URBAN DYNAMIC LAWS AND OUR

DEGREES OF FREEDOM FOR DEVELOPMENT

Francisco J. Martínez Universidad de Chile

Instituto Sistemas Complejos de Ingeniería

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INTERNATIONAL WORKSHOP ON URBAN TRANSPORT SUSTAINABILITY, Sept. 2-4 2013

FROM URBAN DYNAMICS TO SUTAINABLE CITIES

1. STATITICAL EVIDENCE

2. THEORETICAL SUPPORT

3. RELEVANT IMPLICATIONS

FROM URBAN DYNAMICS TO SUTAINABLE CITIES

1. STATITICAL EVIDENCE 2. THEORETICAL SUPPORT

3. RELEVANT IMPLICATIONS

4

Power law in organims’ dynamics

TED Talks (www.ted.com) Geoffrey West: The surprising math of cities and corporations

Blue Whale

2*108 g

Shrew

2g

Elephant

2*106

Mamals’ mass vary 8 orders of magnitud (108)

Shrew 2g Elephant 2*106

Blue Whale 2*108 g Blue Whale 2*108 (National Geographic)

Metabolism rate: B=M3/4

Hemmingson 1960

• Cities’ Goerge K. Zipf’s laws (Gabaix, 1999)

• Power laws (Bettencourt, West, et al 2007,2008)

Evidence of “natural” laws in cities?

America’s Cities Zipf’s laws

Argentina y = -1,1492x + 15,532

R² = 0,98308

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

16,0

18,0

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Canada Colombia

Mexico United States Venezuela

Zipf in graphs America

log(rank ) = b log( population)

Log rank

Log

po

pu

lati

on

Europe Cities Zipf’s laws

Zipf in graphs Europe

Austria Belgium Bulgaria

France Italy Netherlands

log(rank ) = b log( population)

10

Power laws for cities size Y/Y0=Nβ

Y: City variable (input and outputs)

N : Population

Y0 : Base line, specific for each

economy or country

β: Scale factor, common across cities

and countries

DEVELOPED WORLD CITIES’ POWER LAWS

log(Y /Y0) = b log(N )

BETTENCOURT, LOBO, WEST (EUR. PHYS. J, 2008)

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Table1:ScaleparametersforChileancitiesandregions

ObservationUnit

Nrofobservations

IndependentVariable

Parameterestimates

β ln(y0)

County* 19Roadsnetwork

(km)0.85

(9,72)-3.80

(-3,57)

Region** 13Totalenergyconsumption

(GWH)

0.87(4,19)

-4.09(-1,44)

City 148 Area(ha)0.91

(74,41)-3.20

(-26,35)

County 19 Urbanbusfleet0.93

(5,03)-5.38

(-2,37)

County 19 Carfleet 1.00(15,96)

-1.97(-2,58)

Region 15Residentialenergy

consumption(GWh)

1.08(7,17)

-8.96(-4,43)

Region 15Residentialenergy

consumption(GWh)

1.08(7,17)

-8.96(-4,43)

Region 15 Terciaryeducationvacancies

1.16(8,80)

-5.61(-3,17)

County 24Rich

Population1.25

(12,48)-5.79

(-4,72)

County 12 ResearchProjects1.51

(5,35)-17.42(-4,71)

log(Y /Y0) = b log(N )

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Kuhnert, Helbing & West, Physica A363, 96-103 (2003)

Scale Economies Sub linear: gas stations, roads, non-residential, energy…

Increasing returns Super linear: income, PGB, patents, crime

Three cases from empirical evidence statistically constant parameters

i) b ~ 0.8 < 1 INFRAESTRUCTURE (BIOLOGY) SUB-LINEAR ECONOMIES OF SCALE FROM EFICIENTCY

ii) b = 1

LINEAR NOT-INNOVATIVE

iii) b ~ 1.15 >1 SOCIO-ECONOMIC ¿SURPRISING?

SUPER-LINEAR INNOVATION

* Combined forces to concentration in large cities * Power laws accelerate system’s dynamics

CONCLUSIONS from evidence

• Power factors are statistically constant for the set of analyzed cities

• Returns to scale: super-linear creation of wealth and innovation beta>1 .

• Economies of scale: sub-linear infrastructure costs beta<1.

* Combined forces to concentration in large cities

* Power laws accelerate system’s dynamics

FROM URBAN DYNAMICS TO SUTAINABLE CITIES

1. STATITICAL EVIDENCE

2. THEORETICAL SUPPORT 3. RELEVANT IMPLICATIONS

18

TOWARDS A FUNDAMENTAL THEORY OF THE

EVOLUTION OF CITIES

Maurice René Fréchet (1878 - 1973)

Thesis: The power law emerges from agents’ behavior: • Maximization (rational?) choice among options • Random events

Alternative physics based model in two papers: Bettencourt and Batty, both in Science, June 2013

Emil Julius Gumbel (1891 – 1996)

• Agents: individuals/households and firms

• Behavioral perspective:

– Individuals max satisfaction, Firms max profit

– Discrete random choice of activities (McFadden-Gumbel Type I)

– Market prices on goods and services

– Agents’ interaction on economic and social structures

• Land Auction – Competitive auctions (Alonso)

– Agents random bids (Fréchet – Type II)

Power Laws

Welfare power law W ³ A ×N g

R = A ×N g Rents power law

Households’ Optimal Choices conditional on location

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(Ph 'i

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Individual

Activities

Trip Destinations

Pnikj

Pnik

Pni

Prices, Transport

General social and economic equilibrium conditions apply, conditional on location.

Firms’ Optimal Choices conditional on location

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Location

Industry

Firms

Input-output Spatial interaction

Pnikj

D ,S

Pnik

Pni

Agglomeration, Rents

Prices, Transport

Alonso’s Land Allocation: landowners max profit, best bidder rule

r1

w11 w21

r2

w12 w22

h1 h2

Bids

Rents

Agents

Mattson, Weibull, Lindberg (2011)

The Fréchet Model

• Fréchet, extreme value (Type II) of positive variables.

• Agent’s bids: independent θni~Fréchet(β,vni), vni>0, and β>1.

• Auctions: rents are max bids, Fréchet(β,ri)

Remarks: rents power Law

• Rents scale super-linear with population (even for V constant)

• Scale factor defined by beta: agents’ diversity

• Betas from agents’ Gumbel utility: emerges from micro behavior

• Economies of scale (alfa=0)

• Numeraire price factor R0

THEOREM

At the city’s long term market equilibrium, total (consumers plus suppliers) welfare is given by the following power law:

and for economies with non-negative economies of scale it represents a super-linear power law.

with γ = 1 + 1/beta + alpha >1

Population Variance

Economies of scales

URBAN GROWTH

Resources =

• Maintenance: cost of current population

• Investment: grow up babies (education, food,…)

Bettencourt, et al. (2007, 2008)

Maintenance

Growth

Urban Growth Dynamics

C( ) !N t = ¥

T

N(t)

tc

N(t)

t

N(0)

t

• Cities super-linear dynamics

• Time horizon T defines singularity

• T decreases with living cost R0 and N0

Urban Growth Dynamics

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N(t)

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N(t)

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N(t)

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N1(0) .

N(0)

N2(0)

N3(0)

t

• Life time critical point(s)

implies innovations, social and technological reforms

• Pace of life accelerates in next cycle

• Dynamics accelerates with net returns to scale (gama-rho)

Renew system conditions Potentially unstable point

Cycles duration decreases with N0

Cities accelerates with population

Analogy with organism: Biologic vs. Social rhythms

Heart rate v/s mass Social rate vs. population

N= 100K => 4,04 km/hr

N= 1.000K => 4,13 km/hr (2,2%)

N= 10.000K => 4,23 km/hr (4,7%)

Model Remarks

• Scale law emerges from: agents’ behavior, diversity and market conditions

• Diversity: creates super-linear dynamics

• Optimization: fundamental mechanism

• Resources and regulations: define constraints on choice sets and the final equilibrium

• Migrations: extension to a regional system

• The main result only needs bids distributed iid Fréchet and location equilibrium

FROM URBAN DYNAMICS TO SUTAINABLE CITIES

1. STATITICAL EVIDENCE 2. THEORETICAL SUPPORT

3.RELEVANT IMPLICATIONS

Implication Remarks

• Megacities capture returns to scale from: – Diversity: denies equity???

– Population: promote concentration ???

• Urban sustainability: – Potential instability

– Life time critical points

– Accelerated pace of life

CALL FOR LARGE SCALE RESEARCH ON ESTIMATING CURRENT CYCLE

Provocative Remarks

• Organism metaphor: – nutrients-wealth

– cells-agents

HUMANS THINK, HUMANS ARE “FREE” …

SO THEY OPTIMIZE RESOURCES (time and material)

TO END UP FOLLOWING SIMILAR “NATURAL RULES”

ALTHOUGH SUPERLINEAR

GOOD NEWS: THEY ARE PREDICTABLE

FREEDOM OF DEVELOPMENT

• Freedom of choice seems limited to “rational freedom” … i.e. to optimize

• Optimal choice induces scale laws of growth, innovation and economic and social costs

• Scale laws imply predictable development, at macro-scale and system cycles

Phylosofical Remarks: long obsecions (Tzvetan Todorov, Les ennemies intimes de la démocratie, 2012)

• Helvetius (1758, Del espíritu) – “make a moral like an experimental physics”

• Condorcet (1743-1794) – all “human knowledge” may be converted into “objects of

mathematical science”.

– “the legislator is for the social order what the physicist is for nature”

We are far from understanding human behavior and its emerging (dis)order…

but maybe closer to do so,

and to harmonize it with nature