CSIS Discussion Paper No. 154

On the economic geography of an aging society

Takaaki Takahashi
Center for Spatial Information Science,
The University of Tokyo.
5-1-5, Kashiwa-no-ha, Kashiwa, Chiba 277-8568, Japan

March 2018

Abstract
In this paper, we study the impacts of the aging of population upon economic geography, constructing a two-region NEG model with overlapping generations. It has been shown that there isa strong tendency for the elderly to agglomerate in one region and that the increase in the proportion of the elderly promotes the spatial agglomeration of economic activities. Furthermore,
we show that urban costs somewhat weaken this tendency although the elderly’s incentive toagglomerate is still strong.
Keywords: agglomeration; income linkage; location pattern; new economic geography; overlapping generations; surviving rate; urban costs
JEL Classification Numbers: J1; R1; R2
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
I would like to thank Toru Naito for insightful discussions. I also appreciate the comments from seminar participants at
various institutions. This research is partly supported by the Grants in Aid for Research by the Ministry of Education,
Science and Culture in Japan (No. 25285070).

1 Introduction
One of the most significant changes in demography that many industrialized countries have experienced in recent years is the increase in the elderly population. In Japan, for instance, theproportion of elderly persons 65 years old or older was 12.0% in 1990 but exceeded 20% in 2005
and rose to 26.3% in 2015. We can observe a similar change in a number of countries especially in
Asia and Europe, though on a somewhat smaller scale. This change is noteworthy not only in its
scale but also in its speed.
It is doubtless that such a significant change affects a spatial distribution of economic activities.
A main reason is that the incentives behind a location choice for the elderly differ from those forthe young population. In particular, most of the elderly, who do not work, pay little attention to
the spatial difference in wage rates while the younger care deeply about such factors. Therefore,
it is natural that the regions the elderly choose to live in are different from those the younger do.
Having said that, the impacts of the graying of society on the economic geography are notself-evident, because the decision makings of a younger generation and those of an older generation are linked with each other in at least three respects. First, the primary components of theincome for the elderly are savings and pensions, whose amounts are usually independent of theirconcurrent location, but dependent on their locations in the past periods. Therefore, the decision
makings of an older generation have an income linkage with those of a younger generation. Second, both generations need to pay urban costs, which are represented by land rents and include
commuting costs and the costs associated with negative externalities such as congestion and pollution. The amount of such costs depends on the total population, i.e., the sum of the younger andelderly populations, of a region where they live. In this regard, the location choice of the elderly
has a linkage, which we refer to as urban costs linkage, with that of the younger. Third and finally,high migration costs attract the elderly toward the region where they used to live, which yields a
migration costs linkage.
In this paper, we examine how the aging of population affects the economic geography throughthe changes in the working of agglomeration economies, paying special attention to the difference
between the incentives of the elderly and the younger behind location choices. For that purpose,
we elaborate a two-region new economic geography (henceforth, NEG) framework with some elements of an overlapping generation model incorporated, which bears the income and the urban
costs linkages. To begin with, we examine the basic case without urban costs to obtain following
results. First, the elderly are agglomerated in one region while the younger are dispersed over
the two regions. The agglomeration of workers in a certain region invokes harsher competition
among them, which results in a lower wage in that region, ceteris paribus. To the extent that aworking generation seek a higher wage, this works as a centrifugal force. However, it is not relevant to the retired. Second, as we introduce the elderly into the model, the distribution of young
workers becomes more biased to the region where the elderly are agglomerated. This is because1the agglomeration of the elderly in a region brings about a higher demand for the differentiatedproduct in that region. Third and finally, the rise in a proportion of the elderly promotes thespatial agglomeration of young workers. Next, the model is extended so as to incorporate urban
costs. We show that urban costs somewhat weaken the tendency for the elderly to agglomerate
in one region although it is still strong.
Despite the significance of the topic, the impacts of the aging of population on economic geography have been seldom studied through a rigorous analysis.1 One of the most important
exception is the work by Gaigné and Thisse (2009). They show that the increase in the retired
strengthens the tendency of agglomeration, considering the differences between the incentives ofworking and retired generations as in this paper. However, they do not take into account the income linkage. Instead, they assume that the income of the elderly consists entirely of the revenue
from land rent.2 Another exception is Naito and Omori (2017), who show, using an overlapping generation model, that agglomeration of workers becomes more likely to occur as the aging
advances. However, their model lacks the location choices of the elderly. In other words, theirinterests are not in the difference between the incentives of old and young populations behind location choices. In addition, the reason for their results is not related to agglomeration economies,
but rather to an intrinsic asymmetry between regions.3 Finally, Sato and Yamamoto (2005) examine the effects of demographic changes upon population concentration. They do not consider,however, the change in the elderly rate.
The rest of the paper consists of four sections. In the next section, we present a basic framework without urban costs. In Section 3, a formal definition of equilibrium is presented. Some
properties of equilibrium are derived in Section 4. Section 5 extends the model to include urban
costs. Section 6 concludes.

2

Model

Our analysis is based on a new economic geography (henceforth, NEG) framework extended so
as to take into account the difference between younger and older consumers’ incentives in a location choice. As an NEG framework, we use a two-region footloose entrepreneur model originated1 Three reasons are conceivable. First, it may be the case that the change began too recently for researchers to notice

its importance. Second, the change is not much noticeable in the United States, where young population have constantly
flowed in from abroad. Third and last, the effects are elusive to grasp empirically: too many factors are responsible for thelocation choices of the elderly in the real economy and controlling them is not straightforward.
2 This assumption would be questionable in terms of empirical validity. Casual observation tells us that the assumption
holds true only for wealthier retirees: average retirees cannot count on rent revenues for their remaining lives but on theirown savings and pensions. Furthermore, in their setting, the payments for land rents by young workers are received by
the elderly. Thus, an income transfer from a young to an old generation through the rent payments is one of the key
elements in their analysis. In contrast, we focus on the income linkage between the two generations.
3 In their two-region model, capital is used only in an “urban” region, and therefore, the marginal productivity of laborrises as capital increases only in that region while it remains constant in the other “rural” region. When the amount of
capital exceeds a critical level, therefore, all the workers decide to move into the urban region, which offers a higher wage.

2

from Forslid and Ottaviano (2003), which enables us to explicitly solve for an equilibrium. Ourmodel, however, diverges from a conventional model in that the transport cost to ship a differentiated product from one region to the other is assumed so prohibitive that it is indeed geographically immobile. This assumption is also adopted in the work by Gaigné and Thisse (2009), among
others. Furthermore, we incorporate elements of an overlapping generation model in order todiscuss younger consumers’ and older consumers’ location choices separately. In the research of
economic growth, the main arena where the overlapping generation model is used, the capital
accumulation arising from consumers’ savings occupies a central place in the explanation of theevolution of an economy. In our analysis, however, its role is kept minimal. The purpose is tofocus on the role of agglomeration economies.
In an economy, there are two regions (1 and 2); two categories of consumers, which we callworker and entrepreneurs; and two sectors, a constant-returns-to-scale (CRS) sector and a service
sector. The former sector produces a homogeneous good, which we call a CRS good, using capitaland labor of workers while the latter produces a variety of differentiated “services” using laborof entrepreneurs. The CRS good can be shipped between the two regions with no trade costs. Theinter-regional trade costs of services are, in contrast, prohibitive: they cannot be shipped.
Workers and entrepreneurs live for one or two periods. In the first period, they are “young”
and work in the CRS sector and the service sector, respectively. At the end of that period, some ofthem die and the rest survive. The probability that a consumer who was alive at a certain period
is still alive at the following period is a given constant and denoted by s ∈ (0, 1). This survival
rate is known to all agents in the economy. If they survive, they become “old” and retire fromjobs.
In this paper, we concentrate on a simple case where both the population of young workers
and that of young entrepreneurs are invariant over time. In other words, the same masses ofyoung workers and young entrepreneurs are born at each period. We denote these masses by L̄
and H̄.
Workers cannot migrate between regions: their locations are fixed. To focus on the case symmetric ex ante, we assume that half of them live in region 1 and the other half live in region 2.
More specifically, L̄/2 young workers and s L̄/2 old workers live in each region at every period.
Let us turn to entrepreneurs who work in the service sector. They can freely move between
the two regions at the beginning of not only their second period but also their first period. At
time t, λ1t of young entrepreneurs live in region 1 and λ2t of them live in region 2 (λ1t + λ2t = 1).
Out of the surviving entrepreneurs who lived in region 1 at time t − 1, the portion of µ1t choose
to remain in region 1 at the beginning of time t while the rest, the portion of 1 − µ1t , choose tomigrate to region 2. µ2t denotes the counterpart for the surviving entrepreneurs who lived inregion 2 at time t − 1. Then, the population of old entrepreneurs in region i at time t is equal to[]s H̄ µit λit−1 + (1 − µtj )λtj −1 (j ̸= i, i = 1, 2). It goes without saying that the total mass of the oldentrepreneurs at time t becomes equal to s H̄.

3

We adopt a standard setup of macroeconomics that insurance companies get rid of the uncertainties arising from the risk of death by aggregating individual uncertainties. The companies sella fair insurance to young consumers with the following clauses: If the consumer who has bought
the insurance is alive in the next period, the company pays not only what she has paid but also a
dividend (annuity) proportional to her payment. If she is daed, however, all the payment is takenby the insurance company. Let ρ be a dividend rate and i a market interest rate. (For a while, we
suppress the superscript for time because it causes no confusion.) An insurance company raises
from d dollars of insurance sales a profit equal to (1 + i )d − (1 + ρ)sd. For this insurance to be
fair, therefore, ρ must satisfy
1+ρ =

1+i
.
s

(1)

Because this exceeds 1 + i, consumers indeed buy this insurance. Thanks to this insurance, theassets left by the consumers who die at the end of the first period are redistributed to the surviving
consumers.
Workers and entrepreneurs have the same preference over the consumption of a variety ofservices and the CRS good. Consider a representative consumer who lives in region i when young
and lives in region j when old, if she survives. The instantaneous utilities that she obtains fromthe consumption at the young period and that at the old period are given byuyi = α ln Xyi + (1 − α) ln zyi

anduoj = α ln Xoj + (1 − α) ln zoj ,(2)respectively. Here, Xyi and Xoj are aggregate consumptions of services at respective periods:
[∫
Xyi =

ni

xy (v)σ −1
σ] σ−σ 1
dv

[∫
Xoj =

,

0

nj

xo (v)σ −1
σ] σ−σ 1dv,

0

where xyi (v) and xoj (v) are the consumptions of each variant at the two periods, and ni and n j are
the mass of varieties provided in region i and that provided in region j (i ∈ {1, 2} and j ∈ {1, 2}).
Furthermore, zyi and zoj are the amounts of the CRS good at the two periods. α is a share ofspending in the services.
The utility over the two periods that the consumer expects at her young period is equal to
Uij = uyi + βsuoj ,(3)

where β is a discount factor. Note that this is an expected utility, which is the weighted sum of theutility obtained when the consumer survives with probability s and the utility obtained when she
dies with probability 1 − s, which is equal to 0.
The budget constraint that a representative consumer with income y faces is given by
Eyi +
where
Eyi ≡

∫ ni
0

pi (v) xyi (v) dv + piG zyi

Eoj
= y,
1+ρ

and
4

Eoj ≡

(4)

∫ nj
0

p j (v) xoj (v) dv + pG
j zoj

are the instantaneous expenditures at the young and old periods, respectively. pi (v) is a price ofeach variant of services produced in region i and pG
j is a price of the CRS goods in region j.
Consumers make decisions on both locations and consumptions.
At the beginning of their young period, entrepreneurs decide the region to live in during that
period: they choose i to maximize Uij . They can freely change their location at the next period,that is, the decision on the location at the young period does not confine the location at the oldperiod. However, their decision at the young period affects the utility at the old period throughthe amount of wage they earn in the young period. In addition, old entrepreneurs choose theirlocation at the beginning of their second period. That is to say, they choose j so as to maximizeuoj .
As to consumptions, both workers and entrepreneurs make a consumption plan over the two
periods at the beginning of the first period: They decide xyi (v)’s, xoj (v)’s, zyi and zoj to maximize(3) subject to (4). Furthermore, surviving consumers can again choose consumption bundles at the
beginning of the old period right after they find that they have actually survived. They maximize
uoj subject to (4) given the value of Eyi that has been already spent. However, a glance at theutility function reveals that their choices are time consistent. In other words, the maximization
problem of the instantaneous utility, uoj , at the beginning of the old period gives the same valuesof xoj (v)’s and zoj as the maximization problem of the expected utility over the two periods, Uij ,at the beginning of the young period does. It is because the instantaneous utility at each period
is independent of the consumptions at the other period given the location decisions, that is, uyi
does not depend on Xoj nor zoj while uoj does not depend on Xyi nor zyi . Consequently, we do
not have to consider the old generation’s maximization problem.
Now, let us turn our attention to the production side. In the CRS sector, they produce the
CRS good through a CRS technology using capital and young workers’ labor. The output in eachregion is given by Kiθ L1i −θ where Ki is the amount of capital used in region i. The CRS good isshipped between the two regions with no transport costs and sold in a competitive market. Therefore, its price becomes equal to each other: p1G = p2G ≡ pG . Furthermore, capital moves freely
between the two regions and therefore, an equal rental price of capital, denoted by r, prevails in
the two regions. It follows from these arguments that the wage rates of young workers are equalized in the two regions. We take them as a numeraire. Finally, it follows from profit maximization
that
Ki =

θ
L̄
·
1 − θ 2r

and

pG =

1
1−θ

[

]
r (1 − θ ) θ
.
θ

(5)

In the service sector, on the contrary, they produce differentiated services through an IRS technology and sell it in a monopolistically competitive market. Each young entrepreneur owns afirm that produces one variety in the region of his/her residence. Thus, the number of varieties
produced in region i is equal to the number of young entrepreneurs in that region:ni = λi H̄

(i = 1, 2).

5(6)

To produce 1 unit of a variety, each firm uses a units of the CRS good and the skill of its entrepreneur. The revenue that remains after the payment for the input is taken by the entrepreneur,that is, the wage rate of the entrepreneur who produces variety v in region i at period t becomesequal to[]wi ( v ) = p i ( v ) − a q i ( v ),(7)

where qi (v) is the amount of each variant produced in region i. The entrepreneurs sell theirvarieties at the price that maximizes their own wages.
As has been mentioned in the introduction, we minimize the role of capital accumulation,
which occurs as a result of consumers’ savings, in order to focus rather on that of agglomeration
economies. For that purpose, two ad hoc assumptions are introduced. First, our economy is small
and open concerning capital. That is, capital moves freely not only between the two regions there
but also all over the world; and thus its rental price is determined in a global market. Given
this price, the amount of capital used in the economy is determined through (5) no matter how
much capital has been accumulated. The excess of capital flows out abroad while its shortage is
supplemented from abroad. Second, capital is owned by those living outside of our economy andtherefore, the income of a consumer in our economy consists only of wage.
Finally, because of a technical reason, which will become clear in the following analysis, welimit our attention to the case where σ is greater than 2. This corresponds to the result of empiricalstudies that σ is indeed much higher than 2 in the real economy.4
Assumption 1
σ > 2.

3 Definition of an equilibrium
In this section, we define an equilibrium.
To begin with, note that maximizing (3) under the constraint of (4) leads us to the well-known
fact that the price elasticity of demand for each variety is equal to a constant, −σ. Therefore,
maximizing wi (v) in (7) implies that
p1 ( v ) = p2 ( v ) ≡ p ≡

σ
a
σ−1

for any v,(8)that is, the prices of variants are equal to each other and become a constant. Therefore, (7) impliesthatwi ( v ) =4 For instance, Bergstrand et al.

a
q (v)
σ−1 i

(i = 1, 2).

(9)(2013) find out that the value is approximately equal to 7. Anderson and Wincoop(2004) review the literature and conclude that the parameter takes 5-10.

6Since the prices of all the variants are equal to each other, the demands for each variant are also
equal. They are derived from the utility maximization problem as follows:xyi (v) = xyi ≡αy·,1 + βs pni

xoj (v) = xoj ≡αβs(1 + ρ) y·
1 + βspn j

for any v.

(10)From now on, we need to distinguish time periods and thus begin to add a superscript ifnecessary. Aggregating demands for all consumers, we can derive the total demand for eachvariant:
qit (v) = qit ≡

[
]
α
t
t
t
Y
+
βs
(
1
+
ρ
)
Y
oi
pnit (1 + βs) yi

for any v.

(11)Here, Yyit is the aggregate income of young consumers in region i at time t, which is given by
Yyit = λit H̄wit +

L̄
.
2

(12)

Note that all the entrepreneurs in the same region earn equal wage because all the variants are
sold by the same amount: wit (v) = wit for any v (see (7)). Furthermore, Yoit is the aggregate incomeof old consumers in that region at that period. Because the old consumers at time t earned theirincomes at time t − 1,[]L̄Yoit = s µit λit−1 H̄wit−1 + (1 − µtj )λtj −1 H̄wtj −1 +
.
2

(13)

The first term inside the square brackets represents the aggregate income of the old entrepreneurs
who earned their wage in region i at the precedent period and then remained in that region. The
second term represents the counterparts of those who earned their wage in region j and thenmigrated to region i. Lastly, the last term denotes the income of the old workers in region i.
As has been mentioned, this paper concentrates on the steady state equilibrium in which all thevariables do not change over time. First, entrepreneurs’ decisions on which region to live in arealways same: µit = µi and λit = λi for any t (i = 1, 2). In what follows, we may use slightly
different notations, λ ≡ λ1 along with λi ’s. Second, the wage rates also take constant values:
wit = wi for any t (i = 1, 2). Third and last, the world capital market is also in a steady state, thatis, i and r take the same values over time, respectively. It implies that pG and Ki are also constants(see (5)) in addition to ρ (see (1)).
It is straightforward to derive the steady-state equilibrium wage rate by solving (8), (9) and(11) along with (12) and (13):wi =α(1 + η ) L̄ k j + αη (1 − µ j ),·k 1 k 2 − α2 η 2 µ1 µ22λi H̄

(14)

where η ≡ (1 + ρ) βs2 and k i ≡ σ (1 + βs) − α(1 + ηµi ) (i = 1, 2). Here, we can interpret η
as a “social subjective discount factor” in the following sense: If a young consumer postpones
her consumption worth 1 dollar until her old period, she will obtain 1 + ρ dollars when old.
However, she discounts the consumption at the old period to a constant fraction of β. In addition,
she may die with probability s, which further promotes discounting. Finally, if 1 thousand young
7consumers postpone their consumption, only s-thousand of those will actually benefit from it.
This becomes another force of discounting when we aggregate the benefits over the whole society.
Now, note that Assumption 1 implies that k1 > 0, k2 > 0 and k1 k2 − α2 η 2 µ1 µ2 > 0.5 Therefore,
wi ’s are positive.
To discuss entrepreneurs’ decisions on locations, it is necessary to compute indirect utilities.
The utility maximization problem implies that the expected indirect utility for the representative
entrepreneur with income y, who lives in region i in her young period and region j in her old
period, is given by
Vij (y) = vyi (y) + βsvoj (y),whereαln ni + k
σ−1

(16)

[
]
α
ln n j + k + ln (1 + ρ) βs
σ−1

(17)

vyi (y) ≡ ln y +andvoj (y) ≡ ln y +(15)

are the instantaneous indirect utilities associated with Uy and Uo , respectively, and k is a constant.6
When choosing locations, entrepreneurs are myopic in that they care only for instantaneous
differences in utility levels, which is a common assumption in the literature.
[]
First of all, the difference for a young entrepreneur is given by νy ≡ max V11 (w1 ), V12 (w1 ) −[]max V21 (w2 ), V22 (w2 ) . Here, the first maximum represents the lifetime indirect utility of an entrepreneur who lives in region 1 when young, while the second maximum represents the counterpart of an entrepreneur who lives in region 2 when young. If νy > 0, young entrepreneurs
choose to live in region 1 and therefore, λ = 1. Instead, if νy < 0, they prefer living in region 2
and therefore, λ = 0. If νy = 0, they are indifferent and any λ ∈ [0, 1] can be realized.
Next, the difference for an old entrepreneur is equal to vo1 (w1 ) − vo2 (w1 ) if she lived in region
1 when young, and to vo1 (w2 ) − vo2 (w2 ) if she lived in region 2 when young. However, it turns
out that vo1 (y) − vo2 (y) does not depend on y. Therefore, we can write those differences as νo ≡vo1 (y) − vo2 (y) for any y. If νo > 0, old entrepreneurs choose to live in region 1 and therefore,µ1 = 1 and µ2 = 0. If νo < 0, to the contrary, they choose to live in region 2: µ1 = 0 and µ2 = 1.
Finally, if νo = 0, they are indifferent between the two regions and therefore, µ1 and µ2 can takeany values in [0, 1]. However, we limit ourselves to the special case with µ1 = 1 − µ2 , where
the share of the old entrepreneurs who took up residence in region 1 and remain in that region
is equal to the share of those who settled down in region 2 and migrate to region 1. In otherwords, the share of the old entrepreneurs who live in region 1 is the same no matter where they
lived when young. Probably it is because each region has particular intrinsic “attractiveness” that
consumers do not care about when νo ̸= 0. Focusing on this special case enables us to consider
5 First, ki > σ (1 + βs ) − α (1 + η ) since µi ∈ [0, 1] (i = 1, 2). Since α (1 + η ) < 2, however, σ (1 + βs ) − α (1 + η ) > 0 if σ >2. Second, k1 k2 − α2 η 2 µ1 µ2 = [σ(1 + βs) − α(1 + ηµ1 )] · [σ(1 + βs) − α(1 + ηµ2 )]. However, σ (1 + βs) − α(1 + ηµi ) > 0
since α(1 + ηµi ) < 2 (i = 1, 2).
[]
6 k ≡ α ln ( α/p ) + (1 − α ) ln (1 − α ) /p G − ln(1 + βs ).

8that µ1 + µ2 is always equal to 1 since µ1 = 1 and µ2 = 0 if νo > 0, and µ1 = 0 and µ2 = 1 if
νo < 0. Thus, we can remove one variable and denote µ1 ≡ µ and µ2 ≡ 1 − µ.
Let us denote the location pattern by ℓ ≡ (λ, µ). We can use (6), (14), (16) and (17) to obtainthe utility differences. First, tedious computations yield(νy (ℓ)= (1 + βs) ln
= −

w1
w2

)

+

α
ln
σ−1

(σ − 1)(1 + βs) − α
ln
σ−1

where
ω (µ) ≡

(

(

n1
n2

λ
1−λ

)
(18)

)

+ (1 + βs) ln ω (µ),σ (1 + βs) − α(1 + η − 2ηµ)σ (1 + βs) − α(1 − η + 2ηµ)(19)

is a ratio of the total wage bill paid to the young entrepreneurs in region 1 to the bill paid tothose in region 2, that is, ω (µ) ≡ w1 λ1 /w2 λ2 . Note that Assumption 1 implies that both thenumerator and the denominator of (19) are positive. Consequently, ω (µ) > 0. For the following
analysis, in addition, it is useful to introduce a new notation for the product of the numerator anddenominator of (19):[][]A(µ) ≡ σ(1 + βs) − α(1 + η − 2ηµ) σ (1 + βs) − α(1 − η + 2ηµ) > 0.
Second,νo (ℓ) =

α
ln
σ−1

(

n1
n2

)

=

α
λ
ln
,
σ−1 1−λ(20)

which shows that old consumers are interested only in the relative availability of varieties in thetwo regions.
Then, the equilibrium is formally defined as follows:
Definition 1 (equilibirum)

A location pattern ℓ is an equilibrium if it satisfies both the following conditions:


 λ=1
λ ∈ [0, 1] λ=0

if νy (ℓ) > 0
if νy (ℓ) = 0
if νy (ℓ) < 0

and



 µ=1
µ ∈ [0, 1] µ=0

if νo (ℓ) > 0
if νo (ℓ) = 0
if νo (ℓ) < 0.

(21)

One of the simplest ways to discuss the stability of an equilibrium is to introduce an ad hoc
dynamic process. Our scenario goes as follows. Entrepreneurs simultaneously determine theregion to live as soon as they become old. If they are rational, the equilibrium defined above
realizes. Now, suppose that one of entrepreneurs mistakenly chooses to live in the region thatis not prescribed by the equilibrium. Then, we think that other entrepreneurs gradually migrate
across regions in response to the mistake. They migrate from the region where they receive alower level of indirect utility to the region where they receive a higher level, whenever there is9

a difference between those levels. Their migration behavior is thus described by the following
dynamics:{

λ̇ = bλ νy (ℓ)µ̇ = bµ νo (ℓ)(22)

for some bλ > 0 and bµ > 0, which are the adjustment speeds. It is obvious that the equilibrium
defined above is indeed a stationary point of this dynamical system. We consider the equilibrium
stable if the stationary point is locally asymptotically stable.

4

Properties of the equilibrium

In this section, we derive the equilibrium defined in the previous section and examine its properties.

4.1

Benchmark case with only one generation

Before proceeding to the analysis of a general case, it is worth examining a benchmark case with
only young generation existing. Suppose that s = 0 and eliminate variable µ. Then, the utility
difference for the young generation, (18), is degenerated to
(
)
σ−1−α
λ
νy (ℓ) = −
ln
.
σ−1
1−λ

(23)

It is obvious that there is a unique equilibrium, for which λ = 1/2. Because νy (ℓ) > 0 for λ < 1/2
and νy (ℓ) < 0 for λ > 1/2, the equilibrium is asymptotically stable (see (22)).
Proposition 1

In the benchmark case with only young generation (s = 0), there exists a unique stable equilibrium, where young entrepreneurs are equally split between the two regions.

4.2

General case with two generations

Now, let us begin the analysis of the general case with two generations. First of all, note that
Assumption 1 implies that(σ − 1)(1 + βs) − α > 0.

(24)

Therefore, νy (ℓ) goes to negative infinity as λ approaches 1. Consequently, there is no possibility
that νy (ℓ) > 0 at the equilibrium (see the first line in the left half part of (21)). Moreover, νy (ℓ) goesto positive infinity as λ approaches 0. Therefore, we can conclude that there is no possibility thatνy (ℓ) < 0, either. Hence, the only possibility left is that νy (ℓ) = 0 holds at the equilibrium. Let
us denote the value of λ that solves νy (ℓ) = 0 as a function of µ by λo (µ). Then, the equilibrium
must satisfy λ = λo (µ).

10

It is convenient to distinguish three cases, the case with λ > 1/2, that with λ = 1/2 and thatwith λ < 1/2.
For the first case with λ > 1/2, νo (ℓ) > 0 (see (20)): old consumers prefer living in the region
where more varieties are provided. (21) implies that µ = 1: old entrepreneurs are concentrated in()
region 1. Thus, provided that λ > 1/2, the only candidate for an equilibrium is ℓ = λo (1), 1 . To
relieve the burden of notation, let us express the variables evaluated at µ = 1 with a circumflex:ℓ̂(λ) ≡ ℓ(λ, 1), ν̂y (λ) ≡ νy (λ, 1), ω̂ ≡ ω (1) and λ̂o ≡ λo (1).
We examine the existence and uniqueness of λ̂o . First, note that ν̂y (1/2) = (1 + βs) ln ω̂, whereω̂ =σ (1 + βs) − α(1 − η )> 1.
σ (1 + βs) − α(1 + η )(25)

Therefore, ν̂y (1/2) > 0. Second, we have already seen that νy (ℓ) goes to negative infinity as λapproaches 1. Third, (24) implies that ν̂y (λ) is a decreasing function becausedν̂y (λ)(σ − 1)(1 + βs) − α
=−
< 0.
dλ
( σ − 1) λ (1 − λ )(26)These three observations imply that ν̂y (λ) cuts the λ-axis once and only once in interval (1/2, 1).
In other words, ν̂y (λ) = 0 has a unique solution in that interval, that is, there exists uniqueλ̂o ∈ (1/2, 1).
So far, we have shown that there exists a unique equilibrium given by (λ̂o , 1), provided that
{ > }
{ < }
λ > 1/2. The next task is to examine its stability. First, note that ν̂y (λ) = 0 if λ = λ̂o .
<
>
{ > }
{ < }
Therefore, λ̇ = 0 if λ = λ̂o . Second, ν̂o (λ) > 0 if λ > 1/2. Since λ̂o > 1/2, therefore,<>ν̂o (λ̂o ) > 0. It follows from (22) that µ̇ > 0. Fig.1 shows these two observations in a phase
diagram. Note that the locus of νy (ℓ) = 0 becomes upward sloping.7 It follows from the diagramthat the unique equilibrium is stable.
Next, let us move to the second case with λ = 1/2. First, (21) implies that λ = 1/2 canconstitute an equilibrium only if νy (1/2, µ) = 0 or equivalently, ω (µ) = 1. Therefore, µ must
be equal to 1/2. Second, (6) implies that the equal number of varieties are produced in the tworegions. Therefore, old consumers receive an equal level of utility whether they live in region 1or region 2, and consequently, any µ ∈ [0, 1] satisfy the conditions of (21). Consequently, there
is a unique equilibrium given by (1/2, 1/2). However, it is not stable: it is a saddle point. To
see this, we consider the linear approximation of (22) around a stationary point. Let J be thecorresponding Jacobian matrix:∂νy (ℓ) ∂λJ≡ ∂νo (ℓ)∂λ
7 A tedious computation yields

∂νy (ℓ)∂µ .
∂νo (ℓ) ∂µ[]A(µ) (σ − 1)(1 + βs) − αdµ[] > 0.
=dλ
4αηλ(1 − λ)(σ − 1)(1 + βs) σ (1 + βs) − α

11

(27)

SinceTr J = −bλ ·and| J| = −(σ − 1)(1 + βs) − α<0λ(1 − λ)(σ − 1)bλ bµ 4α2 η (1 + βs) [σ(1 + βs) − α]
·
< 0,
A(µ)
λ(1 − λ)(σ − 1)

(28)

(29)

any stationary point is a saddle point. Fig. 2 shows the phase diagram.
Finally, we examine the remaining case with λ < 1/2. Because everything becomes symmetricto that in the case with λ > 1/2, it turns out that there is a unique robust equilibrium given byℓ = (1 − λ̂o , 0).
Thus, we have established the following proposition.
Proposition 2

There are two stable equilibria, ℓ = (λ̂o , 1) and ℓ = (1 − λ̂o , 0). There exists no other stableequilibrium.

Regarding this proposition, two points are important.
First, the old entrepreneurs are always concentrated in one region although young entrepreneurs
are dispersed over the two regions. The reason is that a centrifugal force working in the decision
making process of young entrepreneurs is lacking in the counterpart of the elderly. We can explain this as follows. As the number of young entrepreneurs in a region declines, they are facedby less fierce competition among them, which raises their wage. At the same time, the demand
for their services decreases and their wage falls. However, the decrease in demand is restricted tosome extent because there are a fixed number of immobile workers in that region. Thus, the competition effect dominates the demand effect and as a result, young entrepreneurs are benefited
from the decline in their population. This produces a centrifugal force.
Second, because λ̂0 > 1/2, the distribution of young entrepreneurs is biased toward the region
where the old entrepreneurs are agglomerated. Comparing with the benchmark case, therefore,
we can say that introducing the elderly into the model strengthens the tendency for young entrepreneurs to agglomerate. This is because the concentration of old entrepreneurs in a region
brings about higher demand for the services in that region.
Now, in order to see how aging affects economic geography, let us examine the effects upon
λ̂o of a change in the surviving probability, s, which is equal to the proportion of old consumers
in the total population at the steady state. Totally differentiating ν̂y (λ̂o ) yields[]( )(σ − 1)(1 + βs) − α
w1
w2 dw1 /w2dνy (λ̂ ) = −dλ + β ln+ (1 + βs)
ds. (30)λ(1 − λ)(σ − 1)w2
w1ds
λ = const.
o

To begin with, the first term in the right hand side with dλ is negative: as the share of young
entrepreneurs in region 1 (λ) increases, more variety is produced in that region but at the sametime, the ratio of wage rates (w1 /w2 ) decreases because the competition among them becomes12

severer in region 1. The former effect is dominated by the latter and consequently, the rise in λreduces νy (λ̂o ).
Next, the first term in the square brackets represents a direct effect. Note that ν̂y (λ̂o ) = 0 implies
that

(
ln

w1
w2

)

α
=−
ln
(σ − 1)(1 + βs)(λ̂o
1 − λ̂o

)

<0

(31)

for λ̂o > 1/2: entrepreneurs’ wage rate is lower in the larger region, because a wider variety
of services are provided there. Consequently, the direct effect is negative. As consumers cometo expect a longer life period, income level becomes more important for them. Therefore, theexpected lifetime utility received when they work in a lower-wage region becomes further lower
comparing to the expected lifetime utility received when they work in a higher-wage region.
Finally, the last term in the right hand side represents the effect through the change in wage
rate ratio when the distribution of entrepreneurs is fixed. It is further decomposed into two subeffects:
dw1 /w2ds
λ = const.

==∂w1 /w2∂w1 /w2 dρ+∂s∂ρ
ds λ = const.
ρ,λ = const.
2αη (1 − λ)sλ [σ (1 + βs) − α(1 + η )]

·
2

[{

} {
}]
2(σ − α) + σβs − σ(1 + βs) − α(32)

The first term in the right hand side of the first line, or equivalently, the term in the first pairof braces in the second line captures the effect through the change in a relative wage rate when
the dividend rate, ρ, in addition to λ, remains unchanged. This wage effect is positive. As thesurviving probability rises, the demand for the variety produced in the larger region relatively
increases because old consumers are agglomerated in that region. As a result, the wage rate in
the larger region rises more compared to that in the smaller region. Instead, the second term in
the right hand side of the first line in (32), or equivalently, the term in the second pair of braces,
represents a negative dividend effect. As the surviving probability rises, the amount of dividend
each consumer receives decreases. Because this gives a negative impact on the consumption in an
old period more than that in a young period, the producers in the larger region suffer more thanthose in the smaller region. Therefore, the relative wage rate of the larger region decreases. (32)shows that the positive wage effect dominates the negative dividend effect. Since the direct effectis negative, however, the sign of the term in the square brackets in (30) is ambiguous.
However, it turns out that as s goes to 0, the size of the direct effect, given by the first term inthe square brackets in (30), approaches 0. The reason is as follows. As s goes to 0, η also goes to
0 and therefore, ω̂ approaches 1 (see(25). Therefore, ν̂y (λ) = 0 implies that λ converges to 1/2,
and the wage rate ratio, w1 /w2 = ω̂ (1 − λ)/λ, also converges to 1. On the other hand, the size ofthe sum of the wage effect and the dividend effect, given by (32), approaches a positive constant.
Consequently, the term in the square brackets in (30) becomes positive when s is sufficiently small.
/Then, dνy (λ̂o ) = 0 implies that dλ̂o ds > 0. Thus, we have established the following result.
13

Proposition 3

When s is not too large, the share of young entrepreneurs in the larger region further increases as
the surviving probability rises.

Thus, the aging of a population tends to strengthen the spatial agglomeration of economic
activities.

5 Extension to the case with urban costs
The above result that old entrepreneurs are concentrated in one region follows from the consequence of our model that the elderly prefer living in the larger region however large its population
is. In the reality, however, the concentration of population brings about various sorts of negative
impacts on consumers: It will boost land rent and thus housing price in the region of concentration, raise commuting costs through the geographical expansion of a city, and give rise to negative
externalities such as congestion and pollution, to name a few. We can conjecture that these factors
work as a force against the concentration of old entrepreneurs. In this section, we introduce such“urban costs” into the model. A main finding is that the tendency of concentration is weakened
but still strong. Because the main structure of the extended model is similar to that of the basic
one, we discuss the steady state from the beginning.
The level of the disutility of urban costs for each consumer depends on the population of theregion where she lives. Let Ni be the steady state population of region i. It follows from ourdefinitions thatNi (ℓ) =1+s
L̄ + (λi + sµi ) H̄.
2(33)

The sub-utility of a young consumer living in region i and that of an old consumer living in region
j are now expressed asuyi = α ln Xyi + (1 − α) ln zyi − γ ln c( Ni ) and uoj = α ln Xoj + (1 − α) ln zoj − γ ln c( Nj )(34)

instead of (2). Here, c(·), which is an increasing function, is a measure of the disutility of urban
costs and γ is its relative importance in consumer’s preference. Note that we are focusing on thesimple case where the functional form of the disutility for an old consumer is the same as that fora young consumer (both being equal to c(·)), and furthermore, the weights of the disutility arealso equal between the two types of consumers (both being γ) .
Let us obtain the differences between the utility levels when a consumer lives in region 1 andwhen she lives in region 2. For a young entrepreneur, the difference is given by
[ (
)]
(
)
c N1 (ℓ)(σ − 1)(1 + βs) − αλ) .
νy (ℓ) = −ln+ (1 + βs) ln ω (µ) − γ ln (σ−1
1−λc N2 (ℓ)(35)

Variable ω (µ) is still given by (19). Note that νy (ℓ) approaches positive infinity as λ goes to 0,
and negative infinity as it goes to 1. Therefore, it follows from the definition of equilibrium, (21),14that neither λ = 0 nor λ = 1 can be supported by an equilibrium. Consequently, νy (ℓ) = 0 at theequilibrium. As in the basic model, we denote λ that solves this equation by λo (µ). Because of
the limit properties mentioned above, there necessarily exists such λo (µ) in interval (0, 1). For an
old entrepreneur, furthermore, the difference is equal toανo (ℓ) =
ln
σ−1

(

λ
1−λ

)

[ (
)]
c N1 (ℓ)) .
− γ ln (c N2 (ℓ)(36)

Thus, the consumer prefers living in a region if the number of variants produced in that region isrelatively large and/or the disutility of urban costs to be paid in that region is relatively small.

5.1

Equilibrium with partial agglomeration

First, we examine the equilibrium at which old entrepreneurs are concentrated in one region, sayregion 1, while young counterpart are dispersed in the two regions. As in the basic model, we
denote the variables evaluated at µ = 1 with a circumflex, that is, λ̂o ≡ λo (1), ν̂o (λ) ≡ νo (λ, 1),ω̂ ≡ ω (1), which is still given by (25), and N̂i (λ) ≡ Ni (λ, 1) for i = 1, 2. A sufficient condition
for the concentration in region 1 being supported as an equilibrium is that ν̂o (λ̂o ) > 0 whereasa necessary condition is that ν̂o (λ̂o ) ≥ 0. Similarly, for the concentration of old consumers in()
region 2, a sufficient condition is given by νo λo (0), 0 < 0 while a necessary condition is given()
by νo λo (0), 0 ≤ 0.
/[]
Let us define Ω(µ) ≡ ω (µ) 1 + ω (µ) < 1, for which Ω̂ ≡ Ω(1) = ω̂/(1 + ω̂ ). Note thatΩ̂ > 1/2 since ω̂ > 1 (see (25)). Furthermore, we define
[ (
(
)]
)
c′ N2 (ℓ)
c′ N1 (ℓ)() + () > 0,
Γ(ℓ) ≡ γ H̄ ·c N1 (ℓ)
c N2 (ℓ)(37)

which measures the average of the growth rates of urban costs when the population of each region
increases. In addition, Γ̂(λ) ≡ Γ(λ, 1). Then, we can obtain the following result.
Lemma 1i) If λ̂o > Ω̂, both (λ̂o , 1) and (1 − λ̂o , 0) are stable equilibria.
ii) If λ̂o < Ω̂, there exists no equilibrium with µ = 1 nor equilibrium with µ = 0.
Proof

{ > }
{ > }
/
= Ω̂ if λ̂o (1 − λ̂o ) = ω̂, which holds if
<
<
(
(
)
)
{ > }
λ̂o
λ̂o
α
(σ − 1)(1 + βs) − α
ln
ln
+
−
(
1 + βs) ln ω̂ = 0
(38)
<
σ−1
σ−1
1 − λ̂o1 − λ̂oFor a while, we focus on (λ̂o , 1). Note that λ̂osince ω̂ > 0 by Assumption 1. However, ν̂y (λ̂o ) = 0 implies that the left hand side of (38) is equalto ν̂o (λ̂o ). If λ̂o > Ω̂, therefore, we have ν̂o (λ̂o ) > 0, which is a sufficient condition. Therefore,(λ̂o , 1) is an equilibrium. Instead, suppose that λ̂o < Ω̂. Then, ν̂o (λ̂o ) < 0, which violates thenecessary condition, ν̂o (λ̂o ) ≥ 0. Consequently, there is no equilibrium with µ = 1.
15

Next, the stability results from the following two observations. First, ν̂y (λ) is decreasing at λ̂o :dν̂y (λ̂o )(σ − 1)(1 + βs) − α=− o− Γ̂(λ̂o ) < 0.
(39)dλλ̂ (1 − λ̂o )(σ − 1)
{ < }
{ > }
{ > }
Therefore, for λ = λ̂o , sufficiently close to λ̂o , we have ν̂y (λ) = 0 and therefore, λ̇ = 0.
<><
Second, we have seen that ν̂o (λ̂o ) > 0, which implies that µ̇ > 0.
The similar reasoning applies to (1 − λo (0), 0): Since ω (0) ≡ 1/ω̂, we have N1 (λ, 0) ≡ N̂2 (1 −λ) and N2 (λ, 0) ≡ N̂1 (1 − λ), which implies thatvy (1 − λ, 0) ≡ −ν̂y (λ).

(40)

Therefore, 1 − λ solves vy (1 − λ, 0) = 0 if and only if ν̂y (λ) = 0, and consequently, λo (0) ={ < }1 − λ̂o . In the same manner as for (λ̂o , 1), furthermore, we can show that λo (0) = Ω(0) if
>
{ < }
o
o
o
νy (λ (0), 0) = 0. If λ (0) < Ω(0), we have νy (λ (0), 0) < 0, which is a sufficient condition>
for (λo (0), 0) being an equilibrium. However, λo (0) < Ω(0) is equivalent to λ̂o > Ω̂. Instead, if
λo (0) > Ω(0), or equivalently, λ̂o < Ω̂, then νy (λo (0), 0) > 0; and there is no equilibrium with
µ = 0. Finally, to prove the stability of the equilibrium, it suffices to show that dνy (λ, 0)/dλ < 0at λ = 1 − λo (0), or equivalently, dνy (1 − λ, 0)/dλ > 0 at 1 − λ = λo (0). However, it follows[]/
from (40) that this is the case if and only if d −ν̂y (λ) dλ > 0 at λ = 1 − λo (0) = λ̂o , which
indeed holds as (39) shows. ■To explicate the relative size of λ̂o and Ω̂, which plays a key role in the above result, it isuseful to ask what happens if γ changes when µ = 1. For that purpose, we define λ̄ as thevalue of λ that equates the populations in the two regions, that is, λ̄ ≡ (1 − s)/2. It follows that
{ > }
{ > }
N̂1 (λ) = N̂2 (λ) if and only if λ = λ̄. One of the important properties is that λ̄ < 1/2 for<<s > 0. Therefore, Ω̂ becomes greater than λ̄ since Ω̂ > 1/2.
It is straightforward to obtain following three observations (see the Proof of Lemma ?? for thedetail). First, as γ goes to 0, λ̂o converges to a value that exceeds Ω̂. Second, λ̂o approaches λ̄
as γ goes to infinity. Third and last, λ̂o decreases with γ as long as λ̂o > λ̄. These observations
altogether imply that there exists a critical value of γ, denoted by γS , that determines the relative
size of λ̂o and Ω̂:Lemma 2

There exists γS such that λ̂o

{ > }
{ < }
= Ω̂ if γ = γS .
>
<

Proof
First, as γ converges to 0, λ̂o approaches the solution to()(σ − 1)(1 + βs) − αλln+ (1 + βs) ln ω̂ = 0.
−
σ−1
1−λ
The solution is greater than 1/2. For such a limit value, furthermore, the left hand side of (38)][] [becomes equal to α/(σ − 1) ln λ̂o /(1 − λ̂o ) , which is positive. Therefore, (38) holds with the16

last inequality sign and consequently, the limit of λ̂o exceeds Ω̂. Second, we examine what happens if γ goes to positive infinity. Suppose that λ̂o approaches a constant a, which is greater than()/ ()λ̄. Then, c N̂1 (λ̂o ) c N̂2 (λ̂o ) approaches a constant larger than 1. Therefore, ν̂y (λ̂o ) = 0 requires that the first term in the right hand side of (35) go to positive infinity, which is achieved
when λ̂o approaches 0. However, this contradicts a > λ̄. Instead, suppose that λ̂o approaches a()/ ()
constant b, which is smaller than λ̄. Because c N̂1 (λ̂o ) c N̂2 (λ̂o ) approaches a constant smaller
than 1, ν̂y (λ̂o ) = 0 demands that λ̂o approach 1, which is a contradiction. Consequently, the only
possibility left is that λ̂o approaches λ̄. Third, we have∂ν̂y (λ̂o )/∂γ
dλ̂o
=−
.
dγ
dν̂y (λ̂o )/dλ(41)This is negative as long as λ̂o > λ̄ since dν̂y (λ̂o )/dλ < 0 (see (39)) and ∂ν̂y (λ̂o )/∂γ < 0 for λ̂o > λ̄.
The result immediately follows from these three observations and the fact that Ω̂ > λ̄. ■Indeed, we can solve for γS explicitly. Substituting λ̂o = Ω̂ into ν̂y (λ̂o ) = 0 yields the following
result:
γS =

α
ln ω̂
].
· [
σ − 1 ln c ( N̂ ) /c ( N̂ )
1

(42)

2

Since ω is the ratio of wage bills in the two regions, the right hand side is a measure of the relative
size of economic activities in region 1 discounted by the relative value of urban costs in that
region.
Using the result in Lemma 2, we can restate Lemma 1 as follows:
Proposition 4i) If γ < γS , ℓ = (λ̂o , 1) and ℓ = (1 − λ̂o , 0) are stable equilibria.
ii) If γ > γS , there exists no equilibrium with µ = 1 nor equilibrium with µ = 0.
This is explained as follows. When γ is too high, young entrepreneurs care about urban costs
to such a great extent that many of them choose to live in region 2 rather than region 1 where
old entrepreneurs are concentrated. As a result, the number of varieties produced in region 1
is too small to give old entrepreneurs an incentive to live in that region. As γ declines, urban
costs become less important and region 1 becomes more attractive for the young entrepreneurs.
This raises their population in that region (λ̂o increases), which enlarges the number of varieties
produced there. If this effect is large enough, old entrepreneurs come to choose to live in region
1 and thus, (λ̂o , 1) becomes an equilibrium outcome. Indeed, we have seen that as γ approaches
0, λ̂o converges to a value that exceeds Ω̂. Consequently, (λ̂o , 1) is supported by an equilibrium
when γ is sufficiently small. The basic model in the preceding section describes the extreme casewith γ = 0. We can reiterate these arguments for the equilibrium (1 − λ̂o , 0).
The critical value of γ resembles that of “ϕ” or freeness of trade (the inverse measure of transport costs) in the New Economic Geography literature: both values lie on the boundary of the set
17for which the agglomeration of mobile consumers of a certain type is supported as an equilibrium.
Thus, we refer to it as a sustain point, the term widely used in the NEG literature, of γ.
In general, furthermore, the effect of a change in the elderly ratio upon the critical value isambiguous. However, we can show that it is negative for s that is not too large, if function c(·) isconvex and its curvature is sufficiently large. That is, dγS /ds < 0 for s < s̃ ifc′′ ( N ) >
where
s̃ ≡

[c′ ( N )]2
for any N,c( N )(43)

σ − (1 + i )(σ − α)σβ(1 + i )(for the proof, see the Proof of Proposition 5).
Proposition 5

dγS /ds < 0 for s < s̃ if (43) is satisfied.
Proof
Suppose that λ̂o = Ω̂. First, since

[
]

σ
(
1
+
βs
)
−
α
(
1
−
η
)
1
+
s


 N̂1 ={
} + s H̄L̄ +
22 σ (1 + βs) − α1+sσ (1 + βs) − α(1 + η )


L̄ +
H̄,
 N̂2 =
22 [σ (1 + βs) − α](44)

we have N̂1 > N̂2 . However, (43) implies that c′ ( N )/c( N ) increases with N. Therefore, c′ ( N̂1 )/c( N̂1 ) >c′ ( N̂2 )/c( N̂2 ). Second, note that ∂ N̂1 /∂s = L̄/2 + (1 − ko ) H̄ and ∂ N̂2 /∂s = L̄/2 + ko H̄, where
[
]
α
dη
ko ≡
σβη
−
σ
(
1
+
βs
)
−
α
.
{
}
ds
2 [σ (1 + βs) − α]2
It follows from[σ (1 + βs) − α]2[]− σβαη = (1 + βs) σ(σ − 2α) + α2 + σβ2 s2 [σ − α(1 + ρ)] > 0

and dη/ds = (1 + i ) β > 0 that ko < 1/2. Consequently, ∂ N̂1 /∂s > ∂ N̂2 /∂s. These two observations lead to[]c( N̂1 ) c′ ( N̂1 ) ∂ N̂1c′ ( N̂2 ) ∂ N̂2∂c( N̂1 )/c( N̂2 )
=
−
> 0.
(45)
∂s
c( N̂2 ) c( N̂1 ) ∂sc( N̂2 ) ∂s/
Furthermore, note that ∂ω̂/∂s = 2αk′ [σ (1 + βs) − α(1 + η )]2 , where k′ ≡ −σβ + (dη/ds) [σ (1 + βs) − α].
{ < }
{ < }
{ < }
{ < }
Since k′ = 0 if s = s̃, we have ∂ω̂/∂s = 0 if s = s̃. Putting these findings alto>>>>
gether, (42) gives the result. ■This result says that provided that the following two conditions are satisfied, the agglomeration ofold entrepreneurs becomes more likely to be supported as a stable equilibrium when the elderly
ratio rises. The conditions are that the elderly ratio is not too large and that the urban costs
increase rapidly enough with the population of a region.
18

In addition, it is straightforward to see that s̃ decreases with σ, β and i whereas it increases
with α. Therefore, the tendency that aging of society strengthens the likelihood of agglomeration
is more pronounced when σ, β and i are lower and α is higher.

5.2

Equilibrium with full dispersion

Now, let us turn to the equilibrium at which old entrepreneurs are not concentrated in one region,that is, both regions have some old entrepreneurs: µ ̸= 1 and µ ̸= 0. Then, an equilibrium
necessitates νo (ℓ) = 0, or
α
ln
σ−1

(

λ
1−λ

)

[ (
)]
c N1 (ℓ)) = 0.
− γ ln (c N2 (ℓ)(46)

As we have seen earlier, furthermore, νy (ℓ) must be equal to 0 at the equilibrium. Combiningνy (ℓ) = 0 and (46) yields
λ = Ω ( µ ).

(47)

The equilibrium is the pair that solves (46) and (47) simultaneously. It is obvious that the symmetric distribution, ℓ = (1/2, 1/2) is supported as an equilibrium. There may exist anotherasymmetric equilibrium depending on the functional form of c.
To examine the stability of such an equilibrium, consider again the Jacobin of (27). In the casewith urban costs, we have, instead of (29) and (28),[](σ − 1)(1 + βs) − α
Tr J = −bλ+ Γ(ℓ) − bµ sΓ(ℓ) < 0
λ(1 − λ)(σ − 1)and| J| =]
bλ bµ (1 + βs) [− B1 (ℓ) + B2 (ℓ)Γ(ℓ) ,
A(µ)

(48)

(49)

where
B1 (ℓ) ≡4α2 η [σ(1 + βs) − α]sA(µ)
and B2 (ℓ) ≡+ 4αη [σ(1 + βs) − α] .
λ(1 − λ)(σ − 1)
λ (1 − λ )Because the sign of | J | is ambiguous, we can distinguish two cases: A stationary point is stableif Γ(ℓ) > B1 (ℓ)/B2 (ℓ) and therefore, | J | > 0 at that point. Instead, it is a saddle point if Γ(ℓ) <B1 (ℓ)/B2 (ℓ) and therefore, | J | < 0. Roughly speaking, the equilibrium is stable when urban
costs rises rapidly enough with the increase in population. Unfortunately, these conditions are
too complicated for us to conduct comparative statics. In particular, as a result of a change in aparameter, the equilibrium variables that solve (46) and (47) also change, which further affects
the conditions. Therefore, we focus on the symmetric equilibrium, because ℓ = (1/2, 1/2) is
supported as an equilibrium for any values of parameters.
{ < }
{ < }
For ℓ = (1/2, 1/2), Γ(ℓ) = B1 (ℓ)/B2 (ℓ) if γ = γ B , where
>
>
γB ≡

c( N o )
2α2 η
[ {
]
·
′
o
H̄ c ( N ) (σ − 1) s σ(1 + βs) − α} + αη19/
and N o ≡ (1 + s)( L̄ + H̄ ) 2 is the population of each symmetric region. We have thus establishedthe following result:
Proposition 6i) If γ > γ B , ℓ = (1/2, 1/2) is a stable equilibrium.
ii) If γ < γ B , ℓ = (1/2, 1/2) is an equilibrium but unstable.

This proposition states that the symmetric equilibrium is stable if the parameter for the importance of urban costs is greater than a critical value. However, it “breaks” if the parameter is lower
than the value. We refer to the critical value as a break point considering its similarity to the breakpoint of ϕ, freeness of trade, in the NEG literature.
Note that[{][ ′ o ]2 }c (N )dγ B
sσβL̄ + H̄B′′o= −γ ·c (N ) −+.
ds
2c′ ( N o )c( N o )s {σ(1 + βs) − α} + αη(50)This implies the following result.
Proposition 7dγ B /ds < 0 if (43) is satisfied.

Therefore, the symmetric equilibrium becomes more likely to be stable as the ratio of elderly rises,
as long as the curvature of urban cost function is sufficiently large.

6 Concluding remarks
This paper has studied the impacts of the aging of population upon economic geography, paying
special attention to the difference between the incentives of the elderly and the younger behindlocation choices. For that purpose, we elaborate a two-region NEG model with overlapping generations, which bears the income and the urban costs linkages. It has been shown that there isa strong tendency for the elderly to agglomerate in one region and that the increase in the proportion of the elderly promotes the spatial agglomeration of economic activities. In addition, we
have shown that urban costs somewhat weaken this tendency although the elderly still have astrong incentive to agglomerate in one region.
This piece of research, only one of the first steps to attack the topic, has a number of limitations.
Among them, the following seem to be important. First, we have assumed that it takes no cost to
ship a differentiated product across regions. This assumption makes a dispersion force so strong
for mobile workers (young entrepreneurs in our model) as to prevent them from agglomerating
in one region at an equilibrium. In order to have both their dispersion and their agglomeration
as equilibrium outcomes, the model necessitates a positive transport cost. Second, not only the20transport cost but also the migration cost has been abstracted away in our model. It is arguable,
however, that in the reality, the latter cost is considerably high especially for the elderly. It would
be necessary to incorporate the migration costs linkage into the model.

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