11institutetext: Andres Gomez-Lievano 22institutetext: Analysis Group, Boston,
MA, USA
Growth Lab, Harvard University, Cambridge, MA, USA
22email<EMAIL_ADDRESS>

Michail Fragkias 33institutetext: Department of Economics, Boise State
University, Boise, ID, USA
33email<EMAIL_ADDRESS>
# The benefits and costs of agglomeration: insights from economics and
complexity

Andres Gomez-Lievano and Michail Fragkias

###### Abstract

There are many benefits and costs that come from people and firms clustering
together in space. Agglomeration economies, in particular, are the
manifestation of centripetal forces that make larger cities disproportionately
more wealthy than smaller cities, pulling together individuals and firms in
close physical proximity. Measuring agglomeration economies, however, is not
easy, and the identification of its causes is still debated. Such association
of productivity with size can arise from interactions that are facilitated by
cities (“positive externalities”), but also from more productive individuals
moving in and sorting into large cities (“self-sorting”). Under certain
circumstances, even pure randomness can generate increasing returns to scale.
In this chapter, we discuss some of the empirical observations, models,
measurement challenges, and open question associated with the phenomenon of
agglomeration economies. Furthermore, we discuss the implications of urban
complexity theory, and in particular urban scaling, for the literature in
agglomeration economies.

##  1 Introduction

The study of _agglomeration economies_ explores the economic benefits that
arise when individuals live and work _in close physical proximity_. The topic
originates in the observation of the existence of strong and persistent
geographical concentrations in economic life: production and consumption
activities in particular. The idea of agglomeration economies has been one of
the most central topics studied by urban economists and economic geographers,
if not _the_ most studied one glaeser2008cities .

The reaping of agglomeration economies has historically been identified as a
central issue in our quest for increasing economic productivity and eventual
prosperity at the local, regional and global scale. But shocks in the way we
live and work may have a significant effect in the trajectory of those
economies from spatial concentration. In some cases, we can ask whether these
economies will continue to benefit humanity in the long term. For example, the
COVID-19 pandemic has shown that households and firms are able to re-
negotiating life/work arrangements, affecting the spatial patterns of living
and working. Depending on the outcome of this process, and the economies
present in complementary technologies for the exchange of ideas (e.g.
remotely), we can expect important differentials in the level of agglomeration
economies experienced and resulting economic prosperity.

There is a tendency to think that cities exist _because_ of agglomeration
economies. The reasoning goes that since cities are “the absence of space
between people and companies” (as the urban economist Edward L. Glaeser said
(Glaeser2011Totc, , p.6)), if there are benefits to agglomerating, people will
tend to seek each other’s company, and cities will naturally form. According
to this reasoning, cities are the physical manifestation of the gains from
being close to one another. However, there is a certain circularity here,
leading in turn to a misconception: that one concept (cities) cannot exist
without the other (agglomeration economies). That both are synonyms. But they
are not. So to understand what agglomeration economies are, it is useful to
clarify this difference.

On the one hand, cities can exist because of reasons not associated with any
economic benefits from living and working in high densities. For example,
people can come together inside some walls for religious reasons, to protect
themselves against enemies or nature, or to be next to some natural resource
like a river, forest, mine, or port sullivan2006firstcities . On the other,
there are many disadvantages from living in cities, like traffic, pollution,
high rents, crime, and disease. Thus, agglomeration economies are only an
aspect of cities.

The benefits and costs from agglomerating act as two opposing forces. The
former as centripetal forces that pull people (and companies) together, and
the later as centrifugal forces that push them away. In recent times, cities
have become less dependent on being close to certain natural resources. At the
same time, evidence tell us that the benefits from agglomeration are
dominating its costs, and this has been reflected in the fast pace of
urbanization and the continued growth of cities that we see today around the
world. Hence, agglomeration can be considered one of the most important forces
of our current world, and we better understand its effects (i.e.,
agglomeration economies).

In what follows, we address several dimensions of agglomeration economies and
several open research questions in this area. Section 2 reviews the basic and
advanced notions surrounding agglomeration economies - how they have been
theoretically conceptualized, empirically measured, and mathematically
modeled; the challenge of causal inference and differing experiences across
parts of the world. Section 3 discusses ideas from economics and complexity
science that provide a more nuanced understanding of agglomeration phenomena:
the notion of spatial equilibrium from economics (in contrast to the idea of
disequilibrium emerging complexity science); the microfoundations of
agglomeration economies; artificial increasing returns to scale; and urban
scaling theory. Section 4 explores the possibility of blending mainstream
economic analysis of agglomeration with the complexity science perspective.
Section 5 reviews agglomeration in the context of sociology, cultural and
evolutionary anthropology, pointing to the possibility of a grand synthesis
regarding agglomeration phenomena across fields. Finally, section 6 concludes
and discusses open questions that still remain.

##  2 The empirics of agglomeration economies

###  2.1 Conceptual and technical frameworks

Agglomeration economies, broadly construed, are the benefits that emerge from
the clustering of (economic) activity closely in space. In order to quantify
and estimate them, we will use a more precise definition: _agglomeration
economies_ are the _increasing returns to scale_ (IRS), as well as _external
economies specific to a location_ , related to economic productivity and well-
being, occurring in specific locations, such as cities, as they get larger
and/or denser. In the field of economic geography, increasing returns to scale
play a significant role in the location choice of firms, leading to the
formation of core-periphery patterns krugman1991increasingreturns .

Agglomeration in economics has been a central theme of study in urban and
regional economics and the new economic geography fujita1999spatial .
Specifically in economics, agglomeration economies are economies that are
external to firms, but internal to a location goldstein1984economies . Four
types of agglomeration economies based on scale are conceptualized in this
framework: (a) economies internal to the firm at a given location - that is
economies from mass production, (b) economies external to a firm at a specific
location, (c) localization economies (internal to an industry at that
location) and (d) urbanization economies (external to both firm and industry
at a particular location).

The “returns to scale” of a quantity $F$ (e.g., the total well-being in a
settlement) with respect to a variable $x$ (e.g., the size of the settlement)
is given by the change in the function when $x$ is multiplied a constant
$\lambda>1$.111Since the returns to scale concept in economics originates in
production theory, we utilize this concept in line with a “city as a firm”
analogy fujita1999spatial . That is, in this exposition of the concept, we can
think of a city as represented by a single firm. Increasing returns occur when
$F(\lambda x)>\lambda F(x)$. Let $f(x)=F(x)/x$ be the _per capita_ value.
Then, it is easy to verify that IRS implies that $f(\lambda x)>f(x)$. The
quantity $f$ can be wages, patenting rates, or other measures of individual
productivity in a given city, and the quantity $x$ is the total employment,
the working age population, the total population size or even can be the
population density of the city. The discussion of increasing returns to scale
leads into the topic of economies of scale when the payment of factors of
production is introduced.

Economists perceive external economies of agglomeration as resulting from a
shift in a production function of a firm due to the effects of a change in
scale of the area that a firm exists in. In the simplest form, this concept
can assume $Y=G(N)F(K,L)$, where $N$ is population size, $K$ is capital and
$L$ is labor. Expressed in terms of output per worker - dividing by $L$, we
get $y=G(N)f(k)$.222Here, we are assuming the function $F(K,L)$ to have
_constant_ returns to scale. That is, $F(\lambda K,\lambda L)=\lambda F(K,L)$
for any positive constant $\lambda$. In particular, if $\lambda=1/L$, we get
that $(1/L)F(K,L)=F(K/L,1)\equiv f(k)$, a function of capital per worker
$k=K/L$. The function $G(N)$ is referred to as a ‘Hicks-neutral’ externality
since we make the assumption that it does not affect the optimal choice of
labor and capital in the production process.333A Hicks-neutral externality
raises the marginal product of labor and capital proportionally at a specific
capital to labor ratio. Formally, if $dG/dN>0$, we observe economies of
agglomeration while if $dG/dN<0$, diseconomies of agglomeration. Figure 1
shows economies of agglomeration by plotting a hypothetical production
function of a firm existing in a larger agglomeration ($N^{\prime}$) vs a
smaller agglomeration ($N$). Sveikauskas (1975) Sveikauskas1975  was among the
first to explore the empirical relationship between city size and labor
productivity from this point of view. His paper concluded that average wages
were disproportionately higher in large cities because Hicks-neutral
productivity was substantially higher in large cities.

$k$Input (capital per worker)$y$Outputper
worker$y=G(N)f(k)$$y=G(N^{\prime})f(k)$

Figure 1: External economies of agglomeration as experience by a firm through
a shift in the production function due to an increase in the size of a city
($N$ to $N^{\prime}$).

Before going into models of agglomeration economies, the statistical
technicalities, or the underlying causes, let us look at some data so that we
understand why there is so much interest from urban economists about this
phenomenon. As an example, let us start with data from Metropolitan
Statistical Areas (MSAs)444MSAs are the most common unit of analysis in urban
studies, and the US methodology for how to define them has been adopted by
many other countries. Their delineation is not based on administrative or
political boundaries (like those of counties, municipalities, provinces,
states, countries, etc.), but on commuting patterns. The idea behind using
commuting patterns to define urban boundaries is that cities are the places
where people _live and work_. MSAs account for approximately $86\%$ of the
total US population. in the United States (US) for the year 2020 in Figure 2.
There are 384 dots in the plot, each dot representing an MSA, and shows how
the logarithm of average personal income correlates with the logarithm of
population size. The use of a log-log plot to reveal agglomeration economies
is not accidental. This is because the “returns to scale” of a function $f$
can be quantified by the ratio between the rate of increase of $f$ (defined as
$\frac{1}{f}\partial f/\partial x$) and the rate of increase in $x$ (defined
as $\frac{1}{x}\partial x/\partial x$). This is equivalent to comparing
changes between logarithms, as $\partial\ln(f)/\partial\ln(x)$. As a
consequence, a typical step into assessing the presence of agglomeration
economies is plotting how $\ln(f)$ relates to $\ln(x)$.

Figure 2: Association between income per capita and population size across US
MSAs in 2020. The name of the MSAs with the highest and lowest personal income
are indicated in the figure. All dollar estimates are of current dollars (not
adjusted for inflation). Source: Bureau of Economic Analysis (BEA)
BEApersonalincomedata .

As can be seen in Fig. 2, per capita income tends to be higher in larger
cities. The coefficient of the regression is approximately $0.073$.
Coefficients between logarithmic variables denote “elasticities”. An
elasticity of $0.073$ implies that a $1\%$ change in population size is
approximately associated with a $0.073\%$ change in personal income.555The
actual formula is that an $\epsilon$ percent change in the independent
variable (e.g., city size) leads to a $[(1+\epsilon/100)^{\beta}-1]\times 100$
percent change in the dependent variable (e.g., income). When $\epsilon$ is
small, then $(1+\epsilon)^{\beta}-1\approx\beta\epsilon$, which means that a
“$\epsilon$ percent change in the independent variable corresponds to a
$\beta\epsilon$ percent change in the dependent variable”, as was done above
in the text. Sometimes, however, researchers interpret the elasticities with a
$100\%$ change in the independent variable (e.g., doubling city size). In that
case the “$\beta\epsilon$” formula will overestimate the effect on the
dependent variable. With an elasticity of $7.3\%$, doubling the size of a city
corresponds to income increasing by $5.2\%$.

It is worth noting that the relationship in Fig. 2 between log-population and
log-income seems to be linear, and that the $7.3\%$ elasticity seems to apply
across all city sizes. Such linear relationships in the form of
$\ln(y)=A+B\ln(x)$, where the coefficient $B$ is a constant, describe a
_power-law_ relationship between quantities $x$ and $y$, of the form
$y=ax^{B}$ (where $a=e^{A}$)666If the relationship uses $\log_{10}$ instead of
natural logarithms, then $a=10^{A}$.. At the level of cities, agglomeration
economies are often (though not always) represented with these power-law
relationships. Mathematically, we therefore talk of evidence of agglomeration
economies at the aggregate level when

$y=cN^{\delta},$ (1)

where $c$ is a constant, $N$ is city size, and, crucially, $\delta>0$ for
variables $y$ that measure of economic well-being at a per-capita level.

###  2.2 Measurement issues: productivity and scale

Going back to Fig. 2, the power-law relationship between city size and
personal income suggests the presence of agglomeration economies in US MSAs.
Personal income, however, includes earnings not only from job salaries, but
also from interests, dividends, transfers, and others sources777See
https://www.bea.gov/resources/learning-center/what-to-know-income-saving (last
accessed: May 15, 2022)., which do not necessarily reflect the benefits
accrued specifically due to living at a given location. For example, an
individual might live in San Francisco but their income may come from renting
an apartment in New York. This income would still be counted in San Francisco,
even though the economic activity generating such income is from New York. So
how can we measure the “direct” benefits from living in a particular city? A
better quantity would be a measure of _productivity_.

In economics, _productivity_ is an abstract measure of the performance in a
production process. It has to do with the efficiency by which an agent
(person, firm, industry, city, region, or country) converts inputs into
output. Quantifying productivity is challenging because one must account for
increases in output that are not trivial. For instance, if doubling the inputs
of a particular production process ends up doubling the output produced, then
the productivity of that process has not changed. This suggests productivity
should be quantified as a ratio of $\frac{\text{output}}{\text{inputs}}$. At
times, $\frac{\text{sales}}{\text{worker}}$ act as a proxy of productivity.
But it is not that simple. There are more subtle effects to think about. For
example, rather than considering doubling the inputs, consider substituting
some type of inputs with another type (e.g., substituting workers with
machines). A substitution may or may not induce a change in productivity.
Which substitutions would suggest an increase in productivity? Should
productivity be measured as $\frac{\text{profits}}{\text{costs of inputs}}$?

Figure 2 also raises the question of whether total population size $N$ is the
right measure of agglomeration. Would it be better to use population density
(that is, population per unit area)? Duranton and Puga (2020)
duranton2020economics  argue that neither is a great choice: “For instance, a
highly concentrated but tiny cluster of economic activity is unlikely to
generate strong agglomeration economies. On the other hand, workers located at
the edge of large metropolitan areas are unlikely to benefit from their full
scale in the job-matching process” (p. 7). They also suggest that theory is
not going to resolve this issue. Currently, the main practice in economics is
to utilize total population, but increasingly, with the availability of better
data, local density is also utilized

Thus, the economics of density literature moves away from exploring
agglomeration economies as the relationship that emerges between measures of
productivity and city size, and focuses on the alternative approach exploring
the relationship of productivity against density. Duranton and Puga (2020)
show that urban density and population are highly correlated, assuming non-
naive measures of urban density (e.g. that do not include extensive rural
areas in the calculation). That is, proper density measures should reflect how
density is experienced by the typical urban resident of the MSA.

The answers to the choice of measurement are not easy and change depending on
the specific production process under consideration, and many other
assumptions. We will not delve in such discussions here.888There is a whole
field for how to measure productivity. See, e.g., felipe2003aggregation .
Similarly, for measuring density. Instead, we will use _nominal wages_ , as is
customarily done in the literature, to reflect the productivity of a place.
(We will review the conceptual arguments in Section 3.1). We will mainly adopt
scale as a measure of spatial concentration. As we will show, these choices
allow us to discuss the literature from urban economics and urban scaling
uniformly (see, however, ribeiro2018unveiling  and their use of density in the
context of the urban scaling literature). Given these modeling choices, we can
ask: are wages higher in larger cities?

###  2.3 Empirical estimation

Let us define approximate productivity by taking a measure of total aggregate
output generated by a single metropolitan area (i) during a given year (e.g.,
total annual wage bill), $Y_{i}$ and dividing it by the number of workers in
the city, $N_{i}$,

$y_{i}=Y_{i}/N_{i}$ (2)

The ratio in Eq. (2) can be interpreted as the average productivity of workers
in the city.999In general, when you take any measure of total output (e.g.,
total number of patents, total number cars produced, total value of good
exported) and you divide it by the labor used to produce it, you get a measure
of _labor_ productivity. As mentioned in the previous section, caution needs
to be taken in the interpretation.

Given data for $y_{i}$ and $N_{i}$ across several cities within a same urban
system (like a country), the relationship between productivity and scale can
be assessed using ordinary least squares (OLS) regression analysis. From the
economics perspective, the mathematical model specification is derived by a
competitive firm-level profit maximization problem solution
moomaw1981productivity . The statistical specification usually takes the form
of a log-log model, given that researchers are exploring the relationship in
terms of elasticities or estimating a power law (like in Fig. 2).

$\ln(y_{i})=\ln(c)+\delta\ln(N_{i})+\epsilon_{i},$ (3)

where $\delta$ is the elasticity, and $\ln(c)$ is interpreted as the constant
in standard regression. Some debate exists on whether ordinary least squares
is the appropriate methodology in estimating power-laws, or whether maximum
likelihood estimation or Bayesian econometrics should be utilized (see, e.g.,
shalizi2011scaling ; GomezLievanoYounBettencourt2012 ; leitao2016scaling ).

Alternatively, given the land area of cities ($A_{i}$), one can also estimate
the same model but replace the size of the city with a population density
measure, $\rho_{i}=\frac{N_{i}}{A_{i}}$.

$\ln(y_{i})=\ln(c)+\delta\ln(\rho_{i})+\epsilon_{i},$ (4)

Figure 3: Association between average wages and population size across US MSAs
in 2020. The name of the MSAs with the highest and lowest average wages are
indicated in the figure. Source: Occupational Employment and Wage Statistics
(OEWS), Bureau of Labor Statistics (BLS) BLSwages2020 .

Typical results are shown in Figure 3. Our results show a positive association
between population size and well-being as measured by average wage. In this
case, the estimated elasticity is $0.057$, implying that doubling the size of
a city in this cross-section is associated with an increase of $4.0\%$ in
average wages. This elasticity is often referred to as the _urban wage
premium_.

Similar results are found in the economics of density literature. Considering
the agglomeration benefits when concentration is measured in terms of density,
the literature provides similar results: Ahlfeldt and Pietrostefani
AHLFELDT201993 , in a meta-analysis, measure the elasticity of productivity
with respect to density at 0.04.

Estimates from simple bivariate regressions like Eqs. (3) and (4) have many
limitations. They do not actually identify the _causal_ effect of what being
close to other people does to productivity and well-being. They are aggregate
correlations and, as such, they do not rule out other alternative causes. For
example, the same correlation could emerge if people who are more productive
prefer to be close to each other, and as a consequence, tend to move and
_self-sort_ into (large) cities. As we mentioned in the introduction, another
cause of both agglomeration (i.e., population growth) and higher productivity
would be the presence of some natural advantages (potentially unobserved).

With the above in mind, next, we review the empirical challenges of accurately
measuring agglomeration economies.

###  2.4 Empirical challenges: causal inference

Quantifying agglomeration economies consists of being able to single out the
_causal effect_ that physical proximity has on productivity. We have discussed
this, concretely, in the case of the effect city size has on wages. The
challenges involved can be illustrated in the causal diagram depicted in
Figure 4. Each box in the figure represents a variable, and each arrow
represents a causal effect. Thus, measuring agglomeration economies amounts to
quantifying the strength of the green arrow.

The main challenge in measuring the green arrow is that other arrows exist
that also induce correlations between city size and productivity. These issues
are represented by the top and bottom boxes in Fig. 4. On the one hand, the
top box and its yellow arrows represent variables that simultaneously cause
cities to get larger _and_ firms to have higher productivities. An example is
a local natural advantage, such as a river or a sea port, which may attract
firms to operate there (causing city size to increase), and may be an
important source of resources for the city (causing rises in productivity).
These variables are called “confounders”. They are a problem if they are
unobserved. In general, however, the problem of confounders can be minimized
by including available variables for such local urban characteristics as
“controls” in regressions combes2010estimating . For example, one could modify
Eq. (3) to

$\ln(y_{i})=\ln(c)+\delta\ln(N_{i})+\phi X_{i}+\epsilon_{i},$ (5)

where $X_{i}$ can be variables that stand for local characteristics of the
city.

Figure 4: Diagram of causes and effects that could explain correlations
between city size and wages.

The real challenge comes from the bottom part of the diagram. That is, the
blue arrow in Fig. 4 in which the direction of causality runs in the opposite
direction as the one associated with the concept of agglomeration economies
(green arrow). The fact that the arrow points in the opposite duration is not
meant to suggest that there is something that counteracts or reduces the
effect of agglomeration economies. Instead, the arrows just represent which
variable are affected by changes in other variables. In particular, the green
and blue arrows illustrate the fact that a positive association between city
size and productivity might have two possible explanations: either city size
causes higher productivity or individual cities that are already more
productive tend to systematically grow to large sizes. The bottom part of the
diagram represents the issue of so-called “reverse causality”, in which
economic agents decide to locate themselves close to other agents that are
inherently more productive, thus increasing the size of cities
glaeser2001cities ; Combes2011identification . As a consequence, choices will
create a correlation between size and productivity. According to this
direction of causality, the correlation will emerge not because larger sizes
cause economic activities to be more productive, but the other way around:
city size will be the consequence of the choices that productive individuals,
firms and industries make. If high-productivity firms (or industries) tend to
cluster in cities (increasing their size), the urban wage premium found in
large cities could instead reflect firms’ specific production processes rather
than any benefits derived from geographical proximity.

The issue of reverse causality illustrated in Fig. 4 is sometimes also
referred to as “self-sorting”, and it turns out to be incredibly hard to
adjust for in statistical analyses that seek to measure the causal effect of
agglomeration economies. In other words, one step in measuring the presence
and strength of the green arrow is to “shut down” (statistically speaking) the
effect of the blue arrow. However, difficulties include the fact that the
production function of firms is often hard to observe, as well as the fact
that it is hard to identify the reasons why firms and people migrate from city
to city.

A full review of the details involved in separating the causal effect of
agglomeration economies from other causal sources like confounding and self-
sorting can be found in Combes and Gobillon (2015) CombesGobillon2015handbook
. The literature in urban economics keeps investigating new methodologies to
address these empirical challenges, and the sophistication of the methods
keeps increasing steadily, but most of the solutions have converged on a
specific strategy: using data that tracks single individuals over time. This
strategy was first proposed by Glaeser and Maré in 2001 glaeser2001cities
(see also CombesDurantonGobillon2008 ).

With data at the level of individuals, the basic idea is to compare the
productivity of individual workers before and after moving to a large city.
This strategy needs several additional refinements to control for other
simultaneous effects that can interfere with the causal effect of city size on
productivity, but makes the analysis of agglomeration economies more
transparent and interesting than when one is analyzing aggregate data at the
level of cities. The difficulty becomes finding such data. Data at the level
of individuals is either administrative data (like insurance, tax, or social
security data, which are often private and hard to access) that tend to suffer
from issues of statistical representation (because these are samples from
selected groups of people that are not representative of the overall
population), or census data, which are statistically representative, but offer
only a cross-sectional view of the population and are not longitudinal.

###  2.5 Agglomeration economies in developed and developing countries

Bearing in mind the methodological caveats discussed before, we review here
what researchers have found about the magnitude of agglomeration economies
around the world. Overall, the consensus is that the typical magnitude of
agglomeration economies is approximately $4\%$ (see MeloGrahamNoland2009meta ;
AHLFELDT201993 ; grover2021agglomeration ). There is, however, great
variability around the world, in part due to the methodological choices that
different studies have taken. Unfortunately, most of the research in urban
economics has been disproportionately focused on the developed countries. This
fact can in part be explained by the wider availability of metro-level data in
OECD countries.

Hence, the value of $4\%$ is mostly representative of high-income countries.
In the US, for example, accepted estimates range between $3.6\%$ and $4.3\%$,
while in France is around $3\%$. The few studies that have been conducted in
developing countries tend to find higher elasticities. Thus, researchers have
estimated elasticities of $19\%$ in China, $17\%$ in Africa, $6\%-16\%$ in
Latin American countries (see MeloGrahamNoland2009meta ; AHLFELDT201993 ;
grover2021agglomeration  for recent reviews and meta-analyses of the
literature). This is important, because it means that developing countries
derive huge benefits from agglomeration. It urges decision makers to
understand the underlying causes for the strong need in those countries for
density, and the consequent pitfalls.

In summary, while researchers agree that empirical evidence supports the idea
that agglomeration economies in dense urban areas boost productivity, there is
a significant knowledge gap due to a lack of understanding of agglomeration
economies in developing countries. Meta-analyses and literature reviews
MeloGrahamNoland2009meta ; AHLFELDT201993 ; grover2021agglomeration ;
duranton2020economics  rely heavily on estimates from wealthy countries, and
no systematic attempt has been made to quantify the disparities in estimations
between advanced and poor nations. Filling this knowledge gap is critical
since the majority of people on Earth is projected to live and work in
developing-country cities, which are rising quickly in population.

##  3 Models and explanations from economics and complexity

###  3.1 Economics: the idea of spatial equilibrium

_Spatial equilibrium_ is the organizing idea in urban economics for thinking
about cities. Here, we will focus on the inter-urban implications of spatial
equilibrium (i.e., what happens across cities) rather then the intra-urban
implications (i.e., what happens within cities). The literature of the former
was developed by Rosen (1979) and Roback (1982), which is why economists
sometimes talk about the Rosen-Roback model of urban spatial equilibrium
glaeser2008cities .

Imagine there exists a place where the amenities are so nice that everyone
wants to live there (e.g., for many of us, that would be a city with warm
weather throughout the year, beautiful sunsets, fertile soil, next to both
beautiful mountains and a warm beach, where people can swim in the sea and
have plenty of fish they can eat with little effort). Through the exchange of
information about the available options for establishing a life, people would
start moving to that place. From being close to each other, the right matches
between people form to create happy marriages and successful enterprises. The
place would start producing amazing innovations, many of which could start
being exported to the rest of the world. Wealth would accumulate, the best
firms and the brightest people would want to establish themselves there, wages
would increase, and all this economic success would attract more people and
firms to go there. It is not hard to imagine that there is at least one place
in each country on Earth that is not too far from this description. Why are we
not all living in that place? Why do we live in many cities as opposed to a
single (gigantic) city? What are the underlying trade-offs? Something has to
give. But what? In economics, the concept of a _spatial equilibrium_
formalizes and captures the essential elements at play here.

Mainstream economics is centered around the idea of equilibrium - perhaps to a
degree that actually hurts economic inquiry. In contrast, complexity
scientists tend to think of cities as entities in disequilibrium
johnson_cities_2017 . However, thinking about systems in equilibrium does help
with developing some useful insights, even amid the recognition that agents do
not act as perfect rational optimizers (and are, more likely, ‘boundedly
rational’ simon1955behavioral ). In urban economics, the main organizing idea
is that individuals can move freely within a nation, akin to the ‘no friction’
assumption in physics, which is an idealization that simplifies reality. It is
not true in reality, given that myriad factors tether individuals to their
birth cities (e.g., family, private properties they need to take care of,
health, or simple lack of resources to travel), but it is a good approximation
supported by the fact that migration has been one of the universals of human
nature ReichDavid2018Wwaa . Indeed, despite its costs, moving across space in
search of better opportunities is one of the most effective ways to improve
one’s economic well-being richerson2008migration .

Assuming that individuals are free to move in space (between cities) implies
that people are ultimately _indifferent_ between locations. This may seem
counterintuitive, but such implication arises because we assume individuals
have taken into account all benefits and costs, and have all made their final
decision (i.e., if they moved and didn’t like it, they are free to move again
and find another location, until they are happy). Thus, the concept of a
_spatial equilibrium_. Let us develop some additional intuitions from this key
idea.

From the point of view of individuals, three things matter (again, as a
simplification): wages, rent, and amenities. When spatial equilibrium applies,
an individual cannot have a good wage, a low rent, and great amenities, all
three simultaneously. For example, in the beautiful city we pictured above,
wages and amenities were great. However, demand to live there would make
housing very expensive. Moreover, the increasing density of people may in fact
damage the amenities that were the original appeal of the place (e.g., due to
environmental degradation, or visual, sound, and air pollution). In this
simplified version of the world, things will reach an equilibrium. In this
equilibrium, not all people will decide to move to the most beautiful city
because some people will choose to live in places with fewer amenities if the
cost of housing is cheaper, or if wages are higher.

The assumption of spatial equilibrium therefore implies that high wages in a
place are, actually, an indication of poor amenities, high rent prices, or
both. High wages and high rents do not equilibrate across cities (that is,
wages and rents can continue being different between cities) because high
wages in a given city are, in turn, an indication that the firms have decided
to locate there for a reason. And the reason must be that, by locating
themselves in that city, firms are being able to be more productive (and
therefore more profitable).101010Recall, in standard economic theory, assuming
perfect competitive markets, the wage a firm pays to a worker is determined by
the _additional_ value he or she brings to the firm. In economic language,
wage is the _marginal product of labor_. So, at the end, higher wages are an
indication that the area is benefiting from agglomeration.111111This is
formalized in a theorem first stated by Starrett in 1978 starrett1978market ,
called the Spatial Impossibility Theorem, restated by Ottaviano and Thisse
(see, e.g., ottaviano2004agglomeration ). The theorem implies that there are
three sufficient conditions for economic activities to agglomerate: (i)
heterogeneous space, (ii) positive externalities, or (iii) imperfect markets.
The concept of agglomeration economies is particularly associated with (ii),
but there are mechanisms in which (i) and (iii) also create agglomeration
economies. We discuss the specific mechanisms in Section 3.2.

Through this line of reasoning we have arrived at an important insight: to
reveal agglomeration economies, we should look at _nominal_ wages and check
how they correlate with the density or the size of a city. Looking at _real_
wages, as opposed to nominal wages, would defeat the whole purpose. By
assumption, we have said that wages are offset by the costs of living.
Therefore, adjusting nominal wages to account for the costs of living would
simply reveal that all real wages are equal across all locations, and we would
destroy the information generated by agglomeration economies.

###  3.2 Economics: causes of agglomeration economies - sharing, matching and
learning

Economists have been wondering for a long time about the mechanisms that would
cause economic agents to benefit from crowding together in close physical
proximity. The original classification for the causes of agglomeration
economies is usually attributed to Alfred Marshall - a foundational figure in
economics - first published in 1890 in his work _Principles of
Economics_.121212Citations sometimes refer the 1920’s 8th edition of
Marshall’s book. There, Marshall proposed that agglomeration was a consequence
of reducing the transportation costs for goods, people, and ideas. According
to Marshall, by locating near each other, firms will reduce the shipping costs
of materials (reducing, for example, the interaction costs between buyers and
suppliers), the costs associated with the labor market (e.g., the search and
employment of appropriate workers), and finally, the costs of finding new
ideas (e.g., firms will benefit from “knowledge spillovers”).

Duranton and Puga (2004) DurantonPuga2004micro  re-conceptualized this
classification. According to them, once a firm in a particular industry is
established in a city, it paves the way for other firms to get established
there because the firm has assumed some initial fixed costs, has created a
network of customers and suppliers, and has revealed important knowledge about
how to operate successfully in the city. Thus, once the first firm arrives
into a city, other firms will benefit, and this will show up as increases in
productivity. In Duranton and Puga’s classification, close physical proximity
would entail benefits to the firms because of better

  * •

_sharing_ of local infrastructure, indivisible facilities, suppliers and
customers, leading to savings on fixed costs, as well as firms sharing risk,

  * •

_matching_ between employers and employees, or sellers and buyers, and

  * •

_learning_ opportunities in terms of generation, spread and accumulation of
knowledge.

Since Marshall’s time, the causes for the gains from agglomeration have been
thoroughly researched. However, the relative impact of each of these
mechanisms has been hard to quantify. This is due, in part, to the so-called
“Marshallian equivalence” DurantonPuga2004micro , which states that all
agglomeration mechanisms provide the same prediction: firms involved in
similar activity will tend to cluster since they produce benefits for one
another. The addition of gains due to the three causes makes determining which
mechanism has the most weight very challenging. Identifying how each of these
mechanisms impact agglomeration, however, is important not only from a policy
perspective, but also to anticipate how technological advances and
socioeconomic change will affect the growth of cities.

The current understanding is that these mechanisms affect different industries
in distinct ways, and that these effects have been changing over time. Diodato
et al. (2018) diodato2018industries  carried out an extensive study of these
observations. They show that value chain connections used to be dominant for
most of the twentieth century for most industries. Currently, however, there
has been a shift in the economy into services and skill-sharing has become the
most important motivator driving firms in that sector, while value chain
connections remains the main cause for co-location patterns in manufacturing.

Are Duranton and Puga’s views the final word on the origins of agglomeration
economies? There are reasons to suppose that city size has more to it than
their three mechanisms. For example, there are additional urban phenomena that
do not represent economic advantages but are still amplified in larger cities
(like diseases and crime) in the same manner that economic gains are. Yet, the
mechanisms of sharing, matching, and learning are unlikely to also explain
those other non-economic phenomena. From a complexity point of view, however,
a simple set of mechanisms should be able to explain all these phenomena,
economic and non-economic. One element that could explain why larger cities
potentiate all human activities is the fact that they are more diverse (Sec.
3.4 will discuss this idea).

###  3.3 Complex systems: a criticism of the economics approach - artificial
increasing returns to scale

There is an empirical problem with measuring agglomeration economies that
merits a section on its own here, separate from the empirical challenges that
we have summarized so far. The issue we want to discuss here is one in which
the statistical association that emerges between size and productivity does
not arise from difficulties in disentangling economic causes and effects
(i.e., cannot be represented in Fig. 4) or from one the three economic
mechanisms explained in the previous section. It is an effect from over
relying on the concept of “average” that one often does when analyzing complex
systems.

The diagram in Fig. 4 and the three mechanisms for agglomeration economies
(sharing, matching and learning) both make an assumption that, at first sight,
would be unobjectionable. That in order to quantify the sources of
correlations between city size and productivity using data, one can safely
compute _averages_. For example, in Fig. 3 we looked at the _average_ wage
across cities. Or, when researchers take care of the issues of confounding and
self-sorting, they aim to measure how size will influence the productivity of
individuals _on average_. In all these efforts, researchers assume the Law of
Large Numbers (LLN) applies: that taking the average over a finite sample of
observations approximates the true underlying expected value. This assumption
is fundamental to doing science in general. However, quantities in complex
systems sometimes have the nasty property that their distributions either lack
an expected mean, or their mean is very hard to estimate. Science in these
cases must advance with caution, being mindful of this issue (see, e.g.,
schroeder1991fractals ; Sornette2006 ; NewmanPowerLaws2005 ). How does this
problem affect the analysis of cities?

In simple words, the challenge is that cities are fundamentally unequal,
heterogeneous, places. Income, productivity, wages, and other economic
variables, tend to be unevenly distributed across firms and individuals in
cities. The distributions that describe these quantities are often power-laws,
log-normals, or more generally, heavy-tailed distributions, which limit the
validity of the LLN EmbrechtsKluppelbergMikosch1997 . If the validity of the
LLN is in question, one cannot take averages.

To take an average of a set of observed numbers, one needs to guarantee,
first, that their distribution has a finite variance, and second, that the
sample size is large enough. Gomez-Lievano et al. (2021) gomez2021artificial
calculated that if urban quantities are log-normally distributed (as wages in
cities tend to be), then averages can be taken only if
$\ln(N)\geq\sigma^{2}/2$, where $N$ is the city size and $\sigma^{2}$ is the
variance of log-wages. This relation means that the larger the variance of
log-wages, the larger the size must be in order to guarantee that the sample
average $\frac{1}{N}\sum_{j}y_{j}$ of wages across individuals $j=1,\ldots,N$
makes sense. An interesting effect occurs to the sample average of wages when
$\ln(N)<\sigma^{2}/2$, that is, when LLN does not apply: the sample mean
(e.g., average wage) and the sample size $N$ become positively correlated. If
such situation exists in real data, higher average wages will be observed in
larger cities, _even if the true underlying expected mean is constant across
cities_. In other words, under certain situations in which the LLN does not
apply, an artificial effect emerges that can be mistakenly interpreted as
evidence for agglomeration economies. However, pure independent and
identically distributed random variables can generate this effect, as
illustrated in Figure 5 (we encourage the reader to simulate this phenomenon
and replicate this simple figure).

Figure 5: Correlations between city size and measures of output per capita
like wages emerge without agglomeration economies or self-sorting for pure
statistical reasons due to wages being lognormally distributed. In these
simulations, each dot represents a hypothetical city, and within each city
individuals are assigned a wage sampled independent and identically from a
lognormal distribution with same mean and standard deviation across all
cities. Each panel plots the relationship that emerges for different standard
deviations of log-wages $\sigma$. Source: figure from gomez2021artificial .

This effect stands as a cautionary tale about how we think about cities. Many
mathematical models assume representative agents and apply averages to data to
make statements about such agents. However, the heavy-tailed distributions in
cities imply that such representative agents may not exist, and that computing
empirical averages may be mistaken. Conceptually, our models should instead
start from the assumption that heterogeneity is a fundamental characteristic
of cities. Methodologically, this effect implies that our null hypothesis
should be that observing a positive association between size and productivity
ought to be expected even if there are no _real_ causes correlating size and
productivity.

###  3.4 Complex systems: urban scaling theory and its relationship to
agglomeration economies

As has been discussed elsewhere in this book, cities have been studied using
the lens of Scaling Theory. The idea behind scaling is that associations of
the form $Y=aX^{b}$ between quantities of interest and the scale of the system
may indicate a property of the phenomenon of fundamental importance: scale-
invariance. That is, the rule that explains the phenomenon acts the same
across scales. The success of using scaling in physics and biology to derive
theories for how systems work (see, e.g., west1997general ;
schroeder1991fractals ; barenblatt2003scaling ) encouraged the study of cities
with the goal of finding a unified theory for how cities work
BettencourtPLOS2010 .

Using scaling analysis, complex system scientists have observed that the
relationship between many urban quantities $Y$ and size $N$ is characterized
by a power-law function $(Y/Y_{0})=(N/N_{0})^{\beta}$, suggesting a universal
mechanism at work (for a review of the field see Chapter 11, as well as
bettencourt2021introduction ). Indeed, urban metrics following this law span
several different activities and have even been observed even in
archaeological data (see, e.g., ortman2014pre ; lobo2020settlement ). Of
particular interest for our current chapter are phenomena for which exponents
$\beta$ have been observed to be larger than 1. When expressing $Y$ in per
capita terms, $y=Y/N$, these _super-linear_ power laws indicate that larger
cities have higher $y$, because $y\propto N^{\delta}$, with
$\delta=\beta-1>0$.

What is the connection between the scaling relationships that have been
identified by complexity scientists, and the relationship between productivity
and population size that characterizes the phenomenon behind agglomeration
economies, which is also described by a power-law (see Eq. 1)? The fact that
there are other phenomena, different from income and wages, that occur with
higher intensity in larger cities raises the question of whether the economic
arguments described in Sections 3.1 and 3.2 (ideas of spatial equilibrium,
sharing, matching, or learning) are actually correct. Criminal activities,
such as homicides, or the prevalence of diseases, such as cases of syphilis,
also increase on a per capita basis with population size, following power-laws
(see, e.g., GomezLievanoYounBettencourt2012 ; Patterson2015STDs ). Would
economic arguments make sense in order to explain why homicides, or cases of
syphilis, scale with size? At a first look, it does not seem likely.

Some would argue that the theory on agglomeration economies emerging from
economics should be abandoned given that a more general theory may exist.
Surely, a theory that can explain a large number of urban properties has a
significant advantage. Alternatively, however, future work could examine the
possibility that specific economic theories, such as the ones in the space of
agglomeration economies, are a special case of urban scaling theory.

The promise of a unified theory of cities derived from scaling laws has not
been fulfilled yet. Nevertheless, a first parsimonious explanation for the
super-linear scaling of economic variables from a complex systems point of
view was proposed by Bettencourt (2013) and used the idea that larger
populations are inherently more interconnected Bettencourt2013 .

The argument is as follows. Assume that the output of a given individual in
the city is proportional to the density of interactions that such person has
with other people in the city, $y\propto\rho_{\text{social}}$. The density of
interactions, in turn, is defined to be proportional to $N/A_{\text{social}}$,
where population count is divided not by the surface area of the city, but by
the “social area” in which interactions between people occur (e.g., the
infrastructural network), $A_{\text{social}}$. Now, let people be evenly
distributed throughout the city’s surface are, $A_{\text{surface}}$, and let
the average distance between individuals be $d$. Given this, the surface area
is proportional to the population size multiplied by the area between them,
$A_{\text{surface}}\propto Nd^{2}$. The social area where interactions occur
represents only a small fraction of the city’s surface area. The model assumes
that the social area is actually defined by the _linear_ space between people,
which yields $A_{\text{surface}}\propto Nd$. Thus, if
$\rho=N/A_{\text{surface}}$ is the population density, Bettencourt’s
assumption implies that the population density is proportional to the density
of interactions squared, $\rho\propto\rho_{\text{social}}^{2}$. Going back to
the output per capita, we get that

$y\propto\rho_{\text{social}}\propto\rho^{1/2}$ (6)

According to Equation (6), output per capita scales with the square root of
city population density. In this model, this relationship holds regardless of
the type of output, the physical makeup of the city, or the budget constraints
of individuals. To derive a relation between $y$ and $N$ that does not depend
on the area $A_{\text{surface}}$, however, one does need to make further
assumptions. Thus, Bettencourt (2013) assumes that the minimum output per
capita allowed in the city must be enough to pay for the cost of traversing
the whole city. The minimum output is assumed to be proportional to the
population density, $y_{\text{min}}\propto\rho$ (not $\rho_{\text{social}}$),
while the cost of traversing the city is assumed to be proportional to the
linear extent of the city, $c\propto L\propto A_{\text{surface}}^{1/2}$. The
condition $y_{\text{min}}=c$ implies that $(N/A_{\text{surface}})\propto
A_{\text{surface}}^{1/2}$, such that one can write $A_{\text{surface}}\propto
N^{2/3}$. Putting everything together yields

$y\propto N^{1/6}.$ (7)

In words, the model predicts an elasticity between productivity and size of
$\delta=1/6\approx 16.7\%$, which is far from the estimates for developed
countries we mentioned in Section 2.5, but not too far from the estimates for
developing countries.

Note that Bettencourt’s model is agnostic about how interactions are converted
into output. The model does not indicate whether interactions represent
instances of _sharing_ resources, or _matches_ between employers and employees
(or suppliers and buyers), or cases of _learning_ due to the interchange of
information between individuals when they meet. In that sense, it is more
general. Note, however, that the model has similar limitations in explaining
phenomena like disease or homicides. It assumes social interactions are
necessary and sufficient conditions to generate the different types of output
(cases of crime and disease), and it either predicts a universal relationship
with population density with an exponent of $0.5$ that should hold across all
activities (Eq. (6)), or a relationship with city size arising from economic
assumptions that involve a ‘minimum output per capita’ equating a ‘cost of
transportation’ (Eq. (7)).

Some improvements on Bettencourt’s model have been proposed in which
interactions are necessary but not sufficient (e.g., gomez2016explaining ;
gomez2021estimating ). These models also account for the cognitive constraints
that limit the number of social links that humans can maintain Milgram1970 ;
dunbar1998social ; gonccalves2011modeling , and consider individuals to be
_combiners_ of information, rather than conduits (for a review of the evidence
in this area see fogarty2015cultural ).

We highlight the model by Gomez-Lievano et al. (2017) gomez2016explaining ,
for example, in which people derive value not from the quantity of social
links, but from the diversity that comes from them. In such model, individuals
in larger cities are exposed to more diverse cultural factors, other people’s
know-how, and different pieces of information, which can be combined to
generate value weitzman1998recombinant ; muthukrishna2016innovation . In this
model, therefore, larger sizes lead to more diversity, and more diversity
leads to higher output131313It is worthwhile pointing out that this finding
aligns with the concept of urbanization economies, also known as the
urbanization economies as opposed to the concept of localization economies.
The former involves the spatial concentration of firms due to across-industry
advantages (the richness of economic activity in cities). The latter involves
the clustering of similar firms due to within-industry advantages..
Mathematically, a power-law scaling emerges as the combined outcome of
exponentiating a logarithm, as $Y\propto e^{diversity}\propto
e^{\beta\ln(N)+c}\propto N^{\beta}$. This model is well supported by theory
and data (e.g., see henrich2004demography ; kline2010population ;
YounEtAl2016Universality ; zumel2020innovation ). One of the predictions of
this model is that socio-economic outputs that are more complex (i.e., those
that require a higher diversity of inputs or ingredients to be produced) will
display larger scaling exponents, will be less prevalent in general, and will
be more variable from city to city. These predictions were recently verified
by Balland et al. (2020) balland2020complex .

Ribeiro and Rybski conducted a survey of the main mathematical models of urban
scaling ribeiro2021mathematical . In general, urban scaling researchers tend
to depict cities as more messy, more governed by stochasticity, intricate
interactions, and serendipitous complementarities among heterogeneous actors,
than mainstream urban economists, who like to emphasize cities’ orderly
structure, the economic forces of supply and demand, and the optimal decisions
from rational agents. The actual world is likely to be a blend of these
different perspectives, and a discussion between academics in urban economics
and complex systems science should generate useful insights.

###  3.5 Complex systems: urban scaling theory criticisms

Since the formulation of the urban scaling hypothesis strumsky2005metropolitan
; paulus2006evolutionary ; BettencourtLoboStrumsky2007 , several researchers
from within the complexity field have offered criticisms of the framework and
the approach. The criticisms are mostly directed at the statistical methods
(see, e.g., shalizi2011scaling ; leitao2016scaling ; altmann2020spatial ), but
some are aimed at the concept itself (e.g., arcaute2015constructing ;
cottineau2017diverse ; sarkar2020evidence ). Literature showing
counterexamples or raising contradictions in statistical analyses can be found
Chapter 11. We mostly focus below on conceptual issues related to urban
scaling as they relate to agglomeration economies as discussed in previous
sections.

Shalizi (2011) shalizi2011scaling , for example, argued that the relationship
between economic output and city size is not a power law. Shalizi suggested
that other functions of population $N$ can fit the relationship equally well.
If this is true, then the relationship of urban characteristics and population
is not scale invariant. Bettencourt et al. (2013)
BettencourtEtAl2013hypothesis  respond to this criticism by stating that there
is room for other explanations of urban scaling: “The fact that these other
functions of $N$ can fit urban data well (but not better) than power-laws in
most observed cases […] merely means that other models may be viable
explanations of the scaling behavior of urban indicators, along with scale
invariant functions”. When statistics is not enough to distinguish among
different functional forms, the authors argue one must rely on theory:
“[B]ecause the scale-adjusted logarithmic function does not derive from theory
to the best of our knowledge or fit the data manifestly better, we believe the
hypothesis of urban scaling in terms of a scale invariant function, predicted
by theory, remains at this point the better explanation” (p. 12).

Other criticisms, such as those of Leitão et al. (2016) leitao2016scaling  and
Altmann (2020) altmann2020spatial , have indicated that a misspecification of
the distribution of the dependent variable in urban scaling analysis, or the
independence assumption between observations conventionally made in linear
regressions, lead to wrong conclusions about the functional form relating
outcomes $Y$ with city sizes $N$. These are valid concerns, and researchers
will need to rely on theory to decide which statistical methods and functions
to use to fit the data, as Bettencourt et al. (2013)
BettencourtEtAl2013hypothesis  argue.

To these methodological criticisms, we can add the empirical challenges we
mentioned in Section 2.4, in which the statistical approaches used by urban
scaling researchers to estimate the power-law exponents do not address the
issues of confounders and reverse causality. Complexity scientists might argue
that they are not concerned about reverse causality, since the question in
urban scaling is not necessarily why size _causes_ productivity, but why size
and productivity maintain a very special relationship with one another across
scales. The problem of confounders, however, still holds, since confounding
adds spurious correlations between variables (see Patterson2015STDs  for an
example of urban scaling analysis that does take into account confounders).

But what if the very premise of urban scaling analysis, that aggregate urban
quantities are a function of total size, is mistaken? Such conceptual
questions started with Arcaute et al. (2015) arcaute2015constructing , who
noted that cities are not objects with clear physical boundaries, and
therefore there is no definitive way of quantifying their size, or many other
urban quantities. Thus, the premise of urban scaling may be misplaced. These
authors showed how different ways of delineating the boundaries of cities
return power-laws with different exponents, putting in doubt the very concept
of agglomeration economies. Other studies have followed (e.g.,
cottineau2017diverse ; sarkar2020evidence ) arguing that researchers should be
more cautious about computing aggregate measures like total population size,
or total output. Instead, they should include in their models the specific
intra-city composition of economic and social activity and physical
infrastructure in order to understand urban economic performance.

Note that the criticisms raised above have been addressed by mainstream urban
economists by having a strict definition of cities as “labor markets” (i.e.,
places were people live and work), and by focusing their empirical work to
studying the effects of agglomeration at the micro level of individual firms
and workers. Given this focus, both their mathematical models and their
statistical methods can be fairly detailed about mechanisms, causes and
effects.

##  4 Complexity and economics: convergence for a better understanding of
cities?

Since at least the 1980s, several efforts of bringing a diverse set of
economists, physicists and other scholars in fields related to complexity
science have produced important research andersonarrowpines1988economy . The
field of complexity economics emerged from that work: assumptions of perfect
rationality, well-defined problems and optimal behavior were relaxed.
Heterogeneous agents, each with imperfect information and capable of
adaptation to their environments were introduced. The approach brought about
models of economies that are “organic, always creating itself, alive and full
of messy vitality.” arthur2021foundations . Unfortunately, its impact in the
mainstream of the economics profession is debatable.

The original challenge for the new field involved established research
“cultures” and methodological or modeling choices or traditions. Mainstream
economists have been modeling social phenomena utilizing representative agents
(e.g. households, firms and developers) who solve optimization problems with
the end outcome of the system arriving at an equilibrium. Complexity science-
insprired economics models social phenomena utilizing heterogeneous agents,
with localized knowledge, following rules and adapting to strategies of
neighbors. While both approaches lead to emergent aggregate phenomena, the
complexity economics approach is broader in scope and does not rely on
limiting assumptions made in mainstream economics (representative agents,
rationality, complete markets, etc). Below we summarize and contrasts the
major features typically found in both mainstream urban economics and the
complexity-oriented urban scaling theory approaches (Table 1).

Table 1: Summary of features of urban economics and urban scaling theory
approaches

Feature |  Urban Economics |  Urban Scaling
---|---|---
Agents |  Representative (but N different types) |  Diverse
Behavior |  Optimizing (depending on agent type - utility, profit, rents) |  No goals other than interacting locally
Rationality |  Perfect |  Not defined
Interactions |  Homogeneous (impersonal markets, no network formation) |  Channeled by networks (social and physical) of differing strength and types
Interaction extent |  Global |  Localized
Theme |  Optimal resource allocation and spatial equilibrium |  Formation of structures, emergence of socioeconomic aggregates
Structural change |  Equilibrium shifts |  Endogenous re-organization

Source: arthur2021foundations  (table adapted by the authors)

Thisse (2014) thisse2014new  offers one of the few attempts made from
mainstream economists, exploring the possibility of a synthesis between the
field of urban economics and urban geography (which embodies several themes of
complexity). Reviewing the work by Batty (2013) batty2013new , he reaches a
few conclusions that are revealing. Thisse focuses on the fundamental aspect
of the methodological choices an economist makes and the associated belief
system: “ macroscopic phenomena can best be understood by examining how they
arise from the motivated actions and interactions of individual agents in a
world where various types of uncertainty prevail.” This is not a point that a
complexity scientist would contest. But he takes issue with an urban
complexity view “formed by purposeless agents whose aggregate behavior is
governed by stochastic laws.” (Mainstream) economists model cities “as
emerging from the decisions of many rational agents—firms, households,
developers, and local governments—whose actions are often coordinated by
markets.” He correctly suggests that this difference in approaches is not “an
insurmountable obstacle to interactions between urban geographers and urban
economists”.

While interactions can occur without a significant efforts for unifying our
understanding of the functioning of cities, we argue that what is needed is
parallel shifts in urban modeler methodological positions. For example, on the
one hand, economists need to rely less on representative agents and rational
expectations and allow for agent heterogeneity. The strong emphasis on
microfoundations leading to macro-outcomes has required unrealistic
assumptions to the detriment of whole fields of study in economics
stiglitz2018modern . Furthermore, network structure of interactions between
people is important. Consumer and producers do not only interact in anonymous
market through the price system, but develop relationships of differing
strength in networks. Also, moving away from analytical models and adopting
agent-based simulation modeling approaches in the context of economics could
help bridge some of the gaps that exist. Some progress towards this goals
already exists (behrens2015agglomeration ) but additional efforts will assist
in the convergence of these knowledge domains.

On the other hand, urban scaling researchers (complexity scientists,
physicists, statisticians, urban geographers, etc.) should attempt to further
elaborate the existence of markets and distinct mechanisms of production and
exchange in their models. The emergence of markets or other economic
institutions from interactions within population (who adopt roles of producers
and consumers) would be even more valuable. We are also faced with a paradox
in urban scaling theory. In some ways, the theory appears to not allow for
enough depth in idiosyncratic dimensions. Some people are more talented than
others and could be attracted to bigger cities based on their individual
characteristics. Some cities become diversified and some specialized in terms
of their economic base. The migration of population from rural to urban areas
is also not part of urban scaling theory. Business cycles - periods of
economic booms and busts - are not explained. This richness of the urban
experience is currently somewhat lost in a general theory of urbanization.
Several attempts to close this gap are underway (e.g., gomez2016explaining ;
pumain2018evolutionary ; bettencourt2019towards ; gomez2021estimating ).

Finally, it’s worthwhile pointing out that differences between the two fields
are also due to the way that research questions are asked. In urban scaling
theory, the main question asked is whether the scaling of urban phenomena (an
aggregate outcome) can be attributed to localized agent interactions (micro
behavior). In economics, the questions are much more focused and directed:
what type of economies do we observe? what are their causes within the
operation of a market system? If scaling theory is indeed a ‘General Theory of
Urbanization’, we should have observed similar scaling phenomena across
distinct economic structures (for example, were systems of cities in the USSR
experiencing similar scaling phenomena?) Some evidence exists on scaling of
settlements in ancient societies - which were admittedly much simpler in terms
of the functioning of markets, exchange and production. We argue that more
research is needed when considering the potentially differential effects of
modern economic structure.

##  5 Assessing agglomeration effects beyond economics and complexity

We have summarized the topic of agglomeration economies from the economics and
the complex systems perspectives. However, economists and complexity
scientists are not the only ones who have studied the effects of
agglomeration, and these other perspectives merit a brief mention.

We pointed out in Section 3.2 that the economist Alfred Marshall had
speculated about the causes and the effects of physical proximity back in the
1890’s. He wrote about this question after noticing that industries specialize
geographically. As we explained, he contended that spatial proximity promotes
the transmission of knowledge, reduces transport costs of inputs and outputs,
and allows firms to benefit from a more efficient labor market. At around the
same time, the French sociologist Émile Durkheim also remarked the effects of
agglomeration from a sociological point of view. These and many other of
Durkheim’s ideas were published as part of his 1893 doctoral dissertation,
_The Division of Labour in Society_ durkheim1893division . Durkheim is seldom
cited by economists and complexity scientists, but his influence might be as
important, or more, than Marshall’s. It is interesting to observe that the
central issue of population scale and density have been addressed by
foundational figures in both economics and sociology.

While Marshall emphasized the _immediate_ effects of agglomeration on _firms_
, Durkheim underlined the _collective_ effects on _society_. In Durkheim’s
view, population density and growth were the crucial drivers of social and
economic development. Durkheim’s theory is interesting in that increasing the
number of people in a location would increase the number of social
interactions, but he did not interpret the latter to lead directly to
prosperity. Instead, in Durkheim’s account, the increase in the number of
interactions would lead to more competition, division of labor,
specialization, diversity, and ultimately, conflict. These, in turn, would
force society to develop norms and behaviors to generate, sustain, and improve
social integration. This vision was encapsulated in Durkheim’s concept of
“dynamic density”.

Durkheim’s ideas were further developed by the American sociologist Louis
Wirth (see wirth1938urbanism ), who wrote explicitly about cities, with
clarity and insight. These ideas, however, were not formalized into
mathematical models until the 1970s. Once sociologists started to connect
models with data, a heated debate was sparked about the effects of urban
agglomeration (see, e.g., Milgram1970 ; kasarda1974structural ;
kasarda1979ratio ; Mayhew1976 ; Fischer1977comment ; seiyama1986review ;
rotolo2006population ). On the one hand, some researchers contended that
Durkheim and Wirth’s theories were unnecessarily complicated. The fact that
social interactions scale as the square of population size, $I\propto N^{2}$,
was enough to explain the positive relationships between socio-economic rates
and the size of cities. Mayhew and Levinger (1976) Mayhew1976 , for example,
used this argument to explain the higher rates of crime in large cities. On
the other, such mathematical arguments were viewed as too simplistic, missing
“substantive understanding” of how social systems work. It is instructive to
read, for example, the criticism raised by the sociologist Claude S. Fischer
(1977) Fischer1977comment . For Fischer, in brief, the fact that _potential_
interactions grow super-linearly with size does not imply that _actual_
interactions do (“why do they [the proponents of the interactions-based
theory] not predict increasing marriage or birth rates with city size?”, p.
453). Fischer condemns such models, because they “assume that people in cities
are like numbered marbles in a large urn, constantly being shaken up and
randomly ‘contacting’ one another. [These models are] not even a heuristically
useful approximation of the truth. Casual reading of urban ethnographies—even
a casual look out the window—will show that urbanites’ lives cannot be
described this way. Most city dwellers lead sensible, circumscribed lives,
rarely go downtown, hardly know areas of the city they neither live nor work
in, and see (in any sociologically meaningful way) only a tiny fraction of the
city’s population.”

Finally, a similar debate about the effects of population size on social
systems has taken place, almost in parallel and with no connection to
economics, complexity, or sociology, in the fields of cultural and
evolutionary anthropology, where the issue is referred to as the “population
size hypothesis” (see, e.g., carneiro1967relationship ; boserup1981population
; lee1988induced ; shennandemography2001 ; henrich2004demography ;
kline2010population ; collard2013population ). In anthropology, one of the
central concerns is to explain how cultures have evolved, allowing humans to
occupy almost all ecological habitats on Earth (without subspeciation, which
is the typical mechanism for all other organisms to adapt to new environments)
henrich2015secret . Given the limits to what they can measure from
archaeological and ethnographic data, anthropologists have studied the
“cultural complexity” of societies (rather than productivity), defined by them
as the number of cultural traits (literally, the count of distinct artefacts,
social roles, etc).

When data started to accumulate enough to be amenable to statistical analysis,
a clear relationship between population size and cultural complexity across
several societies was found in 1960s carneiro1967relationship .141414The
anthropologist Robert Carneiro was a strong supporter of adapting insights
from the physical and natural sciences into anthropology. Incidentally, he
introduced an idea from psychology called “scale analysis” carneiro1962scale ,
which should not be confused with “scaling analysis” as used in the urban
scaling literature. His methodologies pre-dated the development of the field
of economic complexity by almost 50 years (see, e.g., HidalgoHausmann2009 ).
That is, societies that had larger population sizes were observed to have more
technologies, deeper divisions of labor, and more political, religious, and
organizational units (for recent analysis of this association see
kline2010population ; collard2013population ). Since then, the question has
been whether population size leads to higher rates of cultural innovation,
vice versa, or whether there are other variables causing both in tandem (i.e.,
the same questions we ask in Fig. 4).

One of the current theories to explain why larger population sizes give rise
to higher rates of cultural complexity is Joseph Henrich’s “collective brain
hypothesis” muthukrishna2016innovation . The argument is very different from
those of economists, complexity scientists, or sociologists that we have
reviewed so far in that Henrich’s explanation is rooted in evolutionary theory
(the field itself is called “cultural evolution”). The basis on which this
theory is built are the recent observations that show that humans are super-
imitators (for a review, see Laland2017 ). This unique cognitive feature,
unmatched across the animal kingdom, allows innovations to spread and pass
down across generations without deterioration henrich2004demography . As a
result of this unique ability to mimic our close peers, cultures can
accumulate an ever-expanding repertory of practices, conventions, tools, and
know-how. The cultural brain hypothesis suggests that as culture evolves and
expands, it exerts evolutionary forces that adjust sociality, imitation, and
transmission variance to cope with the increased complexity of tools,
practices, beliefs, and behaviors. In this view, the characteristics of
individuals remain more or less constant across populations of different
sizes, but not the “collective brain” of those societies. Instead, collective
brains get larger as populations get larger. Hence, agglomeration leads to
more innovation, higher productivity, and even higher IQ scores, but not
because of smarter individual brains, but because of smarter collective
brains.

How do these multi-disciplinary perspectives enrich our understanding of
agglomeration economies in cities? Complexity scientists ought to be in the
position to synthesize the key ideas and propose new ones. We end this chapter
with some final conclusions and questions for further thought.

##  6 Conclusions and open questions

We argue that understanding cities is a very important priority for humanity,
as we increasingly become an urban civilization. This understanding passes
through the understanding of both benefits and costs of agglomerating.

Researchers exploring this fundamental aspect of the function of cities should
be aware of the multiplicity of issues related to measurement of productivity
and scale (population size vs. population density), as well as the definition
of the boundaries of a city or a metropolitan area. Conducting sensitivity
analysis across alternative measurement choices should be an aim of empirical
research.

The economics of density, both in terms of the benefits through agglomeration
economies, and the costs - congestion, crime, etc. - can be viewed both
complimentary but also as a subset of the work occurring through complexity
science. Agglomeration economies, from the economics perspective, is a
phenomenon that relates to firms and the production of goods and services. The
explanations offered by scaling theory could be viewed as a more abstract
level of explanation of agglomeration economies; still, explanations offered
by mainstream economists add a level of detail that is not addressed yet in
the broader scaling context. Economic theory conceptualizes agglomeration
economies as an economic production and cost specific concept while urban
scaling theory discusses it as a broader concept, with application beyond the
economics conceptualization. Unifying the approaches will, in the foreseeable
future, be a challenge due to methodological differences between the fields -
mainstream economists are averse to the idea of agents who are not driven by
some degree of self interest and the lack of modeling of markets and exchange.

Perhaps what would be beneficial for a possible unification would be that the
complexity approach adopts some micro-foundations that align with the
economics approach. This could be amended by developing spatially- and
network-explicit agent-based models of urban economies across scales- modeling
consumption and production processes within and across cities as well as other
dimensions of urban life. These models are expected to produce emergent
power/scaling laws for various urban phenomena in systems of cities. In this
way, explanations of the observed phenomena across the distinct modeling
traditions could be more directly linked.

Urban scaling theory has done a substantial amount of work on the costs of
scale or density - what Glaeser (2011) called the “demons of density”. While
some of the obvious challenges were identified early in the economics
literature pressman1971crime , comparatively, very little attention has been
placed on the microfoundations of costs of agglomeration as compared to the
benefits of agglomeration. Complementarities could be present in this space,
and urban scaling theory could offer guidance. More attention should be placed
in uniting social economics theories to those of urban economics, perhaps
supplemented by social psychology theories on crowding.

It is worthwhile noting that as more detailed models of urban scaling come
about, the connections between the complexity approach and other fields can
begin to emerge. For example, modeling people as derive value not from the
quantity of social links, but from the diversity that comes from them
gomez2016explaining  aligns with the concept of urbanization economies,
emerging in part from what is known as Jacobs knowledge spillovers (for a
review of evidence for Jacobs externalities, see beaudry2009s ). Still, urban
scaling researchers need to address the concept of localization economies and
identify whether scaling theory has a contribution to make in the discussion
of localization vs urbanization economies.

The topic of this chapter presents several opportunities for novel research
and synthesis of the existing approaches. Thus, we end with some open
questions and proposed topics for future research.

Questions for further discussion

  * •

Are the causes of agglomeration economies as explained by mainstream
economists a subset of a complexity science based urban scaling theory? For
example, can we express sharing, matching and learning as production theory
concepts that are special cases of broader concepts in scaling theory? Would
those be: quantity of linkages, quality of linkages and adaptation? Or other
features of scaling/complexity theory?

  * •

Can urban scaling theory incorporate distinct types of agents interacting in
networks in a dynamic setting, through evolving norms and institutions? E.g.
firm formation, land development activity, evolving markets and economic
institutions, etc.

  * •

What are the possible pathways through which mainstream urban economists can
integrate concepts from complexity science into their theory and models?

  * •

How to enrich our understanding of the advantages of agglomeration as arising
from evolutionary processes that involve history and path-dependence?

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