A Spatial Model of Social InteractionsMultiplicity of Equilibria

Pascal Mossay

U of Reading, UK

Pierre PicardU of Manchester, UKCORE, Louvain-la-Neuve

Paris, July 2009

Agglomeration Economies

• Increasing Returns

(New Economic Geography)

Market Mechanism

• Social Interactions

Non-Market Mechanism

Aim of the Paper• Social Interactions

Desire of face-to-face contacts

• FrameworkCommunication Externality

• IssuesEmergence of Multiple CitiesShape and Spacing of Cities

Market for Land

Related Literature• Beckmann (1976), Fujita & Thisse (2002)

“Implication of Social Contacts on Shape of Cities”

• Wang, Berliant, (2006, JET)

“Only 1 Agglomeration in Equilibrium”

• Fujita & Ogawa (1980, 1982, RSUE)

“Multiple center configurations – Multiple Equilibria”

• Hesley & Strange (2007, J.Econ.Geogr.)“Endogenous Number of Social Contacts”

• Tabuchi (1986, RSUE)

“First best city is more concentrated than the eqm”

Plan of the Talk

• Spatial Interaction Model• Model along a Circle• Spatial Equilibria• Characterization and Pareto-ranking• First-best Distribution• Robustness of Equilibria• Local vs. Global Spatial Interactions

Spatial Interaction Model Beckmann (1976), Fujita and Thisse (2002)

• Each agent located in x– Faces some residence cost– Benefits from face-to-face contacts with others– Faces some accessing cost

λ(x)

Spacex

Max U(z,s)

])(),([)(st. I

dyyyxdAYsxrzp

z : consumption goods : land consumptionr(x) : rent in location xλ(x) : population in location x

A : social interaction benefitd(x,y): distance between x and yτ : travelling cost

x

λ(x)

Max U(z,s) = z + u(s)

])(),([)(st. I

dyyyxdAYsxrzp

z : consumption goods : land consumptionr(x) : rent in location xλ(x) : population in location x

A : social interaction benefitd(x,y): distance between x and yτ : travelling cost

xz,s,x

λ(x)

• Indirect Utility

• Spatial Equilibrium

• Trade-off

Residence Cost Accessing Cost

( ) ( ) ( , ) ( )a

aV x x A d x y y dy

( )x

x

( ) , supp(.) :

( ) , otherwise

V x V x

V x V

( ) ( , ) ( )a

ax d x y y dy

1( )

2u s

s

ln( )s

Spatial Equilibrium

• Agents in x=-a: Low Residence Cost High Accessing Cost

Agents in x=0 : High Residence Cost Low Accessing Cost

• Spatial Equilibrium: All agents achieve the same utility level

• Distribution2

( ) cos( )

4with

x c x

xX=0-a a

λ(x)

Proposition: No Multiple-City Configuration

• Consider some agent in xBy moving to his right, lower residence cost

lower accessing cost

Incentive to relocate

x

City 2 City 3City 1

Spatial Interactions Along a Circle

• Agent density • Each agent - faces a residence cost

- benefits from face-to-face contacts

- faces an accessing cost

( ) ,x x C ( )x

( , ) ( )y Cd x y y dy

λ(x)

Out[52]=

A priori: Many Possible Configurations

Large & Small Cities Uneven Spacing

City 1

City 2

City 3

City 2

City 3

City 1

Characterization1) Proposition: Cities can’t face each other

“No Antipodal Cities”• At location x, by moving to the right

marginal residence cost > 0

=> Pop(east)> Pop(west)

• At location x+1/2, by moving clockwise

marginal residence cost > 0

=> Pop(west)> Pop(east)

x

X+1/2

East

West

Characterization2) The number of Cities can’t be even

x1

x2

x3

x4

P1

P2

P3

P4

• At location x1: P4=P2+P3• At location x3: P4=P1+P2

• But then P1=P3

• Similarly, P4=P2

• Thus P1=0

Characterization3) Cities of equal size & evenly spaced

• Proposition: An odd number of Equal & Evenly Spaced is a Spatial Equilibrium.

x

City 1

City 3

City 2

x

x

Out[167]=

Out[167]=

Robustness of Spatial Equilibria

• Proposition: Spatial Adjustments towards higher utility neighborhoods leads back to eqm

Pareto-Ranking

• Consider a Spatial Equilibrium with M cities

V(M)= - (resid. Cost + intra-city cost + inter-city cost)

decreases with M increases with M

• V(M) decreases with M

• Proposition: M=1 is the “best” equilibrium

Equilibrium

First-BestSocial Optimum

Social Optimum

• In the decentralized equilibrium,

agents do not internalize the other agents’ interaction cost

[see Tabuchi (1986), Fujita-Thisse(2002)]

• The Spatial Planner will build a city that is more concentrated than the equilibrium allocation

Localized Interactions (x-n,x+n)

• Consider an agent in location x moving to the right

- faces a higher residence cost- gets closer to people at his right- further away from people at his left- gets access to “new” agents- looses access to some agents

( )x

x X+nX-n

Local vs Global Spatial Interactions

cos x cos x cos x

Global Inter.

Local Inter.

Multiple Spatial Scales

Single Spatial Scale

1cos( )x2cos( )x 3cos( )x

Conclusion

• Along a circle, multiple cities emerge• Characterization: equal-size,

evenly spaced• Pareto-ranking: 1-city > 2-city > 3-city >…• Robustness wrt small initial perturbations• Local Interactions=>multiple spatial scales