Today Chapter 7 –Cities and congestion: economies of scale, urban systems and Zipf’sLaw More on...

• Chapter 7– Cities and congestion: economies of scale, urban systems and

Zipf’sLaw

• More on the role of geographical space– relevance of non-neutral space for urban systems– background paper Stelder (2005) on Nestor

Issues

• Typical outcome of CP model is agglomeration into just a few cities (of equal size)

• reality: urban hierchies=many cities of different size with some regularities accross countries and time

• VL model and the Bell Shape curve better in this respect but does not allow interregional migration

• How can Geographical Economics account for this?

Sources of agglomeration economies

• Marshall (1920) (remember box 2.1)• Knowledge spill-overs

• Labour pool

• Backward linkages– 1: pure/technological– 2: pecuniary

Scope of agglomeration economies

• Rosenberg & Strange (2004)– industrial scope (locailzation<->urbanization)– temporal scope (path dependency)– geographical scope (density inside city<->proximity to other

cities)– organisation and business culture/competitiveness scope

(diversity -> competition)

People or firms?

• Florida (2002)– "creative class" concept – spill-overs between people rather than between firms

• Gleaser (2004)– "bohemien" index insignificant when modelled

together with human capital indices

Scope of scale economies

• MAR (Marschall/Arrow/Romer) externalities: spill-overs between jointly located firms/industries of the same type; also known as localization economies (Sillicon Valley/ Detroit); a firm is more productive in the vicinity of many identical firms

• Jacobs externalities: urban spill-overs between all types; also known as urbanisation economies; a firm is more productive in the vicinity of many firms whatsoever/ in a larger city

• Duranton (2007): current consensus is that both are relevant and of comparable importance

• CP model:– firms group together because local demand is high, and demand is high

because firms have grouped together ->– positive externality is associated with the number of firms, not with

specialization ->– models urbanization rather than localization

Interdependence?

• main focus of empirical studies on (samples of) individual cities

• "free floating islands"• agglomeration forces get more attention than

spreading forces• no role for interdependence between cities and

between cities and their hinterland

Urban versus Geographical economics

• Combes, Duranton & Overman (2005)– wage curve– cost of living curve– net wage curve – labor supply curve

The urban modelprototype Henderson (1974)

• Only cities no hinterland• Industry-specific spill-overs• Counter force: negative economies of scale (congestion) (non-

industry specific)• fig 7.3: will lead to full specialization of each city into one industry

(p285-286)– all cities specializing in industry x must be of the same size

– cities specializing in an industry with higher scale economies will be larger (higher wage can bear more congestion costs)

– inter-city trade

--> urban system of specialized cities of different size trading with each other

Figure 7.3 Core urban economics model

Labor supply curve

Wage curveindustry 2

Wage curveindustry 1

Net Wage A =Net Wage B

W(N)-H(N)

Net Wage Industry 2

Net Wage Industry 1

Cost of Living Curve

Figure 7.4 Core geographical economics model

Nf = NH = 0.5

Low transport cost:Net Wage Home =Net Wage Foreign

Low Transport costHome

Low Transport costForeign

High Transport costForeign

High Transport costHome

High transport cost:Net Wage Home =Net Wage Foreign

Low Transport cost

High Transport cost

Cost of living curves

Wage curves

NH = 1NF = 1

W(N)-H(N)

Net wage curves Home

Foreign

unstable stable

The CP model in fig 7.4

• difference with the urban model of fig 7.3– cost of living falls with city size– real wage may or may not increase with city size,

depending on transport costs

Scale and relevance

Combes, Duranton & Overman:• urban model more relevant for cities, local externalities

more important than long-distance inter-city relations • CP model more relevant for regions and countries,

market access and inter-region/country interdependencies more important\

• critique: is inter-location interdependency more important at larger distances?

Introducing congestion

• L = Nτ(1-τ) ( α + βx) -1 < τ < 1

• if 0 < τ < 1 negative externalities• if -1 < τ < 0 (additional) positive externalities• if τ = 0 no congestion effects

1. Yr = δ λr Wr + φr ( 1 – δ)

2. Ir = ( Σs λs1-τε Trs

1-ε Ws1-ε )1/(1-ε)

3. Wr = λ-τ ( Σs Ys Trs1-ε

Isε-1 )1/ε

with τ =0 (1)-(3) reduces to the CP model

0 1 2 3 4 5

output

total N = 100 average N = 100 total N = 400

average N = 400 total no cong. average no cong.

Figure 7.5 Total and average labor costs with congestion

Parameter values: = 1, = 0.2; = 0.1 for N = 100 and N = 400, = 0 for "no cong."

Figure 4.1 The relative real wage in region 1

2-region base scenario

0 0,2 0,4 0,6 0,8 1

lambda 1

ws = Ws Is-δ

stable unstable

T =1.7

Figure 4.2 The impact of transport costs

Variations in transport costs T

0.000 0.200 0.400 0.600 0.800 1.000

lambda 1

Higher T: spreading more likely

Figure 7.6 The 2-region core model with congestion ( = 5; = 0.4; = 0.01)

a. T = 1.9

0 0,5 1

w 1/w 2

c. T = 1.61

b. T = 1.7

0 0,5 1

Figure 7.6 (cont)-

d. T = 1.4

0 0,5 1

w 1/w 2

e. T = 1.1

0 0,5 1

w 1/w 2

f. T = 1.07

0 0,5 1

w 1/w 2

Figure 7.6 (cont)

g. T = 1.05

0 0,5 1

w 1/w 2

h. T = 1.03

0 0,5 1

w 1/w 2

i. T = 1.01

0 0,5 1

w 1/w 2

CP- congestion model

• range of possible outcomes wider• partial agglomeration as stable equilibrium

possible -> less “black hole” results

Figure 7.7 The racetrack economy with congestion ( = 5; = 0.7; = 0.1)

a. T = 1.21

initial final b T = 1.31

initial final

Figure 7.7 continued

c. Final distribution; T = 1.2

1 3 5 7 9 11 13 15 17 19 21 23

Figure 7.7 continued

d. Final distribution; T = 1.3

1 3 5 7 9 11 13 15 17 19 21 23

Zipf’s Law

• A special case of the power law phenomenon: uneven distribution with few (very) large values and many small values

• log (Mi) = c – q log(Ri)

• Perfect Zipf: q=1• Research: some countries q>1 some q<1, some quite

close to 1 (USA)• Depends strongly on data, definitions and sample size• City proper versus agglomeration• The “tail” of the distribution does not work• The role of the primate city

Table 7.3 Primacy ratio, selected countries*

France (1982) 0.529 UK (1994) 0.703

Austria (1991) 0.687 Egypt (1992) 0.499

Mexico (1990) 0.509 Chili (1995) 0.769

Peru (1991) 0.753 South Korea (1990) 0.532

Indonesia (1995) 0.523 Vietnam (1989) 0.570

Czech Republic (1994) 0.550 Hungary (1994) 0.726

Romania (1994) 0.605 Russian Federation (1994) 0.504

Iran 0.556 Iraq (1987) 0.643

Sample Mean 0.500

Year of observation in between brackets. Source: own calculations based on UN data that can be found at http://www.un.org/Depts/unsd/demog//index/html

Table 7.4 Primacy ratio, selected countries

Table 7.5 Summary statistics for q

City proper Urban agglomeration

Mean 0.88 1.05

Standard Error 0.030 0.046

Minimum 0.49 0.69

Maximum 1.47 1.54

Average R2 0.94 0.95

# observations 42 22

Estimated q for 48 countries

q ≠ 1 more often than not; Zipf's law rarely holds

Figure 7.4 Frequency distribution of estimated coefficients*

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 More

city proper urban agglomeration

A theory on Zipf?

• should accept and explain deviations from q=1• should allow for changing q over time• can congestion-CP model do this?

Other approaches

• Simon (1955): – Random growth on a random distribution predicts that q =1

• Gabaix (1999):– Gibrat' s law: city growth is independent of its size– uniform growth rate with normal distribution variance leads to

q=1– only when either assuming CRS or with DRS and IRS levelling

• problem: no q ≠ 1 possible• Shirky (2005): networks create power laws

– http://www.shirky.com/writings/powerlaw_weblog.html

Zipf simulation

• 24-location racetrack model• feed with random history• three periods:

– pre-industrialization δ=0.5 ; ε=6 ; T=2 ; τ=0.2

– industrialization δ=0.6 ; ε=4 ; T=1.25 ; τ=0.2

– post-industrialization δ=0.6 ; ε=4 ; T=1.25 ; τ=0.33

Figure 7.10 Simulating Zipf

N-shape over time: q increases and later decreases

The role of geography

• Extra literature: stelder.pdf on nestor

• geography=neutral simulate how history matters

• history=neutral simulate how geography matters

• “no history assumption” = “in the beginning there were only little villages” or: initial distribution=equal distribution

• only possible in non-neutral space because no history in neutral space = immediate long term equilibrium (real wage identical everywhere)

symmetric space asymmetric space

Hotelling beach

racetrack

Equilibrium distribution in a racetrack economy

with 12 locations. =0.4, =4, =0.4

Equilibrium distribution in a Hotelling economy

with 12 locations. =0.4, =4, =0.4

with 12 locations. =0.4, =4, =0.4 (no history)

with 371 locations. =0.4, =4, =0.3 (no history)

Equilibrium distribution in symmetric space

with 3x3=9 locations. =0.4, =6, =0.4

with 3x3=9 locations. =0.1, =4.8, =0.52

with 9x9=81 locations. =0.4, =6, =0.4

with 10x10=100 locations. =0.4, =6, =0.4

with 11x11=121 locations. =0.4, =6, =0.4

with 51x51=2601 locations. =0.5, =5, =0.4

with 51x51=2601 locations. =0.5, =5.5, =0.4

with 51x51=2601 locations. =0.5, =6, =0.4

rank size distribution A - B - C

A two-dimensional grid in geographical space

AB=1; AC=2; AF=2+2 2 (shortest path)

Asymmetrical space with 98 locations =0.3, =5, =0.2

The USA model: “going to Miami”

USA =0.3, =6, =0.3

USA =0.3, =6, =0.4

Canada

Latin-America

Europe

Adding foreign trade

USA =0.3, =6, =0.3 with foreign trade

# # # #

# # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #