε and n − 2 cities
If in related trade equilibrium incremental utility
is negative, we say that migration equilibrium
dV1 (N +ε)
evaluated at the point
dε
(n, N ) is (strictly) stable
ε≈0
against agglomeration,
otherwise it is not.
When a symmetric migration equilibrium
(n, N )
migration stable equilibrium.
9
satises both stability conditions, we call it a
0.20
0.15
Φ
0.10
0.05
0
80 000
50
60 000
40 000
L
100
20 000
n
0150
Figure 1: Region of stable equilibria.
The rst requirement of stability is that a small shift of population from a city into a previously
unpopulated area does not create incentives for mass movement to this newly created town. The
second requirement is that small movements of populations from one city to another do not make
the target city more attractive, it makes sense when
n ≥ 2.
We must add that, in reality, there
can be more sophisticated deviations from the stable state: any vector of changes in populations;
thereby real stable equilibria could be narrower that what we call stable. However, our denition
is sucient to show that stability zone is bounded.
Now we formulate the conditions when a symmetric equilibrium is stable in both senses.
Lemma.
(Stability conditions)
persion if and only if
(1) Symmetric migration equilibrium (n, N ) is stable against disr
1−k
− 1 − 1
1 + (n − 1)τ 1−σ σ σ−1
L
>
.
n
nτ 1−σ
(9)
(2) Symmetric migration equilibrium (n, N ) is stable against agglomeration if and only if
√
k L
< √
√ .
1−σ
2
n
−
3k
L
(σ − 1) σ − 1 + σ 1+(n−1)τ
1−σ
2σ − 1
Proof.
(10)
1−τ
See Appendix.
It may be interesting to look at the shape of stable combinations
region is displayed in Fig. 1 for specic values
σ = 11, k = 0.005.
(L, φ, n)
All
from Lemma. This
combinations inside this
shaded area generate stable equilibria, because our Lemma gives necessary and sucient conditions.
10
In Fig. 1, we see that the zone of stable equilibria is
bounded
in all three dimensions
L, n, φ.
This zone has a complex saddle-type shape, it looks like a hill with a grotto underneath (see our
subsequent gures for details). Our further plan is to prove such boundedness for
values
k, σ .
any
parameter
In some sense, this means studying the comparative statics of sections of the shaded
area from Fig.
1.
Specically, to correctly resolve Anas' question about vanishing cities, we
describe now how the
region of stable migration equilibria
changes with the population of the
whole system or/and with trade frictions.
First of all, we simplify the notation by (conventionally) introducing trade freeness
[0, 1].
This measure is decreasing in the elasticity of substitution,
trade. For any number
σ,
and higher
φ
φ ≡ τ 1−σ ∈
implies freer
n of cities, the condition of stability against dispersion can be reformulated
in two alternative ways:
(1) given trade freeness
φ,
total population is bounded from above as
#2
"
1
− σ1 − σ−1
n
1
−
φ
L(n) ≤ Ld (n) = 2 1 − 1 +
;
k
nφ
(2) given total population
L,
freeness of trade is bounded from above as
1
φ(n) ≤ φd (n) =
1+n
Similarly, under any number of cities
1−k
p
.
2σ−1
− σ(σ−1)
L/n
−1
n, the condition of stability against agglomeration requires
that:
(1) given freeness of trade
φ,
total population is bounded from below:
"
#2
n
2
L(n) ≥ La (n) = 2
;
k σ + 2 + (σ−1)σnφ
(2σ−1)(1−φ)
(2) given total population
L,
freeness of trade is bounded from below:
φ(n) ≥ φa (n) =
1
1+n
σ(σ−1)
√
√
L
√
(2σ−1) 2 n−3k
−σ+1
k
.
L
In other words, two kinds of stability conditions provide upper and lower bounds on parameter
values so that any symmetric migration equilibrium can be stable. Using these bounds, the following proposition shows that population growth or trade liberalization (increasing freeness) must
lead to the non-existence of stable equilibria.
11
Proposition 1.
(No stable cities in large/free world)
(1) Maximal stable population L (n) is bounded from above; i.e., there exists such L < ∞,
that any equilibrium (n, L/n) is unstable against dispersion whenever L > L ;
(2) Maximal stable freeness φ (n) is separated from one; i.e., there exists such φ < 1, that
any equilibrium (n, L/n) is unstable against dispersion whenever φ > φ .
Proof.
L (n)
n
d
d∗
d∗
d
d∗
d∗
d
(1) We start with the behavior of
when
goes to innity. Brief inspection reveals that
∞×0 indeterminacy. However, we can apply l'Hospital's rule to rearrangements:
2σ−1
2σ−1
− (σ−1)σ
− (σ−1)σ

−1 1−φ
1−φ
1−φ
2σ−1
− n2 φ 
1 − 1 + nφ
1 + nφ
σ(σ−1)
 p
√
lim k Ld (n) =
=
,
n→∞
1/ n
− 2n13/2
this limit is of type

and get
i
h p
d
lim k L (n) = 0.
n→∞
Then, by continuity
such that
d
limn→∞ L (n) = 0.
∀n ≥ n̄ Ld (n) ≤ Ld (1) > 0.
its maximum
Ld∗
By extreme value theorem,
Ld (n)
on interval
[1, n̄]
n̄
attains
(which is nite) and, therefore, it is bounded by the value of this maximum on
the whole interval
equilibrium
This fact can be interpreted as the existence of some
(n, N )
[1, +∞).
Then, for
is unstable for all
L
greater than the universal critical population
Ld∗ ,
the
n.
(2) The proof of the second part is similar. First, applying l'Hospital's rule to the expression
for
φd (n),
any nal
Thus,
φd∗
it is possible to show that
n,
limn→∞ φd (n) = 0.
and attains some maximum
φd∗ < 1
Further,
at some nite
is separated from 1, so that on the entire interval
φd (n)
n,
is separated from one for
as in the previous argument.
[1, +∞) 3 n
any equilibrium
(n, L/n)
is unstable against dispersion.
if the world is large enough or trade is free enough, the only stable outcome
is the dispersion of the population
In other words,
to villages.
the region of stability (Fig.1) in dimension
equilibria with large number of cities
n
n.
The remaining question is the boundedness of
In other words, we ask whether stable symmetric
exist. The next proposition precludes this possibility and,
therefore, gives additional credibility to the vanishing cities theory.
Proposition 2. (No stable equilibria with many cities).
Under any admissible parameters (L, φ, k, σ),
there exist some n̄ such that any equilibrium with bigger number n > n̄ of cities is unstable.
Proof.
See Appendix.
12
L
L
14 000
14 000
12 000
12 000
10 000
Φ=0.05
10 000
8000
Φ=0.05
8000
6000
6000
Φ=0.1
4000
4000
2000
Φ=0.1
n=1
2000
Φ=0.3
0
5
10
15
20
n=LN
25
Φ=0.3
n
n=LN
0
2000
4000
N
6000
Figure 2: How the region of migration stable equilibria shrinks with respect to trade freeness
φ.
Although our stability conditions reduce the amount of possible equilibrium congurations to
a bounded zone in
(φ, L, n)
space, there is still a continuum of them for admissible parameter
values. Indeed, any existing city creates a lock-in eect, preventing creation of new cities. This
result is akin to that of Fujita, Krugman and Mori (1999), who have found continuum of equilibria
in a city system with explicit linear space structure. Fujita et al. employ evolutionary dynamics
for the selection among equilibria, however, due to complexity of their space structure, they have
been forced to resort to numerical simulations. On the contrary, the simplied spacial structure of
our model allows us to describe all stable equilibria and provide analytical selection by simplied
evolutionary dynamics.
Comparative statics.
How does the system of cities
change
when the population grows or
trade cost decreases? Do the cities in our model grow, gradually decrease or collapse? We rst
explain numerical simulations, interpret them, then develop them into a proposition.
Fig. 2 plots the regions of migration stable equilibria in
freeness:
φ = 0.05,
zone from Fig.
straight line,
0.1, 0.3.
4
coordinates for changing trade
In essence, the left panel of Fig. 2 contains some sections of a stable
1 in (n, L) plane.
n = 1,
(n, L)
The right panel inverts this zone into
(N, L)
space, and the
cuts away the cases with less than one city (the same way the vertical line
with abscissa 1 does in the left panel). The main observation is that
the smaller in terms of area the zone of stable equilibria is
the larger trade freeness is
. Specically, the boundaries of the
zone shrink towards the origin, making the stable area smaller.
4 Our
stability conditions (9) and (10) can be rewritten as L > L˜a (N ) =
L < L˜d (N ) =
trade freeness.
(1−φ)N
φ
1
.
σ(1−σ)
√
(1−k N ) 2σ−1 −1
(2σ−1)(1−φ)N
σ(σ−1)φ
h
2
√
k N
i
− σ − 2 and
It also follows from them that minimal stable city size does not depend on
13
First consider the direction of changes outside stability zone: Does the family of cities move
towards stability or towards collapse? The arrows outside the zone show the migration tendency.
In the right panel we observe that above the upper boundary, the tendency works to
N
of a city (which is shown in the left panel as increasing
n = L/N ).
decrease
size
The arrow in the right side
of the right panel says that when a too-large unstable city decreases its population, it can reach
the region of stability. A similar stable result is shown by the lower left arrow, which is
kink
above the kink
breaking the left boundary. Instead, the upper-left arrow (
below the
) in this panel says
that, under suciently high population, a too-small city further loses its population and
collapses
into a village. This (upper left) boundary of the stable zone is unstable.
With this in mind, we use this gure to express our intuitions about possible changes in the
city system.
To grasp the possible impact of a growing population
consider the following thought experiment. Assume, for instance,
only one settlement (n
= 1)
and with a small population
L = 2:
This historical point of urbanization lies below the critical value
of related stability zone.
L
on cities under given
φ = 0.3.
φ,
Suppose we start with
Adam and Eve. What happens?
La (2), i.e.,
below
the lower bound
So, it cannot happen that the couple live apart, in dierent villages.
Instead, they must agglomerate.
Suppose now that population grows.
Then the agglomeration
tendency remains: everyone lives in the same city. The picture tells us that this single-city pattern
of urbanization will persist until the population exceeds approximately 600.
From this point
onwards, our growing population can either remain in this city or try to settle a new one. A twocities equilibrium becomes possible when the population reaches approximately 1,100. However,
when the population exceeds 1,900, any
n-city
system loses stability; it abruptly collapses into
villages consisting of 1 citizen each (minimal technological size).
Observe that dierent levels of trade cost make a qualitative dierence.
φ = 0.05,
Under
φ = 0.1
or
on the upper border of the stability region there is a weak possibility that a growing
population may result in a
gradually growing number of cities
instead of abrupt dispersion, at
least on the ascending wing of this region (nevertheless, each city decreases in size). Generally,
under typical parameters, this fairy tale and related picture support the idea that
abrupt decrease in city sizes
either gradual or
are possible in response to growing population or/and trade freeness.
(Recall that we study only the cities based on increasing-returns reason for agglomeration, and
5
leave aside other kinds of cities.)
5 On
We formulate this tendency as a proposition.
the other hand, when trade cost is large, after population hits level Ld (1), there is a possibility of the
14
Assume that during growth of the population under xed other parameters, the
number of cities remain stable until the system reaches the border of stability zone, thus, the shape
of the border governs the evolution of city sizes. Then, further evolution can display either gradually
decreasing city size or abrupt collapse, but not an increase.
Proof.
Proposition 3.
In our stability condition (9), as well as in Fig.2, we see that it is the dispersion condition
(not the agglomeration one) that limits the size of the stable world from above. Therefore, this
condition governs the comparative statics.
Its violation triggers the increase in the number of
cities. From the formula we see that the related city size
4
Ld (n)/n
is a decreasing function.
Developers' Stability
Although the notion of migration stability allows us to reduce the number of plausible equilibria
and ensures our impossibility results, the remaining multiplicity of stable equilibria is somewhat
disturbing. Therefore, in this section we develop another notion of stability as a selection criterion:
stability against actions by a developer (local government) who aims to maximize the representative citizen's utility in her city and who has some power to invite in or push out citizens. Of course,
it would be fair to call this actor benevolent local government or city mayor, but if we believe
that the benets of a particular city structure to citizens can be capitalized in the land rent, there
should not be any dierence between the mayor's behavior and that of a developer. However, this
decision-making diers from Anas's global planner. It diers also from simple migration because
it considers all intra-city benets from inviting a new citizen in or forcing some citizen to go out instead of personal benets to a migrant. Nevertheless, we still impose the requirement of stability
against dispersion or creation of a new city, because the developer cannot
force
citizens to stay in
the city.
Wise developer.
We assume in this paragraph that each developer completely predicts all
changes in the trade equilibrium that will happen after inviting a new citizen from some other city.
(Symmetric)
wise developers' equilibrium
local Nash equilibrium among
n
is a system of cities
(n, N )
developers choosing their city sizes.
such that it is a strict
It means that there is an
creation of the another city and the existence of migration stable equilibria with more than one city and total
population larger than Ld (1). Nevertheless, straightforward inspection reveals that maximum stable city size
Ld (n)
n
is decreasing in n, therefore, we conclude that it is implausible to observe an increase in the representative city size
with an increase in the system population in the model economy.
15
ε̂ > 0
such that all possible local
ε-perturbations (ε < ε̂)
bring strictly negative value changes to
a developer.
This notion does not consider the possibility of new cities and other asymmetric situations. By
our assumption of wise predictions, the changes in the equilibrium trade and welfare coincide with
predictions that we made when studying migration, because the wise developer understands that
she can attract a new citizen from some other city. Further, she expects a citizen to join some
other city (not to die) once expelled from hers. Thus, the same equations remain valid, and this
allows us to show that the new concept of equilibrium is a selection from the previous concept.
Namely, under any
φ, the wise developer's equilibrium is the lower border of related equilibria zone
displayed in Fig. 2. Interestingly, a wise developer would chose a system with the largest number
of cities and the smallest city size among migration stable equilibria.
decreases
Indeed, if the migrant goes out and thereby
metry, she
increases
welfare in the destination city, by sym-
welfare in her city of origin. Only when the derivative of welfare with respect
to migration is zero, can the situation be a wise developers' equilibrium. Thus, we come to
(i) A system of cities (n, N ) is a wise developer's
equilibrium only if it satises stability against agglomeration as equality:
Proposition 4.
(Wise developers' equilibrium)
#2
"
n
2
L = La (n) = 2
;
k σ + 2 + (σ−1)σnφ
(2σ−1)(1−φ)
(11)
(ii) It belongs to migration-stable equilibria. Thereby, the developer's equilibrium remain possible
within same three bounds: world population, trade freeness and number of citiesall must be small
enough.
Proof.
Established in text.
Myopic developer.
Although the introduced notion is perfectly rational, we nd our re-
quirements of local government's perfect foresight too demanding.
Indeed, the setup requires a
developer to predict changes not only in her city, but in all cities throughout the country as well.
To relax this requirement, we introduce the notion of a myopic developer. More precisely, we assume that each developer is myopic (boundedlyrational) in predicting outside trade consequences
caused by excluding or inviting a citizen. This means that when maximizing welfare in her city,
she expects no response from all relevant variables in other cities; population, price indexes and
wages are taken as given. Suppose there are
n−1
16
cities of size
N,
whereas #1 developer's city
has size
N1
(to be optimized). Then, given symmetry in
n − 1 other cities,
the (trade) equilibrium
conditions for price index and wage in developer's city can be formulated as:
P11−σ
N θ(N )
=
Fσ
στ c
σ−1
1−σ
N1 θ(N1 )
(n − 1) +
Fσ
σcw1
σ−1
1−σ
(12)
(n − 1)τ θ(N )N [στ cw1 /(σ − 1)]−σ θ(N1 )N1 w1 [σcw1 /(σ − 1)]−σ
= (σ − 1)F/c
+
Pi1−σ
P11−σ
(13)
This form of equilibrium conditions is standard. However, the developer's optimization with
respect to
N1
is dierent, since she takes
N
and
Pi
as given.
Denote elasticities of price index
and wage (perceived by developer) in city #1 with respect to local population
respectively.
Denition.
We call a symmetric equilibrium
(n, N )
N1 = N ,
as
εdP1
and
1
εw
d ,
stable against a myopic developer
elasticity of indirect utility (perceived by developer) is equal to zero
evaluated at
N1
if
P1
1
εVd 1 ≡ εθ + εw
d − εd = 0
and the second derivative of the indirect utility w.r.t.
N1
is negative and
the equilibrium stable against dispersion.
(i) A system of cities (n, N ) is a myopic developer's equilibrium only if it satises the following condition:
Proposition 5.
(Myopic developers' equilibrium)
"
#2
n
2
L = Lm (n) = 2
;
k σ + 2 + (σ−1)σnφ
(2σ−1)
(14)
(ii) It belongs to migration-stable equilibria. Thereby, the developer's equilibrium remain possible
within same three bounds: world population, trade freeness and number of cities all must be
small enough.
Proof.
See Appendix.
The established condition for stability against actions by a myopic developer is similar to
the condition of stability against agglomeration (10). It diers only by multiplier
(1 − φ)
in the
last term of the denominator. Thus, the costlier trade is, the more developer stability behavior
resembles that of stability against agglomeration (and that of a wise developer). The intuition is
straightforward: cities aect each other through trade only. Therefore, the higher trade costs are,
the less impact a developer's city has on other cities, and the smaller is the developer's mistake
17
L
L
14 000
14 000
12 000
12 000
10 000
Φ=0.05
10 000
8000
Φ=0.05
8000
6000
6000
Φ=0.1
4000
4000
2000
Φ=0.1
n=1
2000
Φ=0.3
0
5
10
15
20
Φ=0.3
n=LN
25
n
0
Figure 3: Developer stable congurations in coordinates
in assumption of a lack of change in other cities.
system towards fewer cities, i.e, larger size:
n=LN
2000
4000
N
6000
(n, L) or (N, L) (under k = 0.001, σ = 11).
Moreover, the developer's myopia pushes the
Lm (n) > La (n).
Now we present comparative statics analysis of the stability against a developer's action graphically. Fig.3 is a copy of Fig.2 supplemented with the line of developer's stable equilibria
and its counterpart
Lm (n)
L˜m (N )6 .
Observe that when the world population
L
grows, the related point on the solid curve of the
developer's equilibria moves to the right in the left panel. Its counterpart shifts to the left in the
right panel, which describes the same equilibrium in terms of the city size
means that
decreases
N = L/n.
the number of cities increases in response to population growth
Such behavior
, whereas
the city size
.
A similar conclusion follows for trade freeness, only the comparison does not go along each solid
curve, but is a comparison of three curves. When freeness increases, the point of equilibrium (for
any size of the world
L)
goes to the right in the left panel and to the left in the right panel. This
the number of cities increases in response to decreasing trade costs
city size decreases
L (n)/n
φ
again means that
. Indeed,
m
is again a decreasing function of
, whereas
the
, meaning that the new
equilibrium must have smaller city size and, hence, a larger number of cities. Thus, we have come
to the proposition which was the purpose of introducing the developers' equilibria.
Consider the growth of population L under xed other parameters, or decreasing
trade cost τ under xed other parameters. In each case, both wise and myopic developers' equilibrium displays either gradually decreasing city size or abrupt collapse of cities, but not an increase
Proposition 6.
6 Again
one can easily derive that L˜m (N ) =
(2σ−1)N
σ(σ−1)φ
h
2
√
k N
18
i
−σ−2 .
in city size.
Proof.
See Appendix.
5
Conclusion
We have revisited Anas' (2004) theorem that claims monopolistic competition is insucient to
explain cities: in response to growing population or decreasing trade costs, cities in his model
decline and disappear.
Questioning Anas' assumptions, instead of his normative approach, we
consider migration and developers' equilibria.
Still,
the vanishing eect proves to be robust
to
these realistic modications of the model, as well as to multiple sectors in the economy.
As a result, we interpret the vanishing eect as a realistic outcome. In real life, many industries
that
do not need externalities
from a city indeed relocate to small towns or rural areas. This model
is one of the possible explanations why. Moreover, the comparative statics analysis predicts that
goods with better transportation technology will be more likely to be produced in smaller cities.
19
References
Anas, A. (2004). Vanishing Cities:
What does the new economic geography imply about the
eciency of urbanization? Journal of Economic Geography, 4: 181-199.
Behrens, K., G. Duranton and F. Robert-Nicoud (2010). Productive cities: Agglomeration, selection and sorting. CEPR Discussion Paper 7922.
Fujita, M., Krugman, P., Mori, T. (1999). On the evolution of hierarchical urban systems.
pean Economic Review
Euro-
, 43: 209-251.
Fujita, M., Thisse, J.-F. (2002).
regional growth.
Economics of agglomeration: Cities, industrial location, and
Cambridge: Cambridge university press.
Henderson, J.V. (1974). The sizes and types of cities.
American Economic Review
, 64: 640-56.
Helpman, E. (1998). The size of regions. In D. Pines, E. Sadka and I. Zilcha (eds.) Topics in public
economics. Theoretical and applied analysis. Cambridge: Cambridge University Press, 33-54.
Holmes, T. J., Stevens, J. J. (2004). Spatial distribution of economic activities. In North America.
In V. Henderson and J.-F. Thisse (eds.)
Handbook of regional and urban economics
, volume 4,
2797-2843. Amsterdam: North-Holland.
Kolko, J. (2010). Urbanization, agglomeration and coagglomeration of Service Industries. In E.
Glaeser (ed.) Agglomeration economics. Chicago: National Buro of Economic Research and The
Chicago University Press, 151-180.
Starrett, D. (1978). Market allocations of location choice in a model with free mobility.
of Economic Theory
Journal
17: 21-37.
Tabuchi, T. (1998). Urban agglomeration and dispersion: A synthesis of Alonso and Krugman.
Journal of Urban Economics
, 44: 333-351.
Tabuchi, T., Thisse, J.-F. (2006). "Self-organizing Urban Hierarchy," CIRJE F-Series CIRJE-F414, CIRJE, Faculty of Economics, University of Tokyo.
20
A
Proofs
Proof of Lemma.
ε, ε).
Since rst
(1) Consider a city system perturbed by a new town of size
ε, (n+1, N, N, ..., N −
n − 1 cities are symmetric, we concentrate on trade equilibrium, which is symmet-
ric for those cities and take labor in the rst city as numeraire. This implies wage
these
i = 1, n − 1.
wi = 1
for all
Using pricing rule (6) that uses constant markups, this system has at most six
distinct prices (domestic and export price for normal city, the same for the disturbed city, and for
new city):
pii =
pi0 n =
σc
στ c
σcwn
, p i0 i =
, pnn =
,
σ−1
σ−1
σ−1
στ cwn
σcwn+1
, pn+1,n+1 =
,
σ−1
σ−1
στ cwn+1
.
σ−1
pi0 ,n+1 =
This system can be aggregated into price indexes, evaluated at point
only big cities from new ones (by symmetry, we obtain
"
N θ(N )
Pi =
Fσ
σc
σ−1
"
Pn+1
1−σ
N θ(N )
=
Fσ
ε = 0,
(15)
so that we distinguish
wn = 1):
#1/(1−σ)
(1 + (n − 1)τ 1−σ )
στ c
σ−1
(16)
1−σ #1/(1−σ)
n
(17)
At trade equilibrium, the utilities also depend on wages. To nd the wages, we recall constant
rm size and use market clearing equations (8) for varieties produced in
n big cities and in (n+1)-st
city, which is small:
(1 + (n − 1)τ 1−σ )θ(N )N [σc/(σ − 1)]−σ
= (σ − 1)F/c
Pi1−σ
(18)
nτ θ(N )N [στ cwn+1 /(σ − 1)]−σ
= (σ − 1)F/c.
Pi1−σ
(19)
Taking a ratio of these two conditions, we obtain the equilibrium (shadow) wage in
(n + 1)-st
city:
−σ
wn+1
=
1 + (n − 1)τ 1−σ
nτ 1−σ
21
(20)
Recall that
Vn+1 ≤ Vn
θ(0) = 1. Therefore, we can rewrite the comparison of utilities in small and big cities
wn+1 /Pn+1 ≤ θ(N )/Pn .
as a comparison of real incomes
θ(N ), wn+1
Substituting the denition of
and the ratio of price indexes from above we get result (9).
(2) Consider a city system perturbed (as in the denition of agglomeration stability) by small
migration
ε
from the second city to the rst one,
(n, N + ε, N − ε, N, ..., N ).
We apply the same
method. There are again at most six distinct prices and we can write the equilibrium equations
for the price indexes and wages with labor in cities
"
(N + ε)θ(N + ε)
P1 =
Fσ
"
(N + ε)θ(N + ε)
P2 =
Fσ
"
(N + ε)θ(N + ε)
Pi =
Fσ
σcw1
σ−1
i = 3, n
being a numeraire good:
1−σ
1−σ
(N − ε)θ(N − ε) στ cw2
+
Fσ
σ−1
#1/(1−σ)
1−σ
N θ(N ) στ c
+
(n − 2)
Fσ
σ−1
στ cw1
σ−1
(21)
1−σ
1−σ
(N − ε)θ(N − ε) σcw2
+
Fσ
σ−1
#1/(1−σ)
1−σ
N θ(N ) στ c
(n − 2)
+
Fσ
σ−1
1−σ
(N − ε)θ(N − ε) στ cw2
+
Fσ
σ−1
#1/(1−σ)
1−σ
N θ(N )
σc
+
(1 + (n − 3)τ 1−σ )
Fσ
σ−1
στ cw1
σ−1
1−σ
(N + ε)θ(N + ε)w1 [σcw1 /(σ − 1)]−σ τ (N − ε)θ(N − ε)w2 [στ cw1 /(σ − 1)]−σ
+
P11−σ
P21−σ
τ N θ(N )[στ cw1 /(σ − 1)]−σ
+(n − 2)
= (σ − 1)F/c
Pi1−σ
(22)
τ (N + ε)θ(N + ε)w1 [στ cw2 /(σ − 1)]−σ (N − ε)θ(N − ε)w2 [σcw2 /(σ − 1)]−σ
+
P11−σ
P21−σ
τ N θ(N )[στ cw2 /(σ − 1)]−σ
+(n − 2)
= (σ − 1)F/c
Pi1−σ
Dierentiating this system of equation w.r.t.
ε we aim to sign
22
dV1
at the symmetric point
dε
ε = 0.
Note that: (1) at the symmetric point
we have
dw2
dε
third city (i
1
= − dw
dε
6= 1, 2)
dPi
dε
= 0,
i.e.
for any
the eect from population increase in city #1 is cancels out by the eect from
X
Denote the elasticity of any variable
dX ε
and totally dierentiate equations (21) and (22) with respect to
dε X
with respect to
ε
ε
as
we obtain:
(1 − σ)(1 + (n − 1)τ 1−σ )E P1 = (1 − τ 1−σ )(1 + E θ + (1 − σ)E w1 )
(23)
σ(1 + (n − 1)τ 1−σ )E w1 = (1 − τ 1−σ )(1 + E θ + E w1 + (σ − 1)E P1 )
(24)
We are interested in the elasticity of indirect utility, which can be expressed as
E w1 − E P1 .
N2
(2) by symmetry and denition of
and similar equality applies to price indexes; nally,
exactly same decrease in city #2.
EX ≡
w1 = w2 = 1 ;
E V1 = E θ +
The solution to the system of elasticity equations delivers:
E w1 − E P1 = (1 + E θ )
Therefore, the stability condition
2σ − 1
1−σ
(σ − 1) σ − 1 + σ 1+(n−1)τ
1−τ 1−σ
E V1 ≤ 0
can be rewritten in a form
2σ − 1
≤−
1−σ
(σ − 1) σ − 1 + σ 1+(n−1)τ
1−τ 1−σ
Recall that
√
θ(N ) = 1 − k N
Eθ
.
1 + Eθ
√
and, hence,
k N
√
E θ = − 2(1−k
.
N)
Substituting
Eθ
into previous inequality
delivers result (10). Q.E.D.
Proof of Proposition 2.
proaches zero with a speed
As we have shown in the proof of Proposition 1, critical
1/n.
Ld (n)
ap-
Therefore, consider the following limit and apply to it l'Hospital's
rule:

1−φ
nφ
1− 1+
 p
lim k nLd (n) =
n→∞
1/n
2σ−1
− (σ−1)σ
=
2σ−1
σ(σ−1)
1+
1−φ
nφ
2σ−1
− (σ−1)σ
−1 − n12
− 1−φ
n2 φ

 (2σ − 1)(1 − φ)
=
σ(σ − 1)φ
Similarly,

p
lim k nLa (n) =
n→∞
2
(σ−1)σnφ
σ+2+ (2σ−1)(1−φ)
1/n

=
2n
σ+2+
23
(σ−1)σnφ
(2σ−1)(1−φ)
 = 2(2σ − 1)(1 − φ)
σ(σ − 1)φ
Combining these limits together and applying continuity we obtain
0.
This implies that
such that
n̄
exists such that
La (n) ≤ L ≤ Ld (n),
∀n ≥ n̄ La (n) > Ld (n),
limn→∞ n(La (n) − Ld (n)) >
or equivalently, there is no
L
which is the necessary condition for migration stable equilibrium.
Q.E.D.
Proof of Proposition 5.
Let us perform comparative static exercise w.r.t.
N1 .
Taking
elasticity of the equilibrium conditions, we nd
Here the subscript
(1 − σ)(1 + (n − 1)τ 1−σ )EdPn = 1 + E θ + (1 − σ)Edwn
(25)
σ(1 + (n − 1)τ 1−σ )Edwn = 1 + E θ + Edwn + (σ − 1)EdPn
(26)
d
emphasizes that elasticity is perceived by the developer. The elasticity
equations for the stability against agglomeration (23) and (24) look very much alike, only with
φ) multiplier on the right hand side.
(1−
Therefore, the costlier the trade, the more developer stability
behavior resembles that of stability against agglomeration. The intuition is straightforward: cities
aect each other through trade only.
Therefore, the higher trade costs are, the less impact a
developer's city has on other cities, and therefore, the smaller is the developer's mistake in the
assumption of no change in other cities. Solving these equations and evaluating the elasticity of
indirect utility, we obtain:
EdV1 =
2σ − 1
(1 + E θ ) + E θ
(σ − 1)(2σ − 1 + σ(n − 1)φ)
(27)
Straightforward algebra yields the result. Q.E.D.
Proof of Proposition 6.
We start with the case of a wise developer. First, observe that due
to Bernoulli's inequality
"
2σ−1 #2
− σ(σ−1)
2
n
1
−
φ
n (2σ − 1)(1 − φ)
d
L (n) = 2 1 − 1 +
≤ 2
k
nφ
k
σ(σ − 1)nφ
Second, observe that
La (n)
is a unimodal function with
arg max La (n) = n̄ =
n
Thus, for
n > n̄ La (n)
(28)
is decreasing.
(σ + 2)(2σ − 1)(1 − φ)
σ(σ − 1)φ
Moreover, we now show that for such
(29)
n La (n) ≥ Ld (n).
Indeed,
#2
2 "
2
(2σ
−
1)(1
−
φ)
n (2σ − 1)(1 − φ)
n
2
a
L (n) = 2
≥ 2
≥ Ld (n)
(σ+2)(2σ−1)(1−φ)
k
σ(σ − 1)nφ
k
σ(σ
−
1)nφ
+1
σ(σ−1)nφ
24
where the rst inequality follows from
n > n̄
and denition of
n̄
(29) and the second inequality
follows from (28). Recall that stability against dispersion requires
developer stable equilibrium is possible only on the increasing part of
L
growing, the number of cities
city size
N
declines. Further,
n
grows as well, and since
La (n)
n.
La (n).
La (n)/n
The city size
N
φ
therefore, wise
Thus, with population
is a decreasing function, the
decreases with the increase in freeness
equilibrium must be on the increasing part of the curve, decreasing
in the number of cities
L(n) < Ld (n),
given
φ.
L
Again because the
leads to an increase
declines.
Proof for the myopic developer works along the same lines. Q.E.D.
B
Extension to Two Sectors
A thoughtful reader noticed the discrepancy between our theoretical framework and its empirical
interpretation.
Describing the deurbanization of a particular manufacturing industry among
others, we have dealt so far with a general equilibrium model with one sector only.
However,
this section shows how our setup may be embedded into a multi-sector framework, and maintain
qualitatively similar results.
Model.
Assume our small-city system remains the same. However, now it also trades with
the outside world that produces some aggregate good
sectors.
form:
M
Ǔ = 
n ˆ
X
i=1
mi
produced in our cities (dened as
#σ/(σ−1) µ
(σ−1)/σ
 M 1−µ
xkji
dj
0
denotes overall utility of consumption and
constraint
U
produced outside. Preferences for the two goods have a standard Cobb-Douglas
"
Ǔ
(money), which describes, as usual, other
Our citizens consume two goods: composite good
2) and good
where
M
n ˆ
X
i=1
0 < µ < 1.
It is maximized under the budget
mi
pkji xkji dj + PM M ≤ I(Nk )
0
that includes the outside good and its price
PM ,
that may include the cost of transportation to
our cities. The internal structure of our city system remains the same, however, now cities trade
their local good for the outside good. the outside world consists of some
utilities and producing outside good
M.
L2
consumers with similar
Our rms and developers take the price
PM
of the outside
world as given either because this world is big enough or it exploits linear technology.
25
Analysis.
Under Cobb-Douglas preferences, the indirect utility in any city
i
is dened (up to
a constant multiplier) as follows:
Vi =
θ(Ni )wi
1−µ .
Piµ PM
One can see that the only dierence between the indirect utility in this economy and in the
economy from the main body of the paper is power
µ < 1
over the local good price index
Pi .
All our results are derived from this indirect utility function (it summarizes other elements of the
model). Hence, to be sure that our results apply in the extended version of the model it is sucient
to discuss: What changes when
µ
becomes smaller than 1?
The presence of the outside good
softens
the impact of the local price index on utility. Does
this fact work in favor of or against agglomeration? Other things being equal, one of the positive
eects of a growing city is a decreasing price index, being composed of larger share of cheaper
home-produced varieties relative to more expensive imported ones. This price eect acts towards
agglomeration.
(µ
= 1)
Therefore, among systems with dierent
µ,
the one described in details before
has the greatest driving force for agglomeration. Thereby, our dispersion results for
remain valid for any smaller
µ=1
µ, even being enforced by the decreased agglomeration force (the zone
of stable equilibria must shrink).
This argumentation bridges our economic interpretation with
the developed theoretical setup.
Interpretation.
In comparing the model with reality, the comparative statics in
a testable prediction. Indeed, under Cobb-Douglas preferences, parameter
µ
µ
suggest
represents the share
of income spent on the local good. Recall also (see Fig.3 and related discussion) that the cities'
size decreases in response to decreasing trade costs, or growing world population. A new topic is
comparative statics in parameter
φ,
µ.
It should push cities' size in the same direction as trade freeness
because both work as dispersion forces (more accurate proof requires further study), against
agglomeration. Thus, in a cross-section comparison among industries, one would expect to observe
a
negative
correlation between the urbanization level of each sector and the share of consumer
spending on its product.
Implementing this task, one needs to control only for technological
dierences in transportation and employment. Cost parameters
(c, F )
do not aect city size. To
the best of our knowledge, this hypothesis is novel, and it can be tested in future work.
26