# Innovation through inter-regional interaction in a spatial economic
model††thanks: We are thankful to Sofia Castro, João Correia da Silva, Pascal
Mossay, Pietro Peretto, Anna Rubinchik and Jorge Saraiva for very useful
comments and suggestions. We would also like to thank the participants at the
Fourth International Workshop "Market Studies and Spatial Economics",
Université Libre de Bruxelles, ECARES, and at the 8th Euro-African Conference
on Finance and Economics / Mediterranean Workshop in Economic Theory, Faculty
of Economics, University of Porto. Funding Information: Japan Society for the
Promotion of Science Grant/Award Number: 21K04299; Fundação para a Ciência e
Tecnologia UIDB/04105/2020, UIDB/00731/2020, PTDC/EGE-ECO/30080/2017 and
CEECIND/02741/2017

José M. Gaspar and Minoru Osawa CEGE and Católica Porto Business School,
Universidade Católica Portuguesa. CEF.UP, University of Porto. Email:
<EMAIL_ADDRESS>of Economic Research, Kyoto University. Email:
<EMAIL_ADDRESS>
###### Abstract

This paper analyses a two-region model with vertical innovations that enhance
the quality of varieties of the horizontally differentiated manufactures
produced in each of the two regions. We look at how the creation and diffusion
of knowledge and increasing returns in manufacturing interact to shape the
spatial economy. Innovations occur with a probability that depends on the
inter-regional interaction between researchers (mobile workers). We find that,
if the weight of interaction with foreign scientists is relatively more
important for the success of innovation, the model accounts for re-dispersion
of economic activities after an initial stage of progressive agglomeration as
transport costs decrease from a high level.

Keywords: Innovation; inter-regional spillovers; knowledge diffusion; economic
geography

JEL codes: R10, R12, R23.

##  1 Introduction

This paper aims to explain how the intra-regional and inter-regional
interaction between scientists impacts knowledge creation and affects the
spatial distribution of agents. Moreover, it seeks to elucidate how the weight
of such interactions interplays with economic integration in order to study
the evolution of the space economy as trade barriers decrease.

The production of knowledge affects the firms’ capacity to innovate, which in
turn allows the production of higher quality manufactured varieties in a
region. But the chance of a successful innovation does not only depend on the
number of scientists that live in the same region, but also on the number of
scientists that reside in the other region. The weight of interaction with
“foreign” scientists in the production of knowledge hinges on the strength of
related variety (Frenken et al.,, 2007), which depends on several factors such
as cognitive proximity, cultural factors, diversity of skills and habilities,
among other factors.

Using a two-region spatial economics model, we show a very diversified set of
qualitative predictions regarding the spatial distribution of mobile agents,
and, especially, a very rich gallery of possibilities on how the related
variety affects the agglomeration process as economies become more integrated.
In particular, we uncover predictions of re-dispersion after an initial phase
of agglomeration. But our main novelty is that the process of the well known
bell-shaped relationship between economic integration and spatial imbalances
occurs with very different qualitative properties, not yet described in the
literature, that stem from the dispersive force that is the importance of
inter-regional interaction.

Throughout the last decades there has been a narrow focus of geographical
economics on pecuniary externalities through linkage effects. Other possible
sources of agglomeration economies such as knowledge externalities and
technological spillovers are left out (Gaspar,, 2018). Fujita and Mori, (2005)
argue that this is done out of convenience. Such a narrow focus enables
researchers to design a microfounded model based on the firms’ perspective
using modern tools of economic theory. However, it is true that further
development in geographical economics requires modelling the creation and
transfer of knowledge. In particular, the role of K-linkages has become
increasingly relevant in the economic geography literature. Building upon
pioneering works such as Berliant and Fujita, (2008, 2009, 2012), one should
hope that a new comprehensive economic geography theory fully integrates the
linkage effects among consumers and producers and K-linkages in space.
According to Fujita, (2007), geography is an essential feature of knowledge
creation and diffusion. For instance, people residing in the same region
interact more frequently and thus contribute to develop the same, regional set
of cultural ideas. However, while each region tends to develop its unique
culture, the economy as a whole evolves according to the synergy which results
from the interaction across different regions (i.e., different cultures). That
is, according to Duranton and Puga, (2001), knowledge creation and location
are inter-dependent. Berliant and Fujita, (2012) developed a model of spatial
knowledge interactions and showed that higher cultural diversity, albeit
hindering communication, promotes the productivity of knowledge creation. This
corroborates the empirical findings of Ottaviano and Peri, (2006, 2008).
Ottaviano and Prarolo, (2009) show how improvements in the communication
between different cultures fosters the creation of multicultural cities in
which cultural diversity promotes productivity. This happens because better
communication allows different communities to interact and benefit from
productive externalities without risking losing their cultural identities.
Berliant and Fujita, (2011) take a first step towards using a micro-founded
R&D structure to infer about its effects on economic growth. They find that
long-run growth is positively related to the effectiveness of interaction
among workers as well as the effectiveness in the transmission of public
knowledge.

Therefore, combining the typical pecuniary externalities in economic geography
models with the spatial diffusion of knowledge spawned from intra-regional and
inter-regional interactions alike is important if we want to infer about an
eventual circular causality between migration and the circulation of
knowledge. In other words, NEG may shed light on the importance of knowledge
exchanged between different regions through trade networks compared to
“internally” generated knowledge.

Besides the importance of heterogeneity in knowledge, it is also important to
discern about the _relatedness_ of variety. This relatedness measures the
cognitive proximity and distance between sectors that allows for a higher
intensity of knowledge spillovers. According to Frenken et al., (2007), a
higher _related_ variety increases the inter-sectoral knowledge spillovers
between sectors that are technologically related. This potentially adds a new
dimension to the role of heterogeneity and location in the creation and
diffusion of knowledge. Tavassoli and Carbonara, (2014) have tested the role
of knowledge intensity and variety using regional data for Sweden and found
evidence that the different types of cognitive proximity have an important
weight. This confirms the relevance of the spatial determinants of innovation
and knowledge creation.

The incorporation of knowledge creation and diffusion into geographical
economics could also benefit from the introduction of agglomeration mechanisms
in endogenous growth models with innovation. Particularly, innovations that
affect quality or a firm’s cost efficiency are usually driven by stochastic
processes. Typically, the production of knowledge involves some sort of
uncertainty. Therefore, we can think of quality as a proxy, or at least a
function, of a given firm’s stock of knowledge. In the literature following
Schumpeterian growth models such as Aghion and Howitt, (1992); Aghion et al.,
(1998), Young, (1998), Peretto, (1998), Howitt, (1999), or more recently
Dinopoulos and Segerstrom, (2010), innovations occur with a probability that
depends on factors such as the amount of the firm’s research effort, the
common pool of public knowledge available to all firms, and the individual
firm’s quality level. Other works in Schumpeterian growth theory have used
different technology production functions, such as Peretto, (1998, 2012,
2015). In other settings, firms are assumed to determine their quality levels
optimally such as in Picard, (2015). Introducing geography and worker mobility
in these models allows the success of innovations to depend also on the
magnitude of regional interaction through the exchange of ideas between
workers and producers alike among regions. If each region holds its own set of
ideas, or culture, then more localized spillovers translate into higher
related variety, as innovation benefits more from a regional common pool of
ideas. However, the interaction between researchers hailing from different
cultures is also important for innovation. This adds a potential new role for
transportation costs in NEG. For instance, higher trade integration, as
usually captured by lower transport costs, is likely to foster the inter-
regional communication between researchers, thus adding relevance to the
interaction between researchers from different regions. If cognitive proximity
is relatively less important than the interaction between different cultures,
then this _unrelated_ variety implies that lower transport costs are likely to
induce the dispersion of economic activities.

In the present model, we find that if inter-regional interaction is very
important for the success of firms’ innovation, then an increase in economic
integration from a very low level initially fosters agglomeration in a single
region. However, above a certain threshold, more integration leads to more
symmetric spatial outcomes, because firms find it worthwhile to relocate to
the peripheral regions in order to benefit from higher expected profits due to
the sizeable pool of scientists in the core, which increases the chance of
innovation in the deindustrialized region. Therefore, our model accounts for
re-dispersion after an initial phase of agglomeration. That is, we are able to
uncover a bell-shaped relation between economic integration and spatial
development. For a well documented literature on the causes of a “spatial
Kuznets curve”, we refer the reader to Osawa and Gaspar, (2021).

The rest of the paper is organized as follows. Section 2 introduces the
spatial economic model and describes its short-run general equilibrium.
Section 3 deals with the existence and stability of long-run equilibria
(spatial distributions). Section 4 discusses the relationship between economic
integration and spatial outcomes, which depends on the level of related
variety. Alternatives to the functional form that governs the manufacturing
firm’s innovation process are presented in Section 5. Finally, Section 6 is
left for discussion and concluding remarks.

##  2 The model

The following is an analytically solvable footloose entrepreneur model à la
Pflüger, (2004). The economy is comprised of two regions indexed by
$i=\\{1,2\\}$, two kinds of labour, two productive sectors and one R&D sector.
There is a unit mass of (skilled) inter-regionally mobile agents (which we dub
scientists henceforth) and a mass $l\equiv\lambda>0$ of (unskilled) immobile
workers (just workers, for short) which are assumed to be evenly distributed
across both regions, i.e., $l_{i}=\frac{\lambda}{2}$ $(i=1,2)$. The amount of
scientists in region $1$ is given by $z_{1}\equiv z\in\left[0,1\right]$ and
fully describes the spatial distribution of agents in the economy.

###  2.1 Demand

The utility function of a consumer located in region $i$ is given by:

$u_{i}=\mu\ln\left(\frac{M_{i}}{\mu}\right)+B_{i},\ \ \mu>0$ (1)

where $B_{i}$ is the numéraire good produced under perfect competition and
constant returns to scale. This good is produced one-for-one using $L$ workers
and its price is set to unity as is the wage paid to workers. The quality-
augmented CES composite $M_{i}$ is given by:

$M_{i}=\left[\sum_{j=1}^{2}\int_{s\in
S}\left(\delta_{ij}^{m(s)}d_{ij}(m,s)\right)^{\tfrac{\sigma-1}{\sigma}}ds\right]^{\tfrac{\sigma}{\sigma-1}},$
(2)

where $d_{ij}(m,s)$ is the demand for manufactures in region $i$ produced in
region $j$ for a given variety $s\in S$ of quality
$m\in\left\\{1,....,k\right\\}$, $k$ being the highest quality rung achieved
for any variety, $S$ is the mass of varieties in region $i$ and $\sigma$ is
the elasticity of substitution between any two varieties. The parameter
$\delta>1$ indexes the step size of quality improvements in region $i$ after a
successful innovation and $k$ is the leading quality grade for any given
variety $s$. Since $\delta^{m(s)}$ is increasing in $m$, the utility in (2)
reflects the fact that consumers have a preference for higher quality
(Dinopoulos and Segerstrom,, 2006). However, love for variety implies that
varieties, once adjusted for quality, are perfect substitutes. This means that
each consumer purchases only the good with the lowest quality adjusted price,
$p_{i}(s)/\delta_{i}^{m}$. If any two goods have the same quality adjusted
price, consumers will only buy the highest quality good (Dinopoulos and
Segerstorm 2006; 2010; Davis and Şener,, 2012). The firm responsible for each
quality improvement for a variety $s$ retains a monopoly right to produce that
variety at the highest quality possible, i.e., $k$. Therefore, if the quality
rungs $m=1,...,k$ have been reached, the $k\textit{th}$ innovator is the sole
source of the good of variety $s$ with the quality level $\delta^{k}$ (Barro
and Sala-i Martin,, 2004). In what follows, we can already predict that the
short-run general equilibrium will be comprised solely of firms that produce
the highest quality possible of their variety $s\equiv s_{k}\in S$.

Since individual incomes depend on the distribution of labour activities, we
have $y_{i}=1$ for the workers, and $y_{i}=w_{i}$, which is the the
compensation paid to the scientists that engage in research. Therefore, the
regional income is given by:

$Y_{i}=\frac{\lambda}{2}+w_{i}z_{i}.$

The individual budget constraint is given by $B_{i}+P_{i}M_{i}=y_{i}+\bar{B}$,
where $\bar{B}$ is the initial endowment of the numéraire and $P_{i}$ is the
regional price index. Since total expenditure on manufactures must equal
$P_{i}$ times the quantity composite $M_{i}$, agents maximize (1) subject to
the follwing budget constraint:

$B_{i}+\sum_{j=1}^{2}\int_{s\in S}p_{ij}(s)d_{ij}(s)ds=y_{i}+\bar{B},$

which yields the following optimal individual demands:

$d_{ij}(s)=\mu\frac{a_{i}(s)p_{ij}(s)^{-\sigma}}{P_{i}^{1-\sigma}},\
B_{i}=y_{i}+\bar{B}-\mu,\ M_{i}=\mu P_{i}^{-1},$ (3)

where $a_{i}(s)=\delta_{i}^{m(s)(\sigma-1)}$ is just an alternative measure of
a variety $s$’s quality in region $i$ and $P_{i}$ is the quality adjusted
price index given by:

$P_{i}=\left[\sum_{j=1}^{2}\int_{s\in
S}a_{ij}(s)p_{ij}(s)^{1-\sigma}ds\right]^{\tfrac{1}{1-\sigma}}.$ (4)

We assume that $\bar{B}>\mu$ in order to assure that both types of goods are
consumed. From (1) and (3), we obtain the indirect utility:

$v_{i}=y_{i}-\mu\ln P_{i}-\mu+\bar{B}.$ (5)

###  2.2 Manufacturing firms

For each firm, there is a variable input requirement of $\beta$ workers. A
manufacturing firm in region $i$ thus faces the following cost:

$C_{i}(q_{i}(s))=\beta q_{i}(s),$ (6)

where $q_{i}$ is total production by a firm in region $i$.

Trade of manufactures between regions is burdened by transportation costs of
the iceberg type. Let the iceberg costs $\tau_{ij}\in(1,+\infty)$ denote the
number of units that must be shipped at region $i$ for each unit that is
delivered at region $j$. We have $\tau_{ij}=\tau_{j}\in\left(1,\infty\right)$
for $i\neq j$ and $\tau_{ij}=1$ otherwise. The quantity produced by a firm in
$i$ is thus given by:

$q_{i}(s)=\sum_{j=1}^{2}\tau_{ij}d_{ij}\left(1+z_{j}\right).$

The profit of a manufacturing firm with the leading grade $k$ of variety $s$
in region $i$ is given by:

$\displaystyle\tilde{\pi}_{i}\equiv\pi_{i}(s\equiv s_{k})=$
$\displaystyle\sum_{j=1}^{2}p_{ij}(s_{k})d_{ij}(s_{k})\left(\frac{\lambda}{2}+z_{j}\right)-\beta
q_{i}(s_{k})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\left(p_{ij}(s_{k})-\tau_{ij}\beta\right)d_{ij}(s_{k})\left(\frac{\lambda}{2}+z_{j}\right),$
(7)

where $s_{k}$ denotes the highest quality of a given variety $s$. Given (7)
and the optimal individual demand in (3), the firm’s profit maximizing price
is the usual mark-up over marginal cost:

$p_{ij}=\dfrac{\sigma}{\sigma-1}\tau_{ij}\beta,$ (8)

which does not depend on the quality of the firm’s variety $s$. Under (4), the
regional quality adjusted price index in (4) becomes:

$P_{i}=\dfrac{\beta\sigma}{\sigma-1}\left(\sum_{j=1}^{2}\phi_{ij}A_{j}n_{j}\right)^{\tfrac{1}{1-\sigma}},$
(9)

where $\phi_{ij}\equiv\tau_{ij}^{1-\sigma}\in(0,1)$ is the _freeness of trade_
and $n_{i}$ is the number of manufacturing firms (and manufactured varieties),
and:

$A_{i}=\sum_{s=1}^{n_{i}}a_{i}(s),$

is the aggregate quality index for region $i$. The latter also constitutes a
measure of the aggregate regional knowledge level.

Consider now that the aggregate knowledge level in one region is a public good
such that it becomes readily available to the other region. Then
$A_{i}=\max\left\\{A_{1},A_{2}\right\\}$ so that both regions are able to
reach the highest aggregate knowledge level available. This leads to the
following assumption.

######  Assumption 1.

Let $A_{1}=A_{2}=A$.

The rationale behind Assumption 1 is basically to eliminate first nature
advantages between regions (i.e., regional asymmetries). This not only makes
the analysis much simpler but also allows us to focus on conveying the main
message of the paper, which is the effects of regional interaction on the
spatial distribution of economic activities. Under Assumption 1, the regional
price index becomes simply:

$P_{i}=\dfrac{\beta\sigma}{\sigma-1}\left(A\sum_{j=1}^{2}\phi_{ij}n_{j}\right)^{\tfrac{1}{1-\sigma}}.$
(10)

The higher the aggregate quality, the lower the cost of living in region $i$.

###  2.3 R&D sector

We assume that there is free entry in the R&D sector for each variety $s$. In
order to innovate, a firms decides _ex-ante_ to employ $\alpha$ scientists in
the R&D sector. In doing so, a firm producing variety $s$ at region $i$
reaches a leading quality grade $k$ with instantaneous probability:

$\Phi_{i}(s)=\min\left\\{\dfrac{bz_{i}+(1-b)z_{j}}{a_{i}(s)}\gamma
A,1\right\\},$ (11)

where $b\in(0,1)$ is the _weight of intra-regional interaction in the chance
of innovation_ and defines the importance of the exchange of ideas between
researchers alike among regions, and $\gamma>0$ is an efficiency parameter. We
are assuming that the lowest quality grade possible is $\underline{a}>1$ so
that $a_{i}(s)>1$, for any $s$. In all our analytical and numerical results,
we use a parametrization such that $\gamma
A\left[bz_{i}+(1-b)z_{j}\right]<a_{i}(s)$, guaranteeing that the first term of
$\Phi_{i}(s)$ lies in interval $[0,1]$.

It is worthwhile explaining how the underlying specification governing the
innovation process depends on the magnitude of the interactions (or lack of
them) between different “sets of ideas”. Analogous to the interpretation of
Berliant and Fujita, (2012), we implicitly assume that each region holds its
own set of ideas (or culture). Therefore, production of knowledge (which
amounts to innovation), depends on the amount of “within region” interaction
among scientists, but also on the interaction with scientists hailing from a
different region. As such, we can say that $b$ is sort of a measure of the
_related variety_ of knowledge (Frenken et al.,, 2007). A higher related
variety means that innovation benefits more from a regional common pool of
ideas, that is, from the interaction between researchers and workers in the
same region. To sum up, the location of economic activities plays an important
role in innovation.

Noteworthy, in Berliant and Fujita, (2012), _joint_ knowledge creation is of
multiplicative nature such that no knowledge is created in isolation. But in
their setting joint knowledge creation occurs between just two persons. In the
present setting, innovation is a process of interaction at a regional scale,
so it is unreasonable to assume that the firm’s chance of innovation in region
$i$ is zero when all industry and scientists are agglomerated in a single
region $i$, i.e $(z_{i},z_{j})=(1,0)$,111Which would happen with a more
“common” specification regarding the regional interaction component, such as
e.g. $f(z_{i},z_{j})=z_{i}^{b}z_{j}^{1-b}.$ This case is briefly discussed in
Section 5. because intra-regional interaction alone among scientists is bound
to produce some knowledge. Hence, the additive case seems more reasonable.

In what regards the linearity of (11), its purpose is twofold. First,
simplicity, as linearity makes the model more analytically tractable compared
to other specifications. Second, it is enough to produce interesting new
insights regarding the effect of knowledge interaction between regions on the
spatial distribution of economic activities. Moreover, the fact that
scientists from different regions are substitutes from a firm’s innovative
perspective fits well with the assumption that the aggregate knowledge level
$A_{i}=A$ is a “public good between regions”.222It would however be
interesting to consider a more general functional form of regional
interaction, such as e.g.
$f(z_{i},z_{j})=\left[bz_{i}^{\varphi}+(1-b)z_{j}^{\varphi}\right]^{\tfrac{1}{\varphi}},$
with $\varphi\geq 0.$

It is also reasonable to assume that the firm’s research success is greater
the higher the level of aggregate knowledge in a region $A_{i}=A$, available
to all firms alike. Finally, we assume that the innovation rate is decreasing
in the complexity of each product, as measured by its quality level $a_{i}(s)$
(Li,, 2003; Dinopoulos and Segerstrom,, 2010).

When a manufacturing firm in region $i$ that produces variety $s$ innovates,
it has a probability $\Phi_{i}(s)$ of reaching the leading quality grade $k$.
A firm that successfully reaches quality of grade $k$ becomes the quality
leader and charges the monopolistic competitive price, collecting profits
$\tilde{\pi}_{i}\equiv\pi_{i}(s_{k}).$ Lower quality products are considered
obsolete. Firms who are unable to attain the leading quality grade face
_creative destruction_ and are priced out of the market.

###  2.4 Short-run equilibrium

Given the research intensity $\alpha$, each firm faces the following expected
profit:

$E\left[\pi_{i}(s)\right]=\Phi_{i}(s)\tilde{\pi}_{i}-\left[1-\Phi_{i}\left(s\right)\right]0-\alpha
w_{i},$

where $w_{i}$ is the nominal wage paid to scientists in region $i.$

Labour market clearing implies that the number of varieties (and hence firms)
is given by $S=z_{i}/\alpha$, from where the price index in (10) becomes:

$P_{i}=\dfrac{\beta\sigma}{\sigma-1}\left(\frac{A}{\alpha}\sum_{j=1}^{2}\phi_{ij}z_{j}\right)^{\tfrac{1}{1-\sigma}}.$
(12)

Given free-entry in the R&D sector, in equilibrium expected profits are driven
down to zero, which yields the following condition:

$w_{i}=\frac{\Phi_{i}(s)\tilde{\pi}_{i}}{\alpha}.$

Using (3), (8), this becomes:

$w_{i}=\dfrac{\mu
a_{i}}{\alpha\sigma}\Phi_{i}\sum_{j=1}^{2}\left(\dfrac{p_{ij}}{P_{i}}\right)^{1-\sigma}\left(\frac{\lambda}{2}+z_{j}\right).$
(13)

Replacing (8) and (12) in (13) yields:

$w_{i}=\frac{\mu\gamma}{\sigma}\left[bz_{i}+(1-b)z_{j}\right]\left(\dfrac{\frac{\lambda}{2}+z_{i}}{z_{i}+\phi
z_{j}}+\phi\dfrac{\frac{\lambda}{2}+z_{j}}{\phi z_{i}+z_{j}}\right).$ (14)

Finally, using (14) and (5), we get the indirect utility in region $i$:

$v_{i}=\frac{\mu\gamma}{\sigma}\left[bz_{i}+(1-b)z_{j}\right]\left(\dfrac{\frac{\lambda}{2}+z_{i}}{z_{i}+\phi
z_{j}}+\phi\dfrac{\frac{\lambda}{2}+z_{j}}{\phi
z_{i}+z_{j}}\right)+\dfrac{\mu}{\sigma-1}\ln\left[z_{i}+\phi
z_{j}\right]+\eta,$ (15)

where
$\eta\equiv-\mu\left(\frac{\beta\sigma}{\sigma-1}\right)+\frac{\mu}{\sigma-1}\ln\left(\frac{A}{\alpha}\right)-\mu+\bar{B}$
is a constant.

##  3 Long-run equilibria

Scientists are free to migrate between regions. In doing so, they choose the
region that offers them the highest indirect utility. The long-run spatial
distribution thus depends on the utility differential:

$\Delta v(z)=v_{1}(z)-v_{2}(z).$ (16)

We follow Castro et al., (2021) in the characterization of equilibria and
their stability. There are two kinds of long-run equilibria which should be
dealt with separately.

  1. 1.

_Agglomeration_ of all scientists in a single region
$z^{*}=\left\\{0,1\right\\}$ is an equilibrium if and only if $\Delta v(1)\geq
0,$ or, equivalently, $\Delta v(0)\leq 0$.

  2. 2.

_Dispersion_ of scientists $z^{*}\in\left(0,1\right)$ is an equilibrium if and
only if $\Delta v(z^{*})=0$. If $z^{*}=\frac{1}{2}$ it corresponds to
_symmetric dispersion._ Otherwise, it is called _asymmetric_.

Equilibria are stable if, after a perturbation such that $z=z^{*}\pm\epsilon$,
with $\epsilon>0$ small enough, the utility differential $\Delta v(z)$ becomes
such that agents go back to their place of origin, i.e., $z=z^{*}$. A
sufficient condition for stability of agglomeration is $\Delta v(1)>0$ (or
$\Delta v(0)<0$). A sufficient condition for stability of dispersion is that
$\Delta v^{\prime}(z^{*})<0$.

###  3.1 Existence and multiplicity

Our first result regards the multiplicity of long-run equilibria. Given
symmetry across regions, we focus on the case whereby region $1$ is either the
same size or is larger than region $2$, i.e.,
$z\in\left[\frac{1}{2},1\right]$.

Symmetric dispersion $z^{*}=\frac{1}{2}$ is called an _invariant pattern_ ,
because it is a long-run equilibrium for the entire parameter range (Aizawa et
al.,, 2020). Next, we have the following result regarding possible equilibria
for $z\in\left(\frac{1}{2},1\right].$

######  Proposition 1.

There are at most two equilibria for $z\in\left(\frac{1}{2},1\right].$

###### Proof.

See Appendix A. ∎

We can be more precise regarding the existence of dispersion equilibria with
the following result.

######  Proposition 2.

A dispersion equilibrium $z\equiv z^{*}\in\left(\frac{1}{2},1\right]$ exists
if $b\in\left(\max\left\\{0,\tilde{b}\right\\},\hat{b}\right)$, where:

$\tilde{b}\equiv\frac{\gamma(\sigma-1)(2z-1)\left[(z-1)z\left(\phi^{2}-1\right)+\phi^{2}\right]+\sigma\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi)}\right]}{\gamma(\sigma-1)(2z-1)(\phi+1)\left[2(z-1)z(\phi-1)+\phi\right]},$

and

$\hat{b}\equiv\frac{1+\phi^{2}}{(1+\phi)^{2}}.$

###### Proof.

See Appendix A. ∎

(a) Stable symmetric dispersion: $\phi=0.1$.

(b) Stable asymmetric dispersion: $\phi=0.3$.

(c) Stable agglomeration: $\phi=0.38$.

(d) Stable asymmetric dispersion: $\phi=0.4$.

(e) Stable symmetric dispersion: $\phi=0.8$.

Figure 1: Long-run equilibria and their stability as $\phi$ increases.

Figure 10 shows one scenario with five qualitatively different cases, with
varying freeness of trade $\phi$, for $b\in[0,\frac{1}{2})$, regarding
existence of the model’s long-run spatial distribution, which exhaust all
mathematical possibilities for the parameter values
$\left(\lambda,\gamma,\phi,\sigma,b\right)=\left(2,0.9,0.4,8,0.339\right)$. As
a prelude to the forthcoming Section, Figure 10 also numerically depicts the
local stability of each equilibrium, which is to be analysed analytically in
greater detail in Section 3.2.

In Figure 1(a), only symmetric dispersion exists and is stable for a very
small $\phi$. For a higher trade freeness we have one stable asymmetric
dispersion for $z\in\left(\frac{1}{2},1\right]$ as portrayed in Figure 1(b).
For a greater $\phi$, Figure 1(c) shows that the asymmetric dispersion
equilibrium disappears and symmetric dispersion becomes unstable, whereas
agglomeration becomes stable. For an even greater $\phi$, Figure 1(d)
illustrates an example of two long-run dispersion equilibria $z^{*}$ for
$z\in\left(\frac{1}{2},1\right]$ and symmetric dispersion $z=\frac{1}{2}$,
whereby we can observe that both symmetric dispersion and the more
agglomerated dispersion equilibrium are locally stable, whereas the less
agglomerated equilibrium is unstable. The economy re-disperses and
agglomeration does not exist in this particular case. However, as the trade
freeness increases further, symmetric dispersion remains stable and all other
equilibria disappear. In other words, when $b\in[0,\frac{1}{2})$, the model
accounts for a bell-shaped relationship between economic integration and
spatial development, whereby firms are initially dispersed, then start to
agglomerate in a single region as the trade freeness increases, but then find
it worthwhile to relocate to the peripheral regions in order to benefit from
higher expected profits due to the sizeable pool of scientists in the core
which increases the chance of innovation in the periphery.

The case $b\in(\frac{1}{2},1]$ is much less diversified and can be accounted
for resorting to a subset of the pictures from Figure 10. The history as
economic integration increases is as follows. For a very low trade freeness,
symmetric dispersion is the only stable equilibrium as in Figure 1(a). For an
intermediate value of $\phi$, one asymmetric dispersion equilibrium arises
which is the only stable one and becomes more asymmetric as $\phi$ increases
further. This is akin to the picture in Figure 1(b). Finally, the asymmetric
dispersion equilibrium gives rise to stable full agglomeration in one single
region once $\phi$ becomes very high. This is illustrated in Figure 1(c).

In other words, when intra-regional interaction is relatively more important
($b>\frac{1}{2}$), the model behaves just like the baseline Pflüger, (2004)
model.

In the forthcoming Sections, we will analytically and numerically study in
greater detail the local stability of the spatial distributions and the
qualitative change in the model’s structure as economic integration increases.

###  3.2 Stability

####  3.2.1 Agglomeration

Regarding agglomeration, using (15) in (16), we have that it is stable if:

$\mathcal{S}\equiv\frac{\gamma\left[(b-1)(\lambda+2)\phi^{2}+2b(\lambda+1)\phi+(b-1)\lambda\right]}{2\sigma\phi}-\frac{\ln\phi}{\sigma-1}>0.$

The second term is positive. Hence, agglomeration is always stable if the
first term is also positive:

$b>b_{s}\equiv\frac{(\lambda+2)\phi^{2}+\lambda}{(\phi+1)\left[(\lambda+2)\phi+\lambda\right]}.$

It is easy to check that $b_{s}<\tfrac{1}{2}$ if
$\phi\in\left(\frac{\lambda}{\lambda+2},1\right)$, which means that, if
$\phi\in\left(\frac{\lambda}{\lambda+2},1\right)$ and $b>\frac{1}{2}$,
agglomeration is unstable. In any case, we can conclude that agglomeration is
unstable if related variety is too strong.

Let us now define as _sustain point,_ a value of $\phi$ such that
$\mathcal{S}(\phi)=0.$ We have the following result relating the freeness of
trade and the relatedness of variety.

######  Proposition 3.

If $b<\tfrac{1}{2}$, there exist two sustain points, $\phi_{s1}$ and
$\phi_{s2},$ and agglomeration is unstable for
$\phi\in\left\\{\left(0,\phi_{1s}\right)\cup\left(\phi_{2s},1\right)\right\\}$
and stable for $\phi\in(\phi_{1s},\phi_{2s})$. If $b>\frac{1}{2}$, there
exists a unique sustain point $\phi_{s1}$ and agglomeration is unstable for
$\phi\in(0,\phi_{1s})$ and stable if $\phi\in\left(\phi_{1s},1\right)$.

###### Proof.

See Appendix A. ∎

The result in Proposition 3 suggests that an intermediate level of economic
integration favours agglomeration only if the interaction with foreign
scientists is relatively more important for the chance of successful
innovation. By contrast, if the within region interaction of scientists is
more important, agglomeration is only possible when the freeness of trade is
high enough.

####  3.2.2 Symmetric dispersion

Regarding symmetric dispersion $z^{*}=\frac{1}{2}$, using (15) in (16) we can
say that it is stable if:

$\mathcal{B}\equiv\gamma(\sigma-1)\left[2b(\lambda+1)(\phi+1)^{2}-(2\lambda+3)\phi^{2}-2\lambda-1\right]+2\sigma\left(1-\phi^{2}\right)<0.$
(17)

In fact, it is always unstable if the first term is positive, i.e. if:

$b>\bar{b}\equiv\frac{(2\lambda+3)\phi^{2}+2\lambda+1}{2(\lambda+1)(\phi+1)^{2}}.$

This means that if related variety is prohibitively high, symmetric dispersion
is surely unstable.

We can observe that $\mathcal{B}$ in (17) is a second degree polynomial in
$\phi$ with at most two zeros, i.e.,_break points_ $\phi_{b1}$ and
$\phi_{b2}$, with $\phi_{b1}<\phi_{b2}$ and has a negative leading
coefficient. Therefore if both break points exist, we have that symmetric
dispersion is stable for $\phi<\phi_{b1}$, unstable for
$\phi\in\left(\phi_{b1},\phi_{b2}\right)$, and stable again for
$\phi>\phi_{b2}$. This means that our model accounts for the possibility of
initial agglomeration as trade integration increases from a very low level and
re-dispersion for very high levels of trade integration.

More specifically, the breakpoints are given by:

$\displaystyle\phi_{b1}$
$\displaystyle=\frac{\sqrt{\gamma^{2}(\sigma-1)^{2}\left[8b(\lambda+1)^{2}-4\lambda^{2}-8\lambda-3\right]+4\gamma\sigma(\sigma-1)+4\sigma^{2}}-2b\gamma(\lambda+1)(\sigma-1)}{\gamma(\sigma-1)\left[2b(\lambda+1)-2\lambda-3\right]-2\sigma}$
$\displaystyle\phi_{b2}$
$\displaystyle=-\frac{\sqrt{\gamma^{2}(\sigma-1)^{2}\left[8b(\lambda+1)^{2}-4\lambda^{2}-8\lambda-3\right]+4\gamma\sigma(\sigma-1)+4\sigma^{2}}+2b\gamma(\lambda+1)(\sigma-1)}{\gamma(\sigma-1)\left[2b(\lambda+1)-2\lambda-3\right]-2\sigma}.$
(18)

The break point $\phi_{b1}$ lies in the interval $(0,1)$ if and only if:

  1. (i).

$\gamma\in\left(\dfrac{2\sigma}{(2\lambda+1)(\sigma-1)},1\right)$,

  2. (ii).

$b\in\left[b_{1},b_{2}\right),$

where:

$\displaystyle b_{1}$
$\displaystyle=\frac{\left[\gamma(2\lambda+1)(\sigma-1)-2\sigma\right]\left[\gamma(2\lambda+3)(\sigma-1)+2\sigma\right]}{8\gamma^{2}(\lambda+1)^{2}(\sigma-1)^{2}},$
$\displaystyle b_{2}$
$\displaystyle=\frac{\gamma(2\lambda+1)(\sigma-1)-2\sigma}{2\gamma(\lambda+1)(\sigma-1)}.$

We have that $b_{1}\in\left(0,\frac{1}{2}\right)$ and
$b_{2}\in\left(b_{1},1\right)$. If, additionally, $b<\frac{1}{2}$, then we
have also that $\phi_{b2}\in\left(0,1\right)$.333If
$\gamma<\frac{2\sigma}{\lambda(\sigma-1)}$, then
$b_{2}\in\left(b_{1},\frac{1}{2}\right)$ and the condition is trivially met by
(ii). In this case, both break points exist. This means that the possibility
of re-dispersion following agglomeration as $\phi$ increases requires that
related variety is neither too high or too low.

However, if $b>\frac{1}{2}$, then $\phi_{b2}$ does not exist and we have a
single break point $\phi_{b1}$ if conditions (i) and (ii) are satisfied.

####  3.2.3 Asymmetric dispersion

Although we cannot find an explicit stability condition for any asymmetric
dispersion equilibrium $z^{*}\in\left(\frac{1}{2},1\right)$, we can use
equation (23) that solves the equilibrium condition $\Delta v=0$ given by
implicitly by $\lambda=\lambda^{*}(z)$ in the proof of Proposition 2 (Appendix
A.2). Then the stability condition of an asymmetric dispersion equilibrium is
given by:

$\left.\frac{d\Delta v}{dz}(z^{*})\right|_{\lambda=\lambda^{*}(z)}<0.$

Specifically, using (15) and differentiating (16) with respect to $z$, and
evaluating at (23), we get that an assymetric equilibrium
$z^{*}\in\left(\frac{1}{2},1\right)$ is stable if $\lambda^{*}(z)>0$ and:

$\displaystyle\mathcal{G}\equiv$
$\displaystyle(2z-1)\left(\phi^{2}-1\right)\left[(2b-1)\gamma(\sigma-1)(1-2z)^{2}-\sigma\right]+$
$\displaystyle+\sigma\left[2z^{2}(\phi-1)^{2}-2z(\phi-1)^{2}+\phi^{2}+1\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right]<0.$
(19)

We have the following result.

######  Proposition 4.

If $b<\frac{1}{2}$ an asymmetric equilibrium
$z^{*}\in\left(\frac{1}{2},1\right)$ is stable for a high enough related
variety. If $b>\frac{1}{2}$, an asymmetric equilibrium is always stable when
it exists.

###### Proof.

See Appendix A. ∎

Figure 2: Stability of asymmetric dispersion. The less opaque surface
corresponds to $\mathcal{G}<0$ and $\lambda^{*}(z)>0$ in $(z,\phi,b)$–space
for $\sigma=8$ and $\gamma=0.9$. The black plane corresponds to
$b=\frac{1}{2}.$

Figure 2 illustrates the Proposition by setting $\sigma=8$ and $\gamma=0.9$
and plotting the surface corresponding to
$\left\\{(z,\phi,b):\mathcal{G}<0\cap\lambda^{*}(z)>0\right\\}$. For
$b<\frac{1}{2}$, an asymmetric equilibrium may exist that is not stable, and a
higher $b$ favours its stability. If $b>\frac{1}{2}$, an asymmetric
equilibrium is always stable when it exists, but its existence seems to be
favoured by a lower $b$. In other words, an asymmetric equilibrium exists and
is stable when $b$ is close enough to $\frac{1}{2}$.

Regarding $\phi$, a higher freeness of trade $\phi$ seems to disfavour the
stability of asymmetric dispersion.

##  4 The impact of economic integration

It is common in geographical economics to study the qualitative change of the
spatial economy as economic integration increases. We will now look at some
bifurcation diagrams using the freeness of trade, $\phi$, as the bifurcation
parameter. To provide a complete gallery, we depict 6 qualitatively different
scenarios, keeping most parameter values constant (except for the sixth
scenario) and varying $b$, thus placing emphasis on changes in the value of
the weight of within-region interaction. Two additional illustrations for
different parameter values are provided in Appendix B. The illustrations, 8 in
total, exhaust all mathematical/numerical possibilities.444This can be shown
through the combination of the analysis performed in the previous sections
with various simulations under a very wide range of parameter values. The six
scenarios analysed in this Section are as follows:

  1. (i).

$\left(\lambda,\gamma,\sigma,b\right)=\left(2,0.9,8,0.33\right)$;

  2. (ii).

$\left(\lambda,\gamma,\sigma,b\right)=\left(2,0.9,8,0.338\right)$;

  3. (iii).

$\left(\lambda,\gamma,\sigma,b\right)=\left(2,0.9,8,0.339\right)$;

  4. (iv).

$\left(\lambda,\gamma,\sigma,b\right)=\left(2,0.9,8,0.35\right)$;

  5. (v).

$\left(\lambda,\gamma,\sigma,b\right)=\left(2,0.9,8,0.55\right)$;

  6. (vi).

$\left(\lambda,\gamma,\sigma,b\right)=\left(4,0.9,8,0.55\right)$;

Figure 3: Bifurcation diagram for scenario (i). Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 4: Bifurcation diagram for scenario (ii). Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 5: Bifurcation diagram for scenario (iii). Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 6: Bifurcation diagram for scenario (iv). Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 7: Bifurcation diagram for scenario (v). Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 8: Bifurcation diagram for $(\lambda,\gamma,\sigma,b)=(4,0.9,8,0.55)$.
Filled lines correspond to stable equilbria and dashed lines correspond to
unstable equilibria.

For a prohibitively low related variety, scientists disperse evenly among the
two regions, irrespective of the value of the freeness of trade. The economic
intuition is simple: a lower $b$ implies higher chance of successful
innovation with more scientists living in the other region. Hence, the nominal
wage is higher when the scientists are more evenly distributed.

In scenario (i), shown in Figure 3, related variety is such that within-region
interaction is relatively more important ($b=0.33$) but is still low. For a
low freeness of trade, symmetric dispersion is stable because firms wish to
avoid the burden of a very costly transportation supplying to farmers from
full agglomeration in a single region. As $\phi$ increases, the economy
initially agglomerates, but then re-disperses as $\phi$ increases further.
This re-dispersion process occurs because, for a very high economic
integration, firms find it profitable to relocate to the less industrialized
region in order to benefit from the pool of scientists in the more
agglomerated region, which generates a higher chance of innovation and thus
higher expected profits. Noteworthy, the turning point in the agglomeration
process happens before industry reaches full agglomeration in a single region.

In scenario (ii), illustrated by Figure 4, related variety is just slightly
higher, and the model still accommodates for re-dispersion. However, the re-
dispersion process is not smooth – the economy suddenly jumps to symmetric
dispersion from a fairly asymmetric equilibrium spatial distribution.

Scenario (iii) also just slightly increases related variety compared to the
previous scenario (see Figure 5), and the story as economic integration
increases is very similar, except that in this case, full agglomeration is
stable for a small range of intermediate values of $\phi$, as predicted by
Proposition 3. The parametrization here also corresponds to that illustrated
in Figure 10.

Figure 6 illustrates scenario (iv) and shows that the sudden re-dispersion
process under a slightly higher $b$ now happens from the state of full
agglomeration directly to the state of symmetric dispersion.

Finally, in scenario (v), for a sufficiently high $b>1/2$, within-region
interaction among scientists improves the chances of innovation enough such
that the real wage becomes higher when they are either partially agglomerated
in one region for low values of $\phi$, or completely agglomerated in one
region for a high enough $\phi$. This is portrayed in Figure 7.

We can thus conclude that a higher related-variety is associated with a more
pronounced agglomeration during the industrialization process, for
intermediate values of economic integration, until eventually it becomes so
high that re-dispersion is no longer possible because within-region
interaction among scientists is too important to make any deviation to a
deindustrialized region worthwhile.

In scenario (vi) we illustrate the qualitative change in the spatial structure
of the economy as $\phi$ increases for $b>1/2$, but with a higher $\lambda$,
since, with the parameter values of the previous scenario, agglomeration would
be ubiquitously stable (and hence uninteresting) for higher values of $b$. In
Figure 8, we can observe the typical supercritical pitchfork that we can
observe in the original Pflüger, (2004) model. That is, for low levels of
economic integration, symmetric dispersion is stable. As $\phi$ increases, one
region smoothly becomes more and more industrialized en route to a full
agglomeration whereby that region becomes a core.

Noteworthy, in any scenario for which a break point exists at symmetric
dispersion, it can be shown analytically that the model undergoes a pitchfork
bifurcation, either supercritical or subcritical, depending on the parameter
values).555The conditions for a pitchfork bifurcation in Guckenheimer and
Holmes, (2002) can be shown to hold up to the third derivative of $\Delta
v(z)$ with respect to $z$, whose sign is very difficult to determine
analytically. Additionally, in Figures 4 and 4 (scenarios (ii) and (iii)), a
limit point $\phi\equiv\phi_{l}\in(\phi_{b2},1)$ is discernible at which two
asymmetric equilibria, along a curve tangent to $\phi_{l}$ that lies to its
left, collide and coalesce. This suggests that in both scenarios (i) and (ii)
the model undergoes a saddle-node bifurcation at some asymmetric equilibrium
$z^{*}\in\left(\frac{1}{2},1\right)$. This kind of bifurcation also appears in
the two-region footloose entrepreneur model by Forslid and Ottaviano, (2003)
with heterogeneous agents analysed by Castro et al., (2021) and also in the
Pflüger, (2004) model extended to multiple regions by Gaspar et al., (2018).
This kind of bifurcation seems to be associated with discontinuous jumps
between some asymmetric equilibrium other than agglomeration and the symmetric
dispersion once $\phi$ rises (falls) above (below) some threshold level.

In Appendix B we change the benchmark parameter values and further illustrate
how changes in $b$ can bring about richer implications regarding the
qualitative structure of the spatial economy.

##  5 On the role of regional interaction

It is worthwhile investigating how different types of regional interaction
determining the success of innovation affect the spatial outcomes in the
economy. The process of (de)industrialization as economic integration
increases can be shown to vary quite a lot under different functional forms
for the probability of a successful innovation.

###  5.1 A simple scenario

We first consider a slight modification in the success of innovation that
deems the model even more simple, but leads to very different results
regarding the relationship between regional interaction in the production of
knowledge and the long-run equilibrium spatial outcomes.

Suppose now that a firm producing variety $s$ at region $i$ reaches a leading
quality grade $k$ with probability:

$\Phi_{i}^{S}(s)=\min\left\\{\dfrac{z_{i}+cz_{j}}{a_{i}(s)}\gamma
A,1\right\\},$ (20)

where $c\in(0,1)$ now represents the _weight of inter-regional interaction_ in
the chance of successful innovation. In other words, it is a measure of
_unrelated_ variety. We assume a parametrization that guarantees that the
first term lies in the interval $[0,1]$.

This specification yields at most one asymmetric equilibrium
$z^{*}\in\left(\frac{1}{2},1\right]$ (see Appendix C.1), besides the symmetric
dispersion which remains an invariant pattern. Agglomeration exists and is
stable for a high enough freeness of trade (sustain point $\phi_{s}^{S}$ and
symmetric dispersion is stable for a low enough freeness of trade (break point
$\phi_{b1}^{S}$ (see Appendix C.2). Moreover, it is possible to show that the
model undergoes a supercritical pitchfork bifurcation at the symmetric
dispersion (C.3), just as in Pflüger, (2004).

Consider an increase in the parameter $c$. We have:

$\frac{\partial\Delta v^{S}}{\partial
c}=\frac{\gamma\mu(2z-1)\left[\lambda-2z^{2}+\phi^{2}(\lambda+2(z-1)z+2)+2z\right]}{2\sigma\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]},$

which is negative for $z\in\left(\frac{1}{2},1\right)$. In other words, a
higher $c$ decreases the utility differential. Castro et al., (2021) have
recently demonstrated that an exogenous shock in any parameter that leads to a
higher utility differential favours agglomeration and discourages dispersion.
We have the following result.666The result does not apply to our benchmark
case because it does not comply with the assumptions needed, which are stated
in Castro et al., (2021, pp. 197).

######  Lemma 1.

Under $\Phi_{i}^{S}(s)$, an increase in the weight of inter-regional
interaction in the chance of innovation (i.e. higher $c$) does not favour
agglomeration.

###### Proof.

See Proposition 9 of (Castro et al.,, 2021, p.197). ∎

In other words, a higher $c$ favours symmetric dispersion and discourages
agglomeration.777Note that the sufficient condition provided by the Lemma is
sufficient but not necessary. It is possible, yet much more cumbersome, to
demonstrate that $\frac{\partial S^{S}}{\partial c}<0$,
$\frac{\partial\mathcal{B}^{S}}{\partial c}<0$ and
$\frac{\partial\mathcal{G}^{S}}{\partial c}<0$ by using the corresponding
expressions in Appendix C.2. As illustrated in Figure 9, a higher $c$ shifts
the pitchfork bifurcation rightwards.

Figure 9: Bifurcation diagrams: (i) to the left, in black, we set $c=0.7$;
(ii) to the right, in red, we set $c=0.9$. Parameter values are
$(\sigma,\gamma,\lambda)=(8,1,4).$

The interpretation behind these results is very straightforward: a higher $c$
implies higher chance of successful innovation with more scientists living in
the other region. Hence, the nominal wage is higher when the scientists are
more evenly distributed. So, a higher $c$ makes stability of symmetric
dispersion (agglomeration) more (less) likely for a higher range of $\phi$,
and asymmetric dispersion becomes stable for a higher freeness of trade.

In sum, under this second specification with $\Phi_{i}^{S}(s)$ in (20),
agglomeration is a smooth and progressive progress as the trade freeness
increases, and the unrelatedness of variety measured by $c$ adds an additional
dispersion force to the benchmark model. The most striking contrast with the
first scenario is the absence of a “bubble” shaped relation between economic
integration and spatial imbalances in the second scenario.

###  5.2 Multiplicative case: a numerical analysis

Several other functional forms for $\Phi_{i}(s)$ could be worth checking. In
fact, it is worth considering the case where scientists are imperfect
substitutes from a firm’s innovative perspective. For instance, a Cobb-Douglas
specification such as:

$\Phi_{i}^{CD}(s)=\min\left\\{\dfrac{z_{i}^{b}z_{j}^{1-b}}{a_{i}(s)}\gamma
A,1\right\\},$ (21)

yields a multiplicative scenario, and thus brings our setup closer to Berliant
and Fujita, (2008). Again we shall impose a parametrization that guarantees
that the first term lies in the interval $[0,1]$.

Regarding agglomeration, it _cannot_ be an equilibrium because $\Delta
v(1)=\frac{\mu\ln\phi}{\sigma-1}<0$. Since no innovation occurs, expected
profits are driven down to zero and the only thing that matters is the cost-
of-living, which is positive in the fully agglomerated region. However, if an
agent moves to the “empty” region, innovation occurs through inter-regional
interaction and he will earn a positive nominal wage, which means that
exogenous perturbations always increase the utility of agents. While this may
strike as an implausible outcome, one way to counter this would be to specify
a different probability for corner equilibria, potentially following Berliant
and Fujita, (2008).

The symmetric dispersion can be shown to retain the qualitative properties of
the benchmark case analyzed in Section 3.2.2.888This has been checked
analytically although the proofs are not presented in this paper, for the sake
of space. The formal proofs are available from the authors upon request. That
is, there exist two break points $\phi_{b1}$ and $\phi_{b2}$ given by (18) and
symmetric dispersion is stable for
$\phi\in\left\\{(0,\phi_{b1})\cup(\phi_{b2},1)\right\\}$, and unstable for
$\phi\in\left(\phi_{b1},\phi_{b2}\right)$.

We can grasp the general qualitative behaviour and properties of the model
under $\Phi_{i}^{CD}(s)$ by depicting a gallery of bifurcation diagrams for
several different values of $b$.The benchmark parameter values are
$(\sigma,\gamma,\lambda)=(8,1,4)$, while the value for $b$ is reported in the
caption of each picture.

(a) Stable symmetric dispersion: $b=0.1$.

(b) Stable symmetric dispersion: $b=0.44$.

(c) Stable symmetric dispersion for $\phi<\phi_{b1}$ and $\phi<\phi_{b2}$:
$b=0.45$.

(d) Stable symmetric dispersion for $\phi<\phi_{b1}$: $b=0.5$.

(e) Stable symmetric dispersion for $\phi<\phi_{b1}$ and stable asymmetric
dispersion for $\phi>\phi_{b1}$: $b=0.65$.

(f)
No stable equilibria: $b=0.75$

.

Figure 10:  Bifurcation diagram for $(\lambda,\gamma,\sigma,b)=(4,1,8)$.
Filled lines correspond to stable equilibria and dashed lines correspond to
unstable equilibria.

As $b$ increases from a very small value, initially only symmetric dispersion
exists and is stable (Figure 10(a)). Eventually there emerge two curves of
unstable asymmetric dispersion equilibria (Figure 10(b)), one with a minimum
and the other with a maximum (for the same $\phi$ due to symmetry. For higher
levels of $b$, the extrema collide vertically at symmetric dispersion and two
curves of asymmetric dispersion equilibria branch from two break points.
Symmetric dispersion is stable below the lowest break point and above the
highest break point, and unstable in between (Figure 10(c)). Further increases
in $b$ will make the highest breakpoint disappear and we end up with a
subcritical pitchfork bifurcation (Figure 10(d)), as in Fujita et al., (1999),
whereby symmetric dispersion is stable for low values of $\phi$ and becomes
unstable for high values of $\phi$. The main difference is that there are no
stable equilibria above the break point. If $b$ increases even more, we end up
with a supercritical pitchfork bifurcation whereby a stable symmetric
dispersion loses stability for $\phi$ large enough (Figure 10(e)). This state
encounters a primary branch of stable asymmetric equilibria that, apparently,
undergoes a secondary saddle-node bifurcation.999This qualitative scenario is
very similar to the one encountered by Castro et al., (2021, Fig. 4 in pp.
197) except that the stability of equilibria is “reversed” and the bifurcation
employed by them is a heterogeneity parameter. Finally, for a prohibitively
high value of $b$, there are no stable equilibria in the model, as shown by
Figure 10(f).

This scenario with $\Phi_{i}^{CD}(s)$ as the chance of a successful innovation
departs from the previous two cases in two major aspects. First, it reverses
the predictions about the spatial distribution of economic activities with
respect to increases in $\phi$ (except for the case in Figure 10(c) where re-
dispersion is possible). While this may be hard to explain from an economic
point of view, one may conjecture that this may be due to the multiplicative
nature of the probability of innovation in this scenario that contrasts with
the additive nature in the first benchmark case and in the simpler case of
$\Phi_{i}^{S}(s)$. Second, the last scenario seems to be poorer in terms of
predictions since, e.g., the only stable asymmetric equilibria exist for a
very small range and values of $\phi$.

##  6 Discussion and concluding remarks

In this paper we have analysed a two-region model with vertical innovations
that enhance the quality of varieties of the horizontally differentiated
manufactures produced in each of the two regions. We looked at how the
creation and diffusion of knowledge and increasing returns in manufacturing
interact to shape the spatial economy. Innovations occur with a probability
that depends on the inter-regional interaction between researchers (mobile
workers).

We find that, if the weight of interaction with foreign scientists is
relatively more important for the success of innovation, the model accounts
for re-dispersion of economic activities after an initial stage of progressive
agglomeration as transport costs decrease from a high level. However, the
relationship between economic integration and spatial imbalances is far from
trivial, as we have shown a myriad of different qualitative possibilities that
depend on the weight of inter-regional interaction between scientists.

Our main results so far have been shown to hinge on the functional form for
the probability of innovating. We have studied two simple different additive
cases and provided insights on a multiplicative scenario. However, it would be
worthwhile further investigating how different types of regional interaction
determining the success of innovation affect the spatial outcomes in the
economy. The process of (de)industrialization as economic integration
increases can be shown to vary quite a lot under different functional forms
for the probability of a successful innovation. This becomes noteworthy when
we take the simplest case of knowledge creation depending on scientists across
different regions, and even more so when knowledge creation is of
multiplicative nature as in e.g. Berliant and Fujita, (2012).

##  Appendix A Proofs

This appendix contains the more cumbersome formal proofs that support our main
results.

###  A.1 Proof of proposition 1

Differentiating $\Delta v(z)$ in (16) yields:

$\frac{d\Delta v}{dz}(z)=\frac{\mu
P(z)}{2(\sigma-1)\sigma\left[z(\phi-1)+1\right]{}^{2}\left[z(1-\phi)+\phi\right]{}^{2}},$
(22)

where:

$P(z)=a_{1}z^{4}+a_{2}bz^{3}-2(1-\phi)a_{3}z^{2}+2(1-\phi)a_{4}z+a_{5},$

with:

$\displaystyle a_{1}=$ $\displaystyle
4(1-2b)\gamma\mu(\sigma-1)(\phi-1)^{3}(\phi+1)$ $\displaystyle a_{2}=$
$\displaystyle 8(2b-1)\gamma\mu(\sigma-1)(\phi-1)^{3}(\phi+1)$ $\displaystyle
a_{3}=$
$\displaystyle\gamma(\sigma-1)\left\\{b(\phi+1)\left[(\lambda-2)\phi^{2}-\lambda+18\phi-4\right]-\phi\left[\lambda(\phi-1)\phi+\lambda+6\phi\right]+\lambda-8\phi+2\right\\}$
$\displaystyle+\sigma(\phi+1)(\phi-1)^{2}$ $\displaystyle a_{4}=$
$\displaystyle\gamma(\sigma-1)\left\\{\lambda(\phi-1)\left[b(\phi+1)^{2}-\phi^{2}-1\right]+2\phi\left[b(\phi+1)(\phi+5)-\phi(\phi+2)-3\right]\right\\}$
$\displaystyle+\sigma(\phi+1)(\phi-1)^{2}$ $\displaystyle a_{5}=$
$\displaystyle\gamma(\sigma-1)\left\\{\lambda\left(\phi^{2}+1\right)\left[b(\phi+1)^{2}-\phi^{2}-1\right]+2\phi\left[b\left(\phi^{3}+3\phi^{2}+\phi-1\right)-\phi\left(\phi^{2}+\phi+1\right)+1\right]\right\\}$
$\displaystyle-2\sigma\phi\left(\phi^{2}-1\right).$

The denominator of (22) is positive, which means that the sign of
$\frac{d\Delta v}{dz}(z)$ is given by the sign of $P(z)$, which is a fourth
degree polynomial in $z$. Therefore, $\Delta v(z)$ has at most four turning
points and, thus, at most five equilibria for $z\in[0,1]$. We know that
$z=\frac{1}{2}$ is an invariant pattern. By symmetry, we can establish that
there exist at most two equilibria for $z>\tfrac{1}{2}$, which concludes the
proof. $\square$

###  A.2 Proof of Proposition 2

Proceeding in a familiar fashion as in Gaspar et al. (2018, 2021), the
equilibrium condition $\Delta v(z)=0$ yields:

$\lambda\equiv\lambda^{*}(z)=-2\frac{b_{1}b_{2}+b_{3}\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi)}\right]}{b_{4}},$
(23)

where:

$\displaystyle b_{1}$ $\displaystyle=\gamma(\sigma-1)(2z-1)$ $\displaystyle
b_{2}$
$\displaystyle=\phi^{2}\left[2b(z-1)z+b-z^{2}+z-1\right]+(1-2b)(z-1)z+b\phi$
$\displaystyle b_{3}$
$\displaystyle=\sigma\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]$
$\displaystyle b_{4}$
$\displaystyle=\gamma(\sigma-1)(2z-1)\left[b(\phi+1)^{2}-\phi^{2}-1\right].$

It is easy to note that $\lambda^{*}(z)$ has a vertical asymptote if and only
if $b_{4}=0$, i.e., iff:

$b=\hat{b}\equiv\frac{\phi^{2}+1}{(\phi+1)^{2}}.$

For $z\in\left(\frac{1}{2},1\right],$ the log term of $\lambda^{*}(z)$ is
negative, as is $b_{3}$. Next, we have $b_{1}>0$, and $b_{2}>0$ if:

$b\geq\underline{b}\equiv\frac{(z-1)z\left(\phi^{2}-1\right)+\phi^{2}}{2(z-1)z\left(\phi^{2}-1\right)+\phi(\phi+1)},$

where $0<\underline{b}<\hat{b}$. Since $b_{4}<0$ only if $b<\hat{b}$, we have
that $\lambda^{*}(z)>0$ if $b\in\left[\underline{b},\hat{b}\right)$ and
$\lambda^{*}(z)<0$ if $b\in\left(\hat{b},1\right)$. For
$b\in\left(0,\underline{b}\right)$, we need further inspection.

We have that:

$\frac{\partial\lambda^{*}}{\partial
b}(z)=\frac{2(\phi+1)\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]\left\\{\gamma(\sigma-1)(2z-1)(\phi-1)+\sigma(\phi+1)\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi)}\right]\right\\}}{\gamma(\sigma-1)(2z-1)\left[-b(\phi+1)^{2}+\phi^{2}+1\right]^{2}},$

which is positive for all $z\in\left(\frac{1}{2},1\right].$ The unique zero of
$\lambda^{*}(z)$ in terms of $b$ is given by:

$b=\tilde{b}\equiv\frac{\gamma(\sigma-1)(2z-1)\left[(z-1)z\left(\phi^{2}-1\right)+\phi^{2}\right]-\sigma\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi)}\right]}{\gamma(\sigma-1)(2z-1)(\phi+1)\left[2(z-1)z(\phi-1)+\phi\right]},$

with $\tilde{b}<\underline{b}$ and $\lambda^{*}(z)>0$ for
$b\in(\tilde{b},\hat{b})$. It is possible to show that $\tilde{b}$ is
increasing in $\gamma.$ Moreover, we have $\tilde{b}=0$ if and only if:

$\gamma=\gamma_{c}\equiv\frac{\sigma\left[z(1-\phi)-1\right]\left[z(1-\phi)+\phi\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right]}{(\sigma-1)(2z-1)\left[(z-1)z\left(\phi^{2}-1\right)+\phi^{2}\right]}>0.$

This means that $\tilde{b}\geq 0$ if $\gamma\geq\gamma_{c}$ and $\tilde{b}<0$
if $\gamma<\gamma_{c}$ and $\gamma_{c}\in(0,1]$. Since
$\gamma_{c}\in(0,+\infty)$, we have $\tilde{b}<0$ if $\gamma_{c}>1$. As a
result, we have $\tilde{b}<0$ if
$\gamma\in\left(0,\min\\{1,\gamma_{c}\\}\right)$ and $\tilde{b}\geq 0$ if
$\gamma\in\left[\min\\{1,\gamma_{c}\\},1\right)$. Then $\lambda^{*}(z)>0$ if
$\gamma\in\left(0,\min\\{1,\gamma_{c}\\}\right)$ and $b\in(0,\hat{b})$.
Otherwise, we have $\lambda^{*}(z)>0$ if
$\gamma\in\left[\min\\{1,\gamma_{c}\\},1\right)$ and
$b\in(\tilde{b},\hat{b}).$ Therefore, $\lambda^{*}(z)$ is positive for
$b\in\left(\max\left\\{0,\tilde{b}\right\\},\hat{b}\right)$ and negative for
$b\in\left\\{\left(0,\max\left\\{0,\tilde{b}\right\\}\right)\cup\left(\hat{b},1\right)\right\\}$,
where $\max\left\\{0,\tilde{b}\right\\}$ depends on $\gamma_{c}$ and on the
value of $\gamma$ as described above.

Thus, we can assert that, if
$b\in\left(\max\left\\{0,\tilde{b}\right\\},\hat{b}\right)$, there exists a
value of $\lambda>0$ such that at least one (at most two) dispersion
equilibrium $z\equiv z^{*}\in\left(\frac{1}{2},1\right]$ exists. This
concludes the proof. $\square$

###  A.3 Proof of Proposition 3

We have:

$\lim_{\phi\rightarrow 0^{+}}\mathcal{S}(\phi)=-\infty\text{ and
}\text{$\mathcal{S}(1)=\frac{\gamma(2b-1)(\lambda+1)}{\sigma}.$}$

Therefore, $S(1)>0$ if $b>\tfrac{1}{2}$ and we conclude that
$\mathcal{S}(\phi)$ has at least one zero for $\phi\in\left(0,1\right)$.
Further, we have:

$\dfrac{d\mathcal{S}}{d\phi}(\phi)=\dfrac{1}{2\phi^{2}}\left\\{\frac{\gamma(b-1)\left[\lambda\left(\phi^{2}-1\right)+2\phi^{2}\right]}{\sigma}-\frac{2\phi}{\sigma-1}\right\\},$

whose sign depends on that of the second term, which is a second degree
polynomial and thus has at most two zeros
$\left\\{\phi^{-},\phi^{+}\right\\}$, with $\phi^{+}>\phi^{-}$. However, only
$\phi^{+}$ lies on the interval $\phi\in\left(0,1\right)$:

$\phi^{+}=\frac{\sigma\left[\frac{1}{\sigma-1}-\sqrt{\frac{\gamma^{2}(b-1)^{2}\lambda(\lambda+2)}{\sigma^{2}}+\frac{1}{(\sigma-1)^{2}}}\right]}{\gamma(b-1)(\lambda+2)}.$

Given that the leading coefficient of the polynomial is negative, we have that
$\mathcal{S}(\phi)$ is increasing for $\phi\in\left(0,\phi^{+}\right)$ and
decreasing for $\phi\in\left(\phi^{+},1\right)$. Thus, $\mathcal{S}(\phi)$ has
at most two zeros for $\phi\in\left(0,1\right)$, called sustain points
$\phi_{s1}$ and $\phi_{s2}$ (with $\phi_{s1}<\phi_{s2}$).101010One of which is
given by $\phi=1$ if $b=\frac{1}{2}$. If $b<\tfrac{1}{2}$, there exist two
sustain points $\phi_{s1}\in\left(0,1\right)$ and
$\phi_{s2}\in\left(0,1\right)$ and we have $\mathcal{S}(\phi)<0$ for
$\phi\in\left\\{\left(0,\phi_{1s}\right)\cup\left(\phi_{2s},1\right)\right\\}$
and $\mathcal{S}(\phi)>0$ for $\phi\in(\phi_{1s},\phi_{2s})$. If
$b>\tfrac{1}{2}$, there exists one unique sustain point
$\phi_{s1}\in\left(0,1\right)$ and we have $\mathcal{S}(\phi)<0$ for
$\phi\in\left(0,\phi_{1s}\right)$ and $\mathcal{S}(\phi)>0$ for
$\phi\in\left(\phi_{1s},1\right)$, which concludes the proof. $\square$

###  A.4 Proof of Proposition 4

Taking the derivative of $\mathcal{G}$ in (19) with respect to $b$ we get:

$\frac{\partial\mathcal{G}}{\partial
b}=-2\gamma(\sigma-1)(2z-1)^{3}\left(1-\phi^{2}\right)<0,\ \
z\in\left(\frac{1}{2},1\right)$

Next, solving $\mathcal{G}$ in (19) for $b$ yields:

$b=b_{c}\equiv\frac{(2z-1)\left(1-\phi^{2}\right)\left[\sigma+\gamma(\sigma-1)(1-2z)^{2}\right]-\sigma\left[2z^{2}(\phi-1)^{2}-2z(\phi-1)^{2}+\phi^{2}+1\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right]}{2\gamma(\sigma-1)(2z-1)^{3}\left(1-\phi^{2}\right)}.$
(24)

As a result, we have $\mathcal{G}>0$ for $b<b_{c}$ and $\mathcal{G}<0$ for
$b>b_{c}$. Next, we will prove that $b_{c}<1/2$.

First, notice that $\lim_{\phi\rightarrow 1}b_{c}=\frac{1}{2}.$ Next, we have:

$\dfrac{\partial
b_{c}}{\partial\phi}=\frac{\sigma\mathcal{N}}{2\gamma(\sigma-1)(2z-1)^{3}\left(\phi^{2}-1\right)^{2}\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]},$

where:

$\displaystyle\mathcal{N}=$
$\displaystyle(2z-1)\left(\phi^{2}-1\right)\left[2z^{2}(\phi-1)^{2}-2z(\phi-1)^{2}+\phi^{2}+1\right]-$
$\displaystyle-4\left[z(\phi-1)+1\right]{}^{2}\left[z(1-\phi)+\phi\right]{}^{2}\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right].$

The numerator of the derivative is negative. As for $\mathcal{N}$, observe
that:

$\displaystyle\dfrac{\partial\mathcal{N}}{\partial z}=$
$\displaystyle-4\sigma(2z-1)(\phi-1)^{2}\times$
$\displaystyle\times\left\\{(2z-1)\left(1-\phi^{2}\right)+2\left[z(\phi-1)+1\right]\left[z(\phi-1)-\phi\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right]\right\\}.$

The first term inside the curly brackets is positive and the second one is
negative as is the log term. Therefore, we have
$\frac{\partial\mathcal{N}}{\partial z}<0$. Since
$\mathcal{N}\left(z=\frac{1}{2}\right)=0$ and given that $\mathcal{N}$ is
continuous in $z$, we can conclude that $\mathcal{N}<0$ for
$z\in\left(\frac{1}{2},1\right)$. Thus, we have $\frac{\partial
b_{c}}{\partial\phi}>0,$ which means that $b_{c}<\frac{1}{2}$. Thus, if
$b>\frac{1}{2}$, we have $\mathcal{G}<0$ for any value of $\lambda$ such that
$\lambda^{*}(z)>0$. This concludes the proof.

##  Appendix B Related variety and economic integration: additional
illustrations

In this Appendix we change the benchmark parameter values to find out whether
the value of $b$ can bear more drastic implications to the qualitative
structure of the spatial economy. Particularly, we set
$(\lambda,\gamma,\sigma)=(1,0.9,8)$.

Figure 11: Bifurcation diagram for
$(\lambda,\gamma,\sigma,b)=(1,0.9,8,0.1405)$. Filled lines correspond to
stable equilbria and dashed lines correspond to unstable equilibria.

Figure 12: Bifurcation diagram for $(\lambda,\gamma,\sigma,b)=(1,0.9,8,0.15)$.
Filled lines correspond to stable equilbria and dashed lines correspond to
unstable equilibria.

In Figure 11 we set $b=0.146$, which is considerably lower than in the
previously illustrated cases. We can observe that the re-dispersion process is
quite different. Here, for very low levels of $\phi$ an asymmetric dispersion
equilibrium exists and is stable, and becomes more asymmetric as $\phi$
increases. However, after a certain point, the economy starts to re-disperse
until $\phi$ finds a limit point $\phi_{l}\in(\phi_{b2},1)$ above which the
economy suddenly re-disperses evenly among the two regions. That is, the
asymmetric dispersion undergoes a saddle-node bifurcation (refer to the end of
Section 4 for a more detailed discussion) and there exists locational
hysteresis as both an asymmetric dispersion equilibrium and the symmetric
dispersion equilibrium are simultaneously stable for
$\phi\in(\phi_{b2},\phi_{l})$. The symmetric equilibrium undergoes a
subcritical pitchfork bifurcation at $\phi=\phi_{b2}$.

What is perhaps more striking though is that, as $\phi$ increases further, it
encounters an interval $\phi\in(\phi_{s1},\phi_{s2})$ whereby both
agglomeration and symmetric dispersion are stable, and a curve of unstable
asymmetric equilibria exists in between. The striking feature is that the
curve of agglomeration equilibria is not connected to any other kind of
equilibrium.However, further increases in $b$ will eventually connect the
asymmetric equilibrium curve with the full agglomeration as shown in Figure
12.This apparently strange behaviour may be attributed to the limitations
imposed by the implicitly adopted _asymptotic_ stability as the dynamic
stability criterion in this paper. One way to investigate this issue could be
to employ _strategic_ stability as in Demichelis and Ritzberger, (2003), which
entails additional necessary conditions for stability which restrict the
equilibrium set.111111We thank Anna Rubinchik for this reference on stability
conditions. The potential game’s approach taken recently by Osawa and
Akamatsu, (2020) could also be analysed in this context for equilibrium
refinement. Finally, the connection via an increase in $b$ could hint at the
existence of co-dimension $2$ bifurcations with $\phi$ and $b$ employed as
bifurcation parameters.121212We thank Sofia B.S.D. Castro for pointing out
this potentially relevant issue. Although these are all relevant points, we do
not pursue these issues further in our paper.

In Figure 12, we have $b=0.15$, and the economy reaches full agglomeration
smoothly as $\phi$ increases but then jumps discontinuously to symmetric
dispersion. We have the single breakpoint $\phi_{b2}\in(\phi_{s1},\phi_{s2})$,
which means that there exists locational hysteresis as for
$\phi\in(\phi_{b2},\phi_{s2})$ both agglomeration and symmetric dispersion are
simultaneously stable.

Further increases in $b$ can be shown to lead, first to a situation similar to
that of Figure 7, and then to ubiquitous agglomeration for any level of
$\phi$.

##  Appendix C Results for Section 5

###  C.1 Existence and multiplicity of equilibria

Under this specification, the indirect utility is given by:

$v_{i}^{S}=\frac{\mu\gamma}{\sigma}\left[z_{i}+cz_{j}\right]\left(\dfrac{\frac{\lambda}{2}+z_{i}}{z_{i}+\phi
z_{j}}+\phi\dfrac{\frac{\lambda}{2}+z_{j}}{\phi
z_{i}+z_{j}}\right)+\dfrac{\mu}{\sigma-1}\ln\left[z_{i}+\phi
z_{j}\right]+\eta.$ (25)

Differentiating $\Delta v^{S}(z)\equiv v_{1}^{S}(z)-v_{2}(z)^{S}$ twice with
respect to $z$ yields:

$\frac{d\Delta v^{S}}{dz}(z)=\frac{b_{1}P^{S}(z)}{b_{2}},$

where:

$\displaystyle b_{1}$ $\displaystyle=\mu(2z-1)(\phi-1)$ $\displaystyle b_{2}$
$\displaystyle=(\sigma-1)\sigma(z(\phi-1)+1)^{3}(z(\phi-1)-\phi)^{3},$

and $P^{S}(z)$ is a second degree polynomial in $z.$ The zeros of
$\frac{d\Delta v^{S}}{dz}(z)$ are the zeros of $P^{S}(z),$ which can be at
most two. However, it is possible to check that $\frac{d\Delta
v^{S}}{dz}(\frac{1}{2})=0.$ Given symmetry, no other real zero possibly
exists. This means that $\Delta v^{S}(z)$ changes concavity only at symmetric
dispersion. This implies that $\Delta v^{S}(z)$ has at most two turning points
and thus, at most three equilibria. By symmetry, we can establish that there
exists at most one equilibria for $z>\frac{1}{2}$.

Solving the condition for any (asymmetric) dispersion equilibrium is just as
daunting a task as it is for the benchmark case, For the sake of brevity, let
us just say that a dispersion equilibrium exists if
$\lambda\equiv\lambda^{*S}(z)$, where $\lambda^{*S}(z)$ is a function of $z$
and all the parameters contained in the analogous expression in (23).

###  C.2 Stability of equilibria

First, let us analyze the stability of full agglomeration. Using (25), we get
that agglomeration is stable if:

$\mathcal{S}^{S}\equiv\frac{2(\lambda+1)\phi-c\left[(\lambda+2)\phi^{2}+\lambda\right]}{2\sigma\phi}-\frac{\ln\phi}{\sigma-1}>0.$

It is readily observable that agglomeration is always stable if the first term
is positive:

$c>\frac{2(\lambda+1)\phi}{(\lambda+2)\phi^{2}+\lambda}.$

We have:

$\lim_{\phi\rightarrow 0}\mathcal{S}^{S}=-\infty\ \text{ and }\
\mathcal{S}(\phi=1)=\frac{(1-c)\gamma(\lambda+1)}{\sigma}>0.$

Therefore, but the Intermediate Value Theorem, there exists at least one zero
$\phi$ such that $\mathcal{S}=0.$ Further, we have:

$\frac{d\mathcal{S}^{S}}{d\phi}=\frac{1}{\phi^{2}}\left(-\frac{c\gamma(\lambda+2)\phi^{2}}{\sigma}+\frac{c\gamma\lambda}{\sigma}-\frac{2\phi}{\sigma-1}\right),$

whereby the term in parenthesis is a second degree polynomial with negative
leading coefficient and that only changes sign once. Therefore,
$\mathcal{S}^{S}$ has at most one turning point. Since
$\frac{dS^{S}}{d\phi}(\phi=1)=\frac{1}{1-\sigma}-\frac{c\gamma}{\sigma}<0,$ we
can conclude that the turning point is a maximum and $\mathcal{S}^{S}$ is
strictly concave. Thus, there exists a unique sustain point
$\phi_{s}^{S}\in(0,1)$ and agglomeration is stable when $\phi>\phi_{s}^{S}$

Second, we analyze the stability of symmetric dispersion. Using (25) it can be
shown that $z^{*}=\frac{1}{2}$ is stable if:

$\mathcal{B}^{S}\equiv-\gamma(\sigma-1)\left[\phi^{2}(c(2\lambda+3)+1)+2b\lambda+c-4(\lambda+1)\phi-1\right]+2\sigma\left(1-\phi^{2}\right)<0.$
(26)

We can see that if the first term is positive, symmetric dispersion is always
unstable, i.e., if:

$c<\frac{\phi(4\lambda-\phi+4)+1}{(2\lambda+3)\phi^{2}+2\lambda+1}.$

Moreover, $\mathcal{B}^{S}$ in (26) is a second degree polynomial with at most
two zeros, $\phi_{b1}^{S}$ and $\phi_{b1}^{S}<\phi_{b2}^{S}$, with a negative
leading coefficient, which means that symmetric dispersion is stable to the
left of $\phi_{b1}^{S}$, unstable for
$\phi\in\left(\phi_{b1}^{S},\phi_{b2}^{S}\right)$, and stable to the right of
$\phi_{b2}^{S}$. However, it is possible to show after very cumbersome
calculations that both break points cannot co-exist. As a result, we shall
focus on the more interesting case, that is the stability around the first
break point $\phi_{b1}^{S}$, which is given by:

$\displaystyle\phi_{b1}^{S}=$ $\displaystyle 2\gamma(\lambda+1)(\sigma-1)-$
$\displaystyle\frac{\sqrt{-(c+1)\gamma^{2}(\sigma-1)^{2}\left[c(4\lambda(\lambda+2)+3)-4\lambda(\lambda+2)-5\right]+4(c+1)\gamma\sigma(\sigma-1)+4\sigma^{2}}}{\gamma(\sigma-1)\left[c(2\lambda+3)+1\right]+2\sigma}.$

Finally, we analyze the stability of (asymmetric) dispersion. Differentiating
$\Delta v^{S}(z)$ with respect to $z$ and evaluating at
$\lambda=\lambda^{*S}(z)$, we get that dispersion is stable if:

$\displaystyle\mathcal{G}^{S}\equiv$
$\displaystyle(2z-1)\left(\phi^{2}-1\right)\left[(c-1)\gamma(\sigma-1)(1-2z)^{2}-\sigma\right]+$
$\displaystyle+\sigma\left[2z^{2}(\phi-1)^{2}-2z(\phi-1)^{2}+\phi^{2}+1\right]\ln\left[\frac{z(\phi-1)+1}{z(1-\phi)+\phi}\right]<0.$
(27)

###  C.3 Supercritical pitchfork bifurcation

We can get the whole picture of the dynamic properties of the model by
studying the type of local bifurcation that the symmetric equilibrium
undergoes at $\phi=\phi_{b1}^{S}.$ After some tedious calculations, it is
possible to show the following:

$\displaystyle\dfrac{\partial f}{\partial
x}\left(\dfrac{1}{2};\phi_{b1}^{S}\right)$ $\displaystyle=0;\
\dfrac{\partial^{2}f}{\partial
x^{2}}\left(\dfrac{1}{2};\phi_{b1}^{S}\right)=0;\ \dfrac{\partial
f}{\partial\phi}\left(\dfrac{1}{2};\phi_{b1}^{S}\right)=0$
$\displaystyle\dfrac{\partial^{2}f}{\partial\phi\partial
x}\left(\dfrac{1}{2};\phi_{b1}^{S}\right)$ $\displaystyle>0;\
\dfrac{\partial^{3}f}{\partial
x^{3}}\left(\dfrac{1}{2};\phi_{b1}^{S}\right)<0.$

According to Guckenheimer and Holmes, (2002, pp. 150), the conditions above
ensure that symmetric dispersion undergoes a supercritical pitchfork
bifurcation at $\phi=\phi_{b1}^{S}.$

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