Model-2-Model ‘Validation’
• Example: Axtell-Axelrod-Epstein-Cohen
[1996]
– Modification of Sugarscape to reproduce results
of Axelrod culture model
– Had to modify the agent activation regime
• Sugarscape: uniform activation
• Culture model: random activation
• Model-2-Model workshops…
Multi-Agent Models
of
Cities
Rob Axtell
Joint work with
Richard Florida
(including Ph.D. student Tim Gulden)
Outline
• From firms to cities…
–
–
–
–
Distribution of city sizes
‘New Economic Geography’ explanations
Cities as agglomerations of firms
Modeling the ‘system of cities’
• Intra-city models using agents
– Pedestrian flow
– City growth and development
Motivations
• Positive goals:
– Data on cities are unique in the social sciences:
• Regular: simple functional forms
• Stable: Invariant over 100-200 years
• Robust: Across regions, countries
– Today there exists no microscopic explanation for these data
Zipf’s Law for Cities
Size
Pr(size>S)  1/S
Rank
• Valid in the U.S. for at least 250 years, despite changes in the
number of cities
• Valid in all industrial countries (not Russia!)
• Robust to changes in rank (serial correlation in growth rates)
Deviations from Zipf
• Medieval cities better described by exponential
distribution
– Castle or other security ‘walls’ ostensibly the
determined of city size
• Developing countries may not conform
• Capital cities are too large when they are
simultaneously the (a) political capital, (b)
financial capital, and the (c) cultural capital;
– Examples are Paris and Mexico City
France
France Urban Centers 1999
(centers with population ov er 20,000)
France Urban Centers 1999
(log-log axes)
9,000,000
10,000,000
8,000,000
7,000,000
1,000,000
Populaiton
Population
6,000,000
5,000,000
4,000,000
3,000,000
100,000
2,000,000
1,000,000
0
10,000
0
50
100
Rank
150
200
1
10
100
Rank
1,000
Russia:
Systematic deviation from Zipf
• 67 million people in largest
164 cities
• City size distribution is far
from Zipf
• Too few large cities
• Insufficient human capital
formation?
• We can compute amount of
migration necessary
• Can we compute time needed
for adjustment?
Motivations
• Positive goals:
– Data on cities are unique in the social sciences:
• Regular: simple functional forms
• Stable: Invariant over 100-200 years
• Robust: Across regions, countries
– Today there exists no microscopic explanation for these data
• Normative goals:
–
–
–
–
Governments at all levels want more economic growth
Are they providing the right incentives (e.g., tax breaks)?
Each locality wishes to nucleate its own industrial clusters
Same ideas might be leveraged for development worldwide
Competing Theories of City Formation
• Inter-city structure and function:
– Positive-negative externality trade-off models
(increasing returns vs. congestion)
• e.g., A Marshall, J Jacobs, V Henderson
– ‘Central Place’ theory (Christaller [1933])
• Recent formalization by Fujita, Krugman and Mori
– ‘Nihilistic’ stochastic process models
• For example, Simon [1955], Hill [1974], Gabaix [1999]
• Empirical orientation--these explain Zipf’s law
• ‘New economic geography’: Krugman et al.
Increasing Returns-Congestion
Trade-off Theories
• Marshall--firms (and people) cluster:
– Due to specialized input providers
– To facilitate labor market pooling
– To augment information dissemination
• Henderson--limits to clustering; explore trade-off:
– Workhorse neoclassical model, analytically messy:
• Single city model
• Social planner extremizes welfare functional
– Leads to ‘optimal city size’ type results
• Compatible with any size distribution
Empirical Content of the
‘New Economic Geography’
Simon’s Model of City Formation
• There is some initial distribution of cities
• With probability  « 1, a new city is born
• With probability p = 1 - , a population
lump is added to an existing city in
proportion to the city size (i.e., growth 
size; growth rate independent of size)
• Yields a power law of city size as a
function of rank with exponent 1 + 
• Similar model due to Steindl [1965]
Problems with these approaches
• Stochastic models explain the data but not
‘economically,’ i.e., they have little
economic content
• Models with microeconomic content don’t
explain the data
• The riddle (FKV [1999]): “…at this point
nobody has come up with a plausible story
about the process that generates the ranksize rule…”
The Conceptual Problem
FKV [1999: 225]
“The stochastic models that have been proposed all rely
fundamentally on the assumption of constant returns to city
size, so that a city’s expected growth rate is independent of its
size. Yet all existing economic models of cities involve
returns that are anything but constant….Perhaps there is some
way that we do not currently understand to reconcile the
tension between centripetal and centrifugal forces that we
believe determines city sizes at the micro level, and the as-ifconstant returns dynamics that seem to apply at the macrolevel. We hope that future research will resolve this puzzle”
Story Behind the Model
• People - Firms - Cities:
–
–
–
–
–
–
People live in locations
People come together to form Firms
People migrate to better job opportunities
Local agglomerations of Firms are Cities
Productive Cities attract more People
Larger Cities foster more Firms
• Human Capital Theory:
– Human capital driven growth (Jane Jacobs externalities, Lucas,
Romer, etc.)
• Generates a stable system of cities or urban hierarchy
Methodology
People
 local purposiveness;
 team (joint) production;
 heterogeneous
preferences/human capital;
 adaptive individuals
 constantly adjusting input
 periodically jumping firms
 ability to start-up new
firms
Methodology
People
 local purposiveness;
 team (joint) production;
 heterogeneous
preferences/human capital;
 adaptive individuals
 constantly adjusting input
 periodically jumping firms
 ability to start-up new
firms
Firms
 increasing returns to
human capital;
 dynamic processes of
firm formation and
evolution;
 finite firm lifetimes;
 skewed size distribution;
 successful firms attract
human capital
 firms are emergent
Methodology
People
 local purposiveness;
 team (joint) production;
 heterogeneous
preferences/human capital;
 adaptive individuals
 constantly adjusting input
 periodically jumping firms
 ability to start-up new
firms
Firms
 increasing returns to
human capital;
 dynamic processes of
firm formation and
evolution;
 finite firm lifetimes;
 skewed size distribution;
 successful firms attract
human capital
 firms are emergent
Cities
 cities attract firms
 big cities attract successful
firms
 path-dependent histories
 movement up and down
size distribution
 occasional birth of new
cities
 cities are ‘super-emergent’
Methodology
People
 local purposiveness;
 team (joint) production;
 heterogeneous
preferences/human capital;
 adaptive individuals
 constantly adjusting input
 periodically jumping firms
 ability to start-up new
firms
Firms
Cities
 increasing returns to
human capital;
 dynamic processes of
firm formation and
evolution;
 finite firm lifetimes;
 skewed size distribution;
 successful firms attract
human capital
 firms are emergent
 cities attract firms
 big cities attract successful
firms
 path-dependent histories
 movement up and down
size distribution
 occasional birth of new
cities
 cities are ‘super-emergent’
higher levels of organization
increasing complexity
City Formation Model
• There is a finite set of ‘locations,’ L = {a, b,
c,…, z}
• Each agent’s initial location is random
• When an agent joins a firm it adopts the the
firm’s location (initial location of the founder)
• When an agent starts up a new firm:
– with probability d « 1 it selects a random location
– with probability 1 - d it keeps its present location
Realization #1: Base Case
• 10,000 agents
• Basic firms model:
– increasing returns,  =2
– uniformly distributed
preferences
– equal sharing
– agents start as singletons
• Basic city model:
– 100 locations
– d = 1/2 %
– initial distribution of agents
across locations is uniform
<Run Cities code>
Model Yields Zipf’s Law
Size
Rank
<Run Cities movie>
Income as a Function of City Size
City size (firms)
Income  Size0.10
Prediction:
Dependence of Growth Rate
Variance on City Size
Growth rate standard deviation
City size (firms)
Realization #2:
Growing Population
• Initial population is 10,000, located randomly
• Grow to 100,000 over 100 periods by adding
900 agents per period
 Results:
 Zipf happens faster
 more fluctuations
Analytical Results
• Call gt the growth rate at time t, a r.v. with pdf fg(t|s) at time t
conditional on size s
• Theorem (Kesten [1973]): If growth rates are independent of
size (i.e., fg(t|s) = fg(t)) then for size i the process Sit+1 =
max(Smin, gt+1Sit) generates a power law distribution of city sizes
with exponent 1/(1- Smin/<S>)
• Theorem (Gell-Mann [1992]): If city sizes are:
– Scaling (i.e., ranki = f(Si/population size);
– Additively stable (i.e., when X and Y are iid, Z = X + Y has the same
distribution)
 then they are Zipf distributed, i.e., Pareto with exponent = 1
• Similar results obtained by Solomon and co-workers [19962001]
Realization #3: Quality of Place
• Some cities have intrinsically desirable
features/amenities
• These are accounted for in agents’ relocation
decisions
Summary
• A model of city sizes with explicit
microeconomic foundations
• Reproduces Zipf’s law for cities
• Inherently non-equilibrium at the agent
level
• Example of 4 level MAS: agents -> firms ->
cities -> population of cities
Three Speculations...
1. Microeconomic equilibrium theories will
never explain firm and city size data
2. Many stationary aggregate data do not have
explanations involving agent-level
equilibrium
3. Countries are just agglomerations of cities
which are agglomerations of firms
Firms and Countries: Same
Distribution of Growth Rates!
Canning et al.,
Economics Letters
(1998)
Intra-City Models
• Mostly geographers:
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–
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Intra-city form and function
Distribution of built to green areas (fractal)
Pedestrian movements
‘City as a complex system’ literature
• To learn more:
– Michael Batty’s Cities and Complexity (MIT
Press, 2006)